A Study on the Implications of Spline Methods as Numerical Solution for Partial Differential Equations

INTRODUCTION

Various techniques of spline methods and their application have been developed to obtain the numerical solution of the differential equations. They possess some of advantages and are worth on using in the numerical techniques. So, spline procedures exhibit the following the desirable features: (1) obtained governing system is always diagonal which permits easy algorithms; (2) it provides low computer cost and easy problem formulation. The requirement of the continuity up to the second degree are guaranteed at the mesh points over the domain and the first and second degree of the derivatives are directly evaluated. The mathematical model describing the transport and diffusion processes is the one-dimensional advection-diffusion equation. Mathematical modeling of heat transport, pollutants and suspended matter in groundwater involves the solution of a convection–diffusion equation. When the velocity field is complex and changing in time, transport processes can‘t be analytically calculated, numerical approximations are necessary. The method has widely become popular in recent years thanks to its simplicity of application. The fundamental idea behind the method is to find the weighting coefficients of the functional values at the nodal points by using base functions, derivatives of which are already known at the same nodal points over the entire region. Partial differential equation solves by spline collocation and finite difference method. While numerical solution obtained. Therefore, two common questions are encountered, first is about its acceptance whether it is sufficiently close to true solution or not. If one has an analytic solution then this can be answered very clearly but in either case it is not so easy. One has to be careful while concluding that a particular numerical solution is acceptable when an analytic solution is not available. Normally a method is selected which does not produce an excessive error. We have obtained a numerical solution of the problem by using spline collocation technique. In the investigated mathematical model we consider that the ground water recharge takes place over the large basin of such geological location that the sides are limited by rigid boundaries and the bottom by a thick layer of water table. In this case the flow may be assumed vertically downwards through unsaturated porous media. Here the average diffusivity coefficient of the whole range of moisture content is regarded as constant and the permeability of the moisture content is assumed to have a parabolic distribution. The theoretical formulation of the problem yields a non-

In this equation, a nonlinear dispersion term replaces the nonlinear dispersion term in the Korteweg-de Vries (KdV) equation, coming about with. For certain values of m and n, the K(m, n) equation has solitary waves which are compactly supported. In particula, the variant K(2, 2), has a fundamental "compacton" solution of the form After the first appearance of the compactons, it turned out that similar structures emerge as solutions for a much larger class of nonlinear PDEs, among which is, e.g., which we consider with m = 2, n = 1 as our non-linear model problem. In this work we are mostly intrigued by creating tools for approximating numerically solutions of equations which produce non-smooth structures. Because of the irregularity in the subordinates on the fronts of these developing structures, standard numerical methods, for example, limited contrasts and pseudo-ghostly methods create spurious oscillations on the fronts. Controlling these oscillations requires a numerical separating of the higher modes, which may bring about the disposal of fine scales from the arrangement. Theorem 1. Let and . Let be radially symmetric. If is sufficiently small, then there exists a unique radial solutionsuch that to Moreover, there exist radial functions and such that For GWP we use a fractional integral estimate on the unit sphere such that where. The result of Theorem E corresponds to the case If n = 3, then the finiteness of integralenforces to be less than as in Theorem H since the integral is finite only when. For details see Lemma El and Lemma SI In Theorem El we treated the case for which the integral is not finite if n = 3. However, the three dimensional GWP can be slightly improved up toby using another Strichartz estimate on a hybrid Sobolev space. It will be worthy of trying to fill the gap for n = 3. The Klein-Gordon equation (2) was initially studied by (W. Strauss, 2001 ). In (T. Motai, 2008), the GWP is considered forand. It was proved that the scattering operator for (2) is well-defined on some domain if and . Furthermore, (K. Mochizuki, 2009) showed that if and , then the scattering operator can be defined on some neighborhood near zero in the energy space. In this study the small data scattering of radial solutions is successfully treated below energy space, provided. To state precisely, let us define a weighted spaces and a data space by and , respectively, where is sufficiently small.

Splines provide high-quality approximations, lead to a sparse structure of the system operator represented by shift-invariant separable kernels in the domain, and are mesh-free by construction. Further, high-order bases can easily be constructed from Spline. In order to demonstrate the advantageous numerical performance of Spline methods, we studied the solution of a large-scale heat-equation problem (consisting of roughly 0.8 billion nodes!) on a heterogeneous workstation consisting of multi-core CPU and GPUs. (2015). The Theory of Spline And Their Application. Academic Press, N.Y. 2. Bhatt R. V. (2015). ―An Extended Study of The Spline Collocation Approach To Partial Differential Equations‖, Thesis, South Gujarat University, Surat. 3. Bickley W. G. (2014). Piecewise Cubic Interpolation and Two point Boundary Value Problem. Comp. J, 11, pp. 206. 4. Blue J. L. (2013). ―Spline Function Methods for linear Boundary Value problem.‖Communications of the ACM. 12, No.6, 327. 5. Dr. Grewal B. S. (2015). Numerical Methods in Engineering and Science, Fifth ed, Khanna Publishers. 6. Doctor H. D. and Kalthia N. L. (2013). Spline Approximation of Boundary Value Problems. Presented at Annual Conf.of Indian math.Soc, 49 7. Doctor H. D., Bulsari A. B. And Kalthia N. L. (2014). Spline Collocation Approach To Boundary Value Problems. Int. J. Number, Floids. 4, 6, pp. 511. 8. Fyfe D. J. (2015). The Use of Cubic Splines In Solutions of Two Point Boundary Value Problems, Com.P.J, 12, pp. 188-192. 9. Shastry S. S. (2015). ―Introductory Methods of Numerical Analysis. Prentice-Hall of India Private Limited. New Delhi. 10. P. B. Choksi, A. K. Pathak (2017). ―Solve Non-Linear Parabolic Partial Differential Equation by Spline Collocation Method‖, June 17 Volume 5 Issue 6 , International Journal on Recent and Innovation Trends in Computing and Communication (IJRITCC), ISSN: 2321-8169, pp. 387–391.

Assistant Professor, Mathematics Department, Laloo Mandal College, Gaya Magadh University, Bodh Gaya