Thermal Conductivity Equations via the Improved Adomian Decomposition Methods

Several mathematical models that explain natural phenomena are mostly formulated in terms of nonlinear differential equations. Many problems in applied sciences such as nuclear physics, engineering, thermal management, gas dynamics, chemical reaction, studies of atomic structures and atomic calculations lead to singular boundary value problems and often only positive solutions are vital. However, most of the methods developed in mathematics are used in solving linear differential equations. For this reason, this research considered a model problem representing temperature distribution in heat dissipating fins with triangular profiles using MATLAB codes. MADM was used with a computer code in MATLAB to seek solution for the problem involving constant and a power law dependence of thermal conductivity on temperature governed by linear and nonlinear BVPs, respectively, for which considerable results were obtained. A problem formulated dealing with a triangular silicon fin and more examples were solved and analyzed using tables and figures for better elaborations where appreciable agreement between the approximate and exact solutions was observed. All the computations were performed using MATHEMATICA and MATLAB.


Background of the Study
Several mathematical models that explain natural phenomena are mostly formulated in terms of nonlinear differential equations, both ordinary and partial. Many problems in applied sciences such as nuclear physics, engineering, thermal management, gas dynamics, chemical reaction, studies of atomic structures and atomic calculations lead to singular boundary value problems and often only positive solutions are important. However, most of the methods developed in mathematics are used in solving linear differential equations. For these reasons, a semi-analytical method named Adomian decomposition method (ADM) proposed by George Adomian 1923Adomian 1990 has been attracting the attentions of many mathematicians, physicists, engineers and various graduate researchers. The method has the advantage of converging to the exact solution and can easily handle a wide class of both linear and nonlinear differential and integral equations. The commonly existing problems in various fields of studies mentioned earlier lead to singular BVPs of the form: , ∈ , subject to given boundary conditions where atleast one of the functions , and has a singular point and , , and are finite constants. which results from an analysis of heat conduction through a solid with heat generation. The function % represents the heat generation in the solid, is the temperature and the constant ' is equal to 0,1 or 2 depending on whether the solid is a plate, a cylinder or a sphere. Equation 1) subject to boundary conditions represents temperature distribution inside a triangular extended surface (or fin) with thermal conductivity. Thus, extended surfaces are governed by BVPs; and they are widely used in many engineering appliances which include, but are not limited to, air conditioning, refrigeration, automobile and chemical processing equipments. For this reason this research tried to consider both invariant and power-law dependence of thermal conductivities governed respectively by linear and nonlinear BVPs.
The primary objective of using extended surfaces is to enhance the heat transfer rate between a solid and an adjoining fluid. Such an extended surface is termed a fin. In a convectional heat exchanger heat is transferred from one fluid to another through a metallic wall. The rate of heat transfer is directly proportional to the extent of the wall surface, the heat transfer coefficient and to the temperature difference between one fluid and the adjacent surface. If thin strips (fins) of metals are attached to the basic surface, extending into one fluid, the total surface for heat transfer is thereby increased.
This research tried to consider two triangular profile fin problems by incorporating with MATLAB codes. The modified Adomian decomposition method (MADM) is used to get the component decomposition terms. The Adomian decomposition method (ADM) is a well-known systematic method for practical solution of linear or nonlinear and deterministic or stochastic operator equations, including ODEs, PDEs, integro-differential equations, etc. The method is a powerful technique, which provides efficient algorithms for analytic approximate solutions and numeric simulations for real-world applications in the applied sciences and engineering. It permits to solve both nonlinear IVPs and BVPs without restrictive assumptions such as required by linearization, perturbation, discretization, guessing the initial term or a set of basis functions, and so forth. The accuracy of the analytic approximate solutions obtained can be verified by direct substitution. A key notion is the Adomian polynomials, which are tailored to the particular nonlinearity to solve nonlinear operator equations. The decomposition method has been used extensively to solve effectively a class of linear and nonlinear ordinary and partial differential equations. However, a little attention was devoted for its application in solving the singular two-point boundary value problems (STPBVPs). This project treated some classes of singular second-order two-point boundary value problems both analytically and numerically using the ADM and the modifications Improved Adomian Decomposition Method (IADM) and MADM focusing on the Dirchlet and mixed boundary conditions; and applied the symbolic softwares MATLAB and MATHEMATICA to facilitate computing.

Statement of the Problem
Numerous methods have been applied to determine numerical solutions of scientific problems that obviously result in variety of approximate solutions.
To perform this, it was tried to answer the research questions: 1. How effective are the decomposition method and its modifications to treat nonlinearity behaviors in thermal conductivity problems? 2. How do the modifications remove the singularity nature of the problems?

Objectives
The main objective of this research was to treat thermal conductivity equations through the improved ADMs. Specifically speaking, the study: 1) determined approximate solutions of thermal conductivity equations using improved ADMs. 2) performed error analysis to compare the results obtained accordingly.

