Cubic B-spline Collocation Method for One-Dimensional Heat Equation

In this paper we discuss cubic B-spline collocation method. We have given the derivation of the B-spline method in general. We have applied the method for solving one-dimensional heat equation and the numerical result have been compared with the exact solution.

This problem is one of the well-known second order parabolic linear partial differential equation [1,3,4]. The heat equation is a very important equation in physics and engineering. It shows that heat equation describes the distribution of heat (or variation in temperature) in a given region over time. The heat equation is of fundamental importance in diverse scientific fields. In mathematics, it is prototypical parabolic partial differential equation. In probability theory, the heat equation is connected with the study of Brownian motion via the Fokker -Planck equation [5]. Numerical solutions of those equations are very useful to study physical phenomena. One of the linear evolution equation which we deal with the numerical solution is the heat equation [2]. In financial mathematics it is used to solve the Black -Scholes partial differential equation. The diffusion equation, a more general version of the heat equation, arises in connection with the study of chemical diffusion and other related processes [5]. In history, the heat equation proposed by Fourier in 1822 has been applied to investigating a temperature distribution in materials [6]. The heat equation is used in probability and describes random walks. It is also applied in financial mathematics for this reason. It is also important in Riemannian geometry and thus topology: it was adapted by Richard S. Hamilton when he defined the Ricci flow that was later used by Grigori Perelman to solve the topological Poincaré conjecture. The heat equation arises in the modeling of a number of phenomena and is often used in financial mathematics in the modeling of options. The famous Black -Scholes option pricing model's differential equation can be transformed into the heat equation allowing relatively easy solutions from a familiar body of mathematics. Many of the extensions to the simple option models do not have closed form solutions and thus must be solved numerically to obtain a modeled option price. The equation describing pressure diffusion in a porous medium is identical in form with the heat equation. Diffusion problems dealing with Dirichlet, Neumann and Robin boundary conditions have closed form analytic solutions (Thambynayagam 2011). The heat equation is also widely used in image analysis (Perona & Malik 1990) and in machine-learning as the driving theory behind scale-space or graph Laplacian methods. The heat equation can be efficiently solved numerically using the implicit Crank -Nicolson method of (Crank & Nicolson 1947). This method can be extended to many of the models with no closed form solution, see for instance (Wilmott, Howison & Dewynne 1995). An abstract form of heat equation on manifolds provides a major approach to the Atiyah -Singer index theorem, and has led to much further work on heat equations in Riemannian geometry [5].
In this study the cubic B-spline collocation method is used [7,9,10] for solving the heat equation (1) and the solutions are compared with the exact solution. In the section two, we have given the derivation for the B-spline method and uniform convergence for the method has been discussed. Finally, we have solved the problem (1) using the method, the numerical results and graphs have also been shown.
From the above Eq. (2) we can simply check that each of the functions is twice continuously differentiable on the entire real line, also and that = 0 for ≥ 6/ and ≤ $/ . Similarly we can show that and Each is also a piece-wise cubic with knots at T, and ∈ U. The values of , Q and QQ at the nodal points ′W are shown in Table 1.
Then force ) to satisfy the collocation equations plus the boundary conditions.
We have and On solving Eq. (7), we get by using equations (3) and (5) where hE F G = h F and dE F G = d F . The given boundary conditions (8) become and Eqs. (10), (11) and (12) lead to a + 3 × + 3 tridiagonal system with + 3 unknowns C = $0 , B , … , C60 (where t stands for transpose). Now eliminating $0 from the first equation of (10) and (11), we find Similarly, eliminating C60 from the last equation of (10) and from (12), we find from (10) The above equations lead to the system of + 1 linear equations u C = v C in the where z = −6g + hh ? / , Since h > 0, it is easily seen that the matrix u is strictly diagonally dominant and hence nonsingular. Since u is nonsingular, we can solve the system u C = v C for B , 0 , … , C and substitute into the boundary equations (11) and (12)  Hence this proves the lemma.
Denote the value of at the representative mesh point E F , ‡ G by The forward difference approximation for is by using equations (3) and (5) we get The above equations lead to the system of + 1 linear equations u C = v C in the + 1 unknowns C = B , 0 , … , C of the form where z = −6• + h ? / , We can see that the system is strictly diagonally dominant and hence nonsingular. So we can solve the system for B , 0 , … , C and substitute into the boundary conditions (11) and (12) to obtain $0 and C60 .
The table 2 below illustrates the numerical, exact solution and error for the heat equation