3) made comparison between the numerical and analytical
solutions of the problems by using plots of the equations on the same coordinate plane.

Extended Surfaces
Scientists and engineers are interested in singular BVPs because the problems arise in a wide range of applications, such as in chemical engineering, mechanical engineering, nuclear industry, thermal management and nonlinear dynamical systems. For this reason, these kinds of problems have been studied by many researchers. To be specific, as extended surfaces are found in many engineering appliances, numerous studies have been performed on different fin configurations with constant and variable thermal properties. Fins are echo-friendly and economic means of convective heat transfer enhancement. They are encountered quite often in practice: from industrial compact heat exchangers to CPU heat sink modules of personal computers. Finned structures, better known as heat sinks, have well served thermal management of electronic systems for many years. Kraus, A. D., Aziz, A. and Welty, J. provided a review on this subject devoted to various aspects of extended surface convective heat transfer [13]. Mokheimer Esmail, M. A. investigated the performance of annular fins with different profiles subject to variable heat transfer coefficient [17]. The performance of the fin is expressed in terms of fin efficiency as a function of the ambient and fin geometry parameters.

Numerical and Analytical Treatments of the Bvps
Zavalani, G. used Galerkin-finite element method (G-FEM) to treat two point BVPs numerically [22]. The method uses three basic procedures. Making weak formulation is the first step to determine, the second step is to develop finite element formulation. Lastly, weighted average is used for full discretization; that are all required by neither the ADM nor its modifications. Moreover, the method fails to treat nonlinear and singular BVPs.
Khuri, S. A. and Sayfy, A. proposed a new approach, modified decomposition method in combination with the cubic B-spline collocation technique introduced for numerical solution of a class of singular boundary value problems arising in physiology [11]. The main thrust of this approach is to decompose the domain of the problem into two subintervals. The singularity, which lies in the first subinterval, is removed via the application of a modified decomposition procedure based on a special integral operator that is applied to surmount the singularity. Then, in the second sub-domain, which is outside the vicinity of the singularity, the resulting problem is treated employing the Bspline collocation technique. The approach is applicable to treat nonlinear singular BVPs and in addition, the approximate solutions converge faster to the exact solution but very vast computational work is needed.
Noor, M. A. and Mohyud-Din, S. T. used modified variational iteration method (MVIM) to determine solutions of singular and non-singular initial and boundary value problems [18]. The proposed technique is applied on equations of higher order boundary layer problem, unsteady flow of gas, third-order dispersive and fourth-order parabolic partial differential equations. The authors made the modification by introducing ADM to VIM. They used the Pade approximants technique (PAT) in order to make their work more concise and for better understanding of the solution behavior but needs caution to select the initial value because the approximants are heavily dependent on the initial value. Though they concluded that the proposed frame work is very powerful and efficient in finding the analytical solutions for singular and nonsingular boundary value problems, the method requires a huge computational work as it is incorporated from the ADM, VIM, and PAT.
Ravi, A. S. V. and Aruna, K. used an analytical method called DTM to obtain the exact solutions for some linear twopoint boundary value problems with singular nature [19]. Although these analytical methods are effective in the linear case, their applicability for nonlinear problems was not examined in the paper.

Treatments of Related Problems Using ADM and Its Modifications
Adomian formally introduced ADM concept and formulas for generating Adomian polynomials for all forms of nonlinearities since early 1980's. Several authors have been focusing on this area to develop a practical method for the calculation of Adomian polynomials.
Adomian, G. reviewed the method in Applied Mathematics; and concluded that the decomposition method can be an effective method for solution of a wide class of problems providing generally a rapidly convergent series solution [1]. Based on the author, it has some distinct advantages over usual approximation methods in that it is computationally convenient, provides analytic, verifiable solutions not requiring perturbation, linearization, or discretization required by methods like FDM which results in massive computations.
Inc, M. and Evans, D. J. obtained an approximate solution for only one nonlinear example by using the ADM-Padê technique [9]. The technique they proposed requires the computation of undetermined coefficients.
Hasan, Y. Q. and Zhu, L. M.. introduced an efficient modification of the ADM for solving singular initial value problem in the second-order ordinary differential equations [8]. The scheme is tested for some examples and the obtained results demonstrate efficiency of the proposed method. The study showed that the decomposition method is simple and easy to use and produces reliable results with few iterations. The results show that the rate of convergence of the modified decomposition method, MADM is higher than the standard ADM for initial value problems.
Several other researchers also have developed modifications to the ADM. The modifications arise from evaluating difficulties specific for the type of problem under consideration. The modification usually involves only a slight change and is aimed at improving the convergence or accuracy of the series solution.
Duan, J.-S, Rach, R., Baleanu, D. and Wazwaz, A.-M. reviewed the ADM and its modifications including different modified and parameterized recursion schemes, the multistage ADM for initial value problems as well as the multistage ADM for boundary value problems, new developments of the method and its applications to linear or nonlinear and ordinary or partial differential equations, including fractional differential equations [5]. The authors presented a contemporary review of the ADM and discussed its utility and advantages for solving linear or nonlinear and deterministic or stochastic operator equations without any restrictive assumptions, including ODEs, PDEs, integral equations and integro-differential equations for IVPs or BVPs. They also concluded that the ADM is the method of choice for solving nonlinear differential equations with a wide class of analytic nonlinearities including product, polynomial, exponential, trigonometric, hyperbolic, composite, negative-power, radical and decimal-power nonlinearities. Furthermore, the ADM has been shown to be a reliable method for the solution of nonlinear fractional ODEs and PDEs for both IVPs and BVPs. In summary, based on Duan, J.-S, Rach, R., Baleanu, D. and Wazwaz, A.-M. the ADM is a powerful and efficient technique for the solution of nonlinear ordinary, partial and fractional differential equations [5]. It provides the analyst with an easily computable, readily verifiable and rapidly convergent sequence of analytic approximate functions for the solution.
Ravi, A. S. V. and Aruna, K. proposed two modified recursive schemes for solving a class of doubly singular twopoint boundary value problems [20]. The authors illustrated how the modified recursive schemes can be used to solve a class of doubly singular two-point boundary value problems.
Based on the authors the accuracy of the numerical results indicates that the method is well suited for the solution of this type problem. They included that the main advantage of the approach is that it provides a direct scheme to obtain approximate solutions, and they have also shown graphically that these approximate solutions are almost identical to the analytic solution. They also stated another advantage of the modified recursion scheme as: it does not require the computation of undetermined coefficients, whereas most of previous recursive schemes like by Inc, M. and Evans, D. J. require the computation of undetermined coefficients [9]. The method provides a reliable technique which requires less work compared to the traditional techniques such as FDM, Cubic spline method, and standard ADM. The numerical results of the examples are presented and they believe that only a few terms are required to obtain accurate solutions. By comparing the results with other existing methods, it has been proved that proposed modified ADM is a more powerful method for solving the singular problems.
Lin, Y. and Chen, C. K. proposed a modification of the Adomian decomposition method, MADM [16]. The authors demonstrated that nonlinear double singular BVPs of second order can be handled without difficulty. Furthermore, the authors stated that MADM can also be used to solve singular boundary value problems where the traditional method ADM fails. The numerical computation gives a more precise approximation of the solution. The reported results show a greater improvement over the traditional method.
Kaliyappan, M. and Hariharan, S. recall that they presented a simple way of computing Adomian polynomials by applying the decomposition of positive integers as a subscript of the variable for nonlinear terms [10]. They developed MATLAB codes that can generate Adomian polynomials with single variable. This paper used MATHEMATICA instead.

Materials and Methods
1) convectional heat transfer is considered to study temperature distribution along a triangular fin profile by formulating a singular two-point BVP. 2) relevant journals and books were addressed to gather information about singular BVPs and the methods to treat the problems.
4) The inverse operator, 7 (. ) = 9 * # 9 # * 9 …  were used in IADM to treat problems with Dirchlet boundary conditons. (described in subsection 5.1) 5) Furtheremore, MATLAB and MATHEMATICA were applied suitably to facilitate the computations with the improved operators; and graphs were plotted using those programs.

Singular Points
It has been required to solve the differential equation ?( ) ( ) + @( ) ( ) + A( ) ( ) = 0 (6) or, in standard form, in the neighborhood of a singular point, as the behavior of the solutions there may be among their most important features. When the singularities are not too wild, a modification of the technique of power series can be used to calculate the solutions there.
To simplify the discussion, attentions are restricted to equations of the form (6) where, ?, @ and A are polynomials, that they may be assumed to have no common factors.
The solutions to the linear second order differential equations of the form (6) in the case with ?, @, and A real analytic in a neighborhood of a point = 4 and ? ≠ 0 can be expanded as power series of the form: ( ) = ∑ C * , 5 = 0,1,2,3, … and with the knowledge of C 4 and C , the coefficients C 2 with D ≥ 2 can be determined successively by certain recurrence relations.
There are many important linear second order differential equations (with non-constant coefficients) that arise in mathematical physics which do not satisfy the above conditions, and power series solutions would like to be able to obtained for some of these.
The method discovered by the German mathematician F. G. Frobenius in the 1870's and is therefore often called the Frobenius method is used to determine the singular points.   To test whether example (4.1) is regular, divide the given equation by the factor (1 − ) to obtain Here, it can be observed that the multiplier of has the form 1 − 1 Z 2 1 + [ and the multiplier of has the form 1 − 1 \ − ( + 1) 1 + ].
The denominators of the factors in the parentheses above are not zero at = 1, so they are analytic at = 1. Hence, = 1 is a regular singular point. A similar test for = −1 shows that it also is a regular singular point.

Boundary Value Problems
Definition 4.4. A boundary value problem for second order ordinary differential equations is a differential equation with conditions specified at the extremes of the independent variable in the equation.
is called two-point boundary value problem. Two-point boundary value problems are BVPs with boundary conditions given at two specified points. Linear BVPs can mainly be classified as homogeneous and inhomogeneous. They are called homogeneous if %( ), 4 and are all zero. They are said to known as inhomogeneous if atleast one of %( ), 4 and are nonzero.
A boundary value problem for a differential equation consists of finding a solution of the differential equation that also satisfies the boundary conditions. A boundary value problem may or may not have a solution; and if it has a solution it may be unique or infinitely many. Definition 4.6. A two-point BVP with a singular point = 4 subjected to two specified boundary conditions at and in an interval I as ( ) = and ( ) = , where , , and are some constants, the two-point BVP is called a STPBVP.

Operators
An operator is a function that takes a function as an argument instead of numbers as it is used to dealing with in functions. Here are some examples of operators.
Or, if a function is plugged in, say ( ), in each of the above, then the following can be obtained. and is another differential operator for a particular second order differential equation (see the example below). The operator 7 in second order differential equations is a twice differentiable function. The domain of 7 is the twice differentiable functions on an open interval F . The terminology 7 of the function is used to describe 7( ) or 7 or, simply 7 and the range of the functions on F (and hence 7 is itself a function on F). Generally, 7 is chosen for the f >g order differential equations and thus its inverse 7 follows as the f-fold definite integration operator from If and are twice differentiable functions on F and C and C are constants, then Definition 4.7. An operator 7 satisfying property 17) is called linear operator.
Example 4.2. The differential equation can be re written in compact form as c So from this an operator can be generated to be: so that (18) can be written in an operator form as: Beyond this, it can be verified that the operator 7 is a linear operator. Bearing the basic properties of derivation of ordinary differential equations in mind and plugging in the expression C + C into the operator above it can be shown as: Thus the operator at (19) is linear operator. An operator that is not linear is known as nonlinear operator. In this research nonlinear operators in which nonlinear functions are plugged are symbolized by some various representations like / and f( ) . These operators are used to determine the Adomian polynomials by the help of Adomian formula which is briefly discussed in the following chapter.

Thermal conductivity equations via
Adm, Iadm and Madm

General Description of the Adomian Decomposition Method
In the 1980's, George Adomian introduced a new method to solve nonlinear functional equations. This method has since been termed the ADM and has been the subject of many investigations. The method involves splitting the equation under investigation into linear and nonlinear portions. The linear operator representing the linear portion of the equation is inverted and the inverse operator is then applied to the equation. The nonlinear portion is decomposed into a series of Adomian polynomials. This method generates a solution in the form of a series whose terms are determined by a recursive relationship using the polynomials. In reviewing the basic methodology involved, consider a general differential equation in an operator form: where 7 is an operator representing the linear portion which is easily invertible, / is the nonlinear operator representing the nonlinear term and A is a linear operator for the remainder of the linear portion. Applying the inverse operator 7 , the equation then becomes Since 7 is linear, 7 would represent integration and with any given boundary conditions, will give an equation for incorporating these conditions. This gives where %( ) represents the function generated by integrating and using the boundary conditions. The Adomian decomposition method introduces the solution ( ) and the nonlinear function / by the infinite series as: The desired Adomian polynomials can be determined using similar procedure. . . .
It can be observed that ) 4 depends only on 4 , ) depends only on 4 and , ) depends only on 4 , and and so on.
Optionally, a simple way of computing Adomian polynomials of any type of nonlinearity is presented by applying the decomposition of positive integers 5 as a subscript of the variable for nonlinear terms through the use of MATHEMATICA software in the appendix C.
The recursive relationship is found to be And hence from (28), . . .

= − 7 A − 7 ) 2
for D = 1,2,3, … . So, having determined the components * , 5 ≥ 0 the solution in a series form follows immediately by (24). For instance, as a simple example consider the nonlinear differential equation: Recall that the ADM involves separating the equation under investigation into linear and nonlinear portions. The linear operator representing the linear portion of the equation is inverted and the inverse operator is then applied to the equation. Any given conditions are taken into consideration. The nonlinear portion is decomposed into a series of Adomian polynomials. This method generates a solution in the form of a series whose terms are determined by a recursive relationship using these Adomian polynomials.
Following the method described above, the desired linear operator is defined as The inverse operator is then Rewriting (29) in operator form, we have 7 + / = 1 (32) where / is nonlinear operator such that / = .
Next apply the inverse operator for 7 to the left hand side of the equation the following way Using the initial condition this becomes Returning this to (33) it becomes ( ) = 7 (1) − 7 (/ ) = 9 : Next the Adomian polynomials, ) * would be generated as follows: Let be expanded as infinite series ( ) = and define Then So from (29) the ADM invites the recursive relation: Now using the Adomian formula given by (26) the polynomials become: Hence the approximate solution for ( ) obtained using the first five components denoted z y look like the following.
The decomposition method and its modifications can be applied suitably to treat singular BVPs. How the ADM treats a class of singular BVPs is discussed below. For illustrative purpose, the following class of singular BVPs is considered which can be handled more easily, quickly and elegantly by implementing ADM for the exact solutions without making massive computational work. To begin, Consider (1) in the form subject to the boundary conditions ( ) = • and ( ) = Equation (41) in an operator form can be rewritten as: where the linear differential operator 7 is defined by The inverse operator 7 is therefore defined by 7 (. ) = 9 ~9 . : Now decompose the solution as in (24), It is important to note that the standard ADM suggests that the zeroth component 4 usually defined by the function: that represents the terms arising from integrating the source term ℎ( ) and from using the given conditions; and all are assumed to be prescribed. Accordingly, the solution can be computed by using the recurrence relation: According to the standard ADM, the solution can be computed by using the recurrence relation: Thus, determining the components * , 5 ≥ 0 using (49) and/or (55) the solution in a series form follows immediately. The series may be summed to provide the solution in a closed form. However, for concrete problems, the 5 term partial sum may be used to give the approximate solution as: The classical ADM is very powerful in treating nonlinear BVPs, though this is one of the qualities of the method over some other methods it has its own shortcomings like its failure to treat some nonlinear singular boundary value problems. The following subsection addresses some merits and demerits of the method.

MADM
Since the introduction of the method in early 1980's, ADM has led to several modifications made by various researchers in an attempt to improve the accuracy and expand the application of the original method. As pointed out above, the rate of convergence of the series solutions is one of the potential shortcomings of the decomposition method. To improve on this, the authors tried to introduce different modifications of the method. To begin with, based on Ravi, A. S. V. and Aruna, K. the standard ADM is modified in such a way that the function % in (16) can be divided into two parts as follows to increase rate of convergence of the series solution and minimize the size of computations [21]. This modification is applicable irrespective of the types of the BVPs under consideration. % = % 4 + % (57) Accordingly, a slight variation was proposed only on the components 4 and . The suggestion was that only the parts % 4 be assigned to the component 4 , whereas the remaining part % be combined with other terms given in (21) to define to get the recursive relation: Although this variation in the formation of 4 and is slight, however it plays a major role in accelerating the convergence of the solution and in minimizing the size of calculations. In many cases the modified scheme avoids unnecessary computations, especially in calculation of the Adomian polynomials. In other words, sometimes there is no need to evaluate the so-called Adomian polynomials required for nonlinear operators or if needed to evaluate these polynomials the computation will be reduced very considerably by using the modified recursive scheme. There are two important remarks related to the modified method. First, by proper selection of the functions % 4 and % , the exact solution may be obtained by using very few iterations, and sometimes by evaluating only two components. The success of this modification depends only on the choice of % 4 and % , and this can be made through trials, that are the only criteria which can be applied so far. Second, if % consists of one term only, the scheme (28) should be employed in this case.
Another modification of the standard ADM which alleviates the deficiency of treating some singular boundary value problems like, BVPs subjected to the form of mixed boundary conditions is MADM presented by Hasan, Y. Q. and Zhu, L. M. [8]. In fact, it is a slight refinement to the original ADM; it only modifies the involved differential operator. Generally, MADM by the authors mentioned proposes the differential and inverse operators: The operators above are reduced so as to treat second order BVPs and incorporate symbolic programmings into them to obtain solutions to a triangular fin problems involving both constant and power-law dependent thermal conductivity (highly nonlinear).
Kim, W. and Chun, C. came up with another modification of the standard ADM to solve singular 5 + 1 order boundary value problems [12]. This scheme is designed in such a way that BVPs with singular nature can easily be treated. In this project the operators are used to treat singular second-order BVPs with mixed boundary conditions. Generally, to see what MADM by the authors mentioned look like, consider the singular boundary value problem of 5 + 1 order ordinary differential equation (61) Where / is nonlinear differential operator of order less than 5 , ( ) is a given function, 4 , , … , … , C 4 , C , … , C * … , are given constants, where ' ≤ ≤ 5, ≥ 1.
in order to overcome the singularity behavior at 0.

.
This leads to the complete determination of the components * of ( ). The series solution ( ) defined by (24) follows immediately. For numerical purposes, the 5 term approximant defined by (56) can be used to approximate the exact solution.

IADM
Ebaid, A. 2010. made improvements of operators developed earlier by Lesnic, D. 2001 for the purpose of treating the heat equation with Dirchlet boundary condition [6], [15]. In this work the IADM is used to deal with linear and nonlinear STPBVPs with Dirchlet boundary conditions. The improvement is based on the ADM and Lesnic's work later developed by Ebaid, A. 2010 [6]. Lesnic where •( ) is to be determined such that 7 $$ ( ( )) can be expressed only in terms of the boundary conditions given in (2). Using this definition, thus: In order to express 7 $$ in terms of the two boundary conditions only, the coefficient C has to be eliminated by setting Using (84) in (82), the operator below proposed by Ebaid, A. 2010 is obtained to solve the singular two-point Dirchlet BVPs [6]. Thus 83 is reduced to: From ( 85 ) it is noted that 7 $$ is already expressed in terms of the given boundary conditions without any restrictions on C. So, the choice of the value that C can take depend properly on the singular point of the equation under consideration. For example, if the equation has a singular point say at 4 , C will be chosen to be any real value except the value of 4 . Moreover, if the equation has two singular points at and , then C will be considered any real value except these values of and . In general, if the equation has 5 singular points , ,..., * , then C takes any real value except the values of these singular points.
For solving linear and non-linear singular two-point boundary value problems under the Dirchlet boundary condition, IADM is established using (85) together with the standard ADM.
Consider 1 in the form: Applying the operator 7 $$ . given by 86 ) on both sides of 72 , it is observed that: The Adomian decomposition method introduces the solution and the nonlinear function by infinite series as in 24 and 25 , respectively.
Substituting the results into 88 and according to the ADM, the solution can be smartly computed by using the recurrence relations constructed based on the following cases.
Case 1:-If , i.e., linear function, then the solution can be computed by using the recurrence relation:

Numerical and Analytical Illustrations
Triangular fin problem that involves both invariant and power-law dependent thermal conductivities is considered in this section. The modified operators by Hasan, Y. Q. and Zhu, L. M. shown in sub section 5.1.3 and MATLAB codes are assimilated to obtain the numerical solutions [8]. For the sake of illustrations, additional examples are shown.

A Fin Problem
Consider a wall at temperature ' ; transferring heat by convection to an ambient at temperature ' l 'D5 as shown in the figure below.
where, " is the heat flux, € coefficient of heat transfer, ) the cross section, ' ; temperature at the base /wall temperature/ and ' l the ambient temperature. One of the prime objectives of the study of heat transfer is to find ways of controlling this ". For example, the design of a heat exchanger is often based on achieving the smallest possible heat transfer area or the largest possible amount of heat transfer for any given size heat exchanger.
As can be observed from 91 one of the ways of increasing " is by increasing the heat transfer area. The surface area of a wall may, in principle, be increased in two ways as shown in figures below. In Figure 2 C the extended surfaces are integral parts of the base material, obtained by a casting or extruding process. In Figures 2 and the extended surfaces, that may or may not be made from the base material, are attached to the base by pressing, soldering, or welding. The same geometry is obtained, though less frequently, by machining the base material. In practice, manufacturing technology and cost dictate the selection of the most desirable form.
Applications of extended surfaces are numerous, particularly in heat transfer to gaseous media. Since in this case the corresponding heat transfer coefficient is low, a small, compact heat exchanger may be achieved only by the use of extended surfaces.
Since the temperature of an extended surface does not remain constant along its length, because of transversal heat transfer by convection to the surroundings, the heat transfer from extended surfaces cannot be evaluated from 91 . Thus it is unquestionable to evaluate the temperature distribution in extended surfaces, which enables to determine the heat transfer in terms of the temperature distribution. So this research studied the temperature distribution in a triangular profile as shown in figure below considering both constant and variable thermal conductivities across a variable cross section as follows. This equation is elaborated in two cases considering constant and variable thermal conductivities in the following sub sections.

(i) Constant Thermal Conductivity
Assuming constant thermal conductivity and measuring temperature above the ambient, 92 may further be redesigned to give: , where is the temperature measured above the ambient temperature, subscript stands for fin base and 7 is the fin length. In addition, ™ is the fin parameter defined as: where € and € are convective heat transfer coefficients of fin's either sides, ' is the thermal conductivity and is fin's vertical dimension at its base. For this formulation, the origin of coordinates is placed at the tapered end of the fin.
From 59 and 60 , appropriate differential and inverse operators to treat 95 respectively are, This in its operator form can be expressed as: Applying 98) on both sides of (100), the following is found -( ) = - (7)   So by (56) the approximate solution, say using the first ten components can be determined as: A relative error which covers part of a silicon fin length is defined to see how the approximate MADM solutions to the exact solution are.
An absolute relative deviation (ARD) from exact solution that considers ten different points of the interval is defined and computed the following way. Note that larger temperature difference does not imply less heat loss. Arpaci, S. Vedat shares this by stating 'In terms of heat transfer from extended surfaces, it is more appropriate to compare the exact and the approximate heat losses than the exact and approximate temperatures, to determine the limitation of the approximate solution' [2].

(ii) Power Law Dependence of Thermal Conductivity
Assuming general power-law dependence under steady condition for thermal conductivity of the fin material in the form ' ' 4 ' ® (106) and averaged heat transfer coefficient, the succeeding governing equation can be derived the following way.

Numerical and Analytical Treatments
Model problem Silicon, being an efficient thermal conductor, has been of extensive interest in fabrication of cooling fins and packed heat sink modules especially for thermal management in microelectronics. For temperatures ranging within 300 − Now it can be taken the advantage of the described computational work in section 5.2 based on MADM to investigate the temperature distribution in a triangular silicon fin with dimensions L=0.05m and b=0.005m, subject to a constant base temperature of ' ; 423· 150 ¶ 4 , and ambient temperature of ' l 298· 25 ¶ 4 . The results to this model problem from the first eight decomposition solutions by MADM are used to plot the graph as shown in figure 4. The figure is all about variant thermal conductivity (the power law dependence) of the silicon fin considered above taking the first eight decomposition solutions of the problem. It can be observed from the figure that the temperature distribution decreases as one goes from the source to the tapered end of the triangular fin. So, unnecessary heat from a system can be dissipated to the surrounding atmosphere through extended surfaces. Compared to the FDM by Mokheimer Esmail, M. A temperature along the fin determined by MADM is observed to decline faster though only the first eight components of the decomposition were used [17]. Hence, overheating of operating machines can be protected by plugging in many extended surfaces to the machines. All these confirm that fins can serve as effective coolants. Outside this, it is not difficult to understand how powerful the scheme, modification of ADM to treat highly nonlinear BVPs.

I. ADM solution
As the given equation is not inhomogeneous, ADM fails to provide the self-cancelling noise terms that ADM favors where sum of noise terms vanishes in the limit. Self-cancelling noise terms appear for some inhomogeneous differential equations only. Thus, if terms in 4 are cancelled by terms in , even though includes further terms, then the remaining non cancelled terms in are cancelled by terms in , the remaining terms of are cancelled by some terms of ! and so on. Finally, the exact solution of the equation is readily found for the inhomogeneous case by determining the first two or three terms of the solution and by keeping only the non-cancelled terms of 4 .
Moreover, it can be observed that the standard ADM fails to treat such problems since its operators are designed so as to give no attention to the boundary conditions of the BVPs. Such difficulties are resolved perfectly by the IADM in which the operators are designed in such a way that all the boundary conditions are included. But the exact solution of the problem as given by Arpaci, S. Vedat is [2] 2 ln 2 ln 2 ln 2

II. IADM solution
Rewriting (119) in the standard form gives: For 5 0 and the given terms above at the second equation in 120 , reduces to: which is the approximate solution using the first three iterations only. But the first seven iterative results are used to get better numerical results in the table 1 that shows the numerical illustration using both the IADM and exact solutions together. In order to verify how close the IADM solution Φ ! to the exact solution , the graph of the two solutions are plotted in figure 5. For the IADM solution it is enough to use only the first three components of the decomposition series solution. All the computations and the plot are done by MATHEMATICA.  In figure 5, the IADM solution Φ ! and the exact solution 2 ln 2 ln 2 ln 2 are plotted together but to get better approximate solution, MATHEMATICA is used to evaluate up to the 7 th iteration as shown in table 1. It can be concluded from the figure that an accurate IADM solution is obtained considering only few terms of the decomposition solution. Also, in order to compare the approach with another modification of the ADM at Benabidallah, M. and Cherruault, Y the numerical results for the absolute errors / Φ ¦ÃÄÅ / are presented in table 1 [3]. In the journal mentioned, the authors used the 7terms of the ADM in indirect way in which a huge amount of computational work is needed to obtain the numerical solution with absolute errors † 6 » 10 { . Hence, in comparison, the improved Adomian decomposition method is not only more accurate but also more simple and direct.  and hence substituting the given constants and AE 1, in these equations one can easily get   By investigating the approximate solutions Φ , Φ ! and Φ { , it can be observed that they agree with the Maclaurin expansion of the function w $ Ê up to Ç , Ç and !Ç respectively. So, by evaluating more terms of the decomposition series, the exact solution is found to be w $ Ê .

II. ADM solution
The following (figure 6) is plot by MATHEMATICA where the IADM solutions obtained using only the first four components of the decomposition solutions and the exact solution (for 0.5, AE 1).  Table 2 shows the numerical illustration using the IADM and exact solutions for the problem. Unlike the figure, the first seven terms of the decomposition solution are used. To verify how close the approximate solution is to the exact one: w $ , the IADM solution considering the first four approximate results and the exact solution are plotted in figure 6 for 0.5 and AE 1. It is shown from this figure that the approximate solution obtained through the IADM is very close to the exact one using very few terms. Furthermore, numerical results are shown in the table using the first seven IADM solutions of the example considered. From the absolute error in the table it is observed that the IADM solution is closer to the exact solution.. Lin, Y. and Chen, C. K. used a three-point finite difference method in which a huge amount of computational work is needed to obtain the numerical solution [14].

I. MADM solution
To treat second order BVPs with mixed boundary conditions of this form, the operators at (67)  . : : (126) Hence, from 106 and 107 one can define the operators below to treat the problem considered.
Using the decomposition series for the linear function and the Adomian polynomial series for the nonlinear term y , the following expression can be obtained.
Now to determine the Adomian polynomials for the nonlinear term, call it f y , the Adomian formula given by 3 is used in 132 as follows.

II. Exact solution
The exact solution of the nonlinear BVP is given by Kim, W. and Chun, C. is [12]:   Table 3 shows numerical illustrations of the example that deal with the exact solution given by Kim, W. and Chun, C and the approximate solution determined by MADM to see how close they are using the absolute error [12].

I. IADM solution
As can be expected, it is necessary to represent the nonlinear part by the Adomian polynomials. Here the nonlinear term is The required recurrence relation is (90) which is given by: But for fast convergence to the exact solution, MADM by Wazwaz, A.-M. helps to rearrange the result obtained above so that 4 assumes to be zero and all the existing terms obtained to be added to the following way [21]: For 5 = 0, = −7 $$ $ * + ) 4 The Adomian polynomial ) 4 can be determined either  The terms of the decomposition components are getting too vast to solve by hand but MATHEMATICA facilitates computing. So it can be used to list as many decomposition terms as desired. Though more than ten terms of the decomposition are used in the IADM solution, it is believed not economical to write the next iterative results, it is found important to list only up to the fifth iteration below.  II. Exact solution Consider the given equation Re writing it in the form of (41) and using (56 one can get:   In the overlapping plots shown above, one can easily observe that the IADM and the exact solutions are nearly identical. Furthermore, numerical results are shown in table 4 in which an absolute error ≤ 10 š is obtained. This shows that the IADM converges faster to the exact solution. Unlike that of the DTM, in this example it can easily be observed that it is not difficult to obtain exact solutions for nonlinear inhomogeneous BVPs using the standrad ADM.

Summary
Convective triangular fins with invariant and power-law temperature-dependent thermal conductivities were studied by MADM. Through MATLAB and MATHEMATICA, reliable parametric solutions for both problems was offered by the help of the modification. A realistic model problem regarding a silicon fin of specific dimensions was carried out as a numerical illustration. The MADM solutions were compared with an exact analytical solution by Arpaci, S. Vedat for the constant ' (governed by linear BVP) and a numerical solution via FDM by Mokheimer Esmail, M. A for power-law temperature-dependent ' cases (governed by nonlinear BVP) [2], [17]. Moreover, it was observed that ADM and its modifications; MADM and IADM are comfortable to treat the problems. The approximate solutions obtained by MADM and IADM converge faster to the respective exact solutions. Some numerical approximations by IADM are compared with the results in some other methods like reproducing kernel space method by Cui, M. and Geng, F in which magnificent power of fast convergent behavior of IADM is observed [4].
Further more, this research also considered second order STPBVPs both numerically and analytically using the standard and improved ADMs. The results were compared by plotting the graphs of the approximate and exact solutions on the same coordinate plane by using the symbolic softwares MATLAB and MATHEMATICA. Error analysis was performed to see how close the approximate solutions to the exact solutions are.

Conclusion
Temperature distribution with constant and variable thermal conductivity along triangular fin profile governed respectively by linear and nonlinear two-point BVPs can be treated directly using the improved ADMs.
Thermal equations of linear and nonlinear BVPs can be treated directly through the standard ADM and its modifications effectively.
Although the classical ADM is very powerful, it fails in treating some singular boundary value problems due to the existence of singular point at = 0. So this difficulty is alleviated by the modifications MADM and IADM; and shown by treating STPBVPs with Dirchlet and mixed boundary conditions holding singular feature both numerically and analytically.
Though the methods are very convenient for software treatments, it is noted that the modified decomposition methods (especially IADM) encounters difficulties in obtaining each component for some complex nonlinear problems even if symbolic packages are used since each component is obtained by cumbersome definite integrals.

Recommendations
Based on the analysis of thermal equations via the improved Adomian decomposition methods the following basic recommendations are suggested: 1) To study heat transfer from extended surfaces it is undisputable to study the temperature distribution along the fins length. So individuals who are interested to work on heat transfer in these areas can benefit from the results and the MATLAB codes presented. 2) Two-point BVPs with single singular point were treated by the modifications. It would be worthwhile to expand application of the operators so that the method could be used to treat two-point BVPs with multiple singularities.
3) The ADM is not only used in a straight forward manner but also requires less computational works in comparison to some other methods like HAM and MVIM. To the contrary, though the series can be rapidly convergent in a very small region, it has very slow convergence rate in wider regions and the truncated series solution is an inaccurate solution in that region which will seriously restrict the application area of the method. An investigation into this claim would greatly benefit the scientific community. +h^20*'u20'; Ck=(1/factorial(k))*subs(diff(s^ (-Z)