NUMERICAL SOLUTIONS OF SOME INITIAL AND BOUNDARY VALUE PROBLEMS USING B-SPLINES

Abstract

Collocation method is an emerging popular technique to solve initial and boundary value problems. Collocation method was developed to seek solution of problem in form of a linear combination of coordinate functions with linear coefficients. The collocation method together with B-spline approximation represents an economical alternative, since it only requires the evaluation of the unknown parameters at the grid points. B-splines are piecewise defined polynomial functions. The degree of B-spline used depends upon the order of the differential equation. Three methods are presented in this thesis, collocation method with cubic B-spline basis functions, collocation method with quartic B-spline basis functions and collocation method with quintic B-spline basis functions. The aim of this research is to explore the implementation of collocation method using B-spline functions to solve both linear and nonlinear initial and boundary value problems. The use of various degrees of the B-splines in getting the numerical solution of some partial differential equations is shown to provide easy and simple algorithm. In case of nonlinear equations the method cannot be applied directly so the quasilinearization technique is used to reduce the nonlinear equation to linearized form and then the equation is solved by collocation method with B-spline basis functions. All the Chapters include several examples and problems that have been solved by computational methods. 11 In this thesis, some well known equations like Fisher's Equation, Coupled Viscous Burgers' Equation, One-dimensional Hyperbolic Telegraph Equation, Benjamin-Bona-Mahony-Burger Equation, Kuramoto-Sivashinsky Equation, extended Fisher-Kolmogorov Equation and Swift-Hohenberg Equation are solved that have widespread applications in several branches of engineering and applied science. The thesis comprise of eight Chapters. Chapter 1 is introductory in nature. Besides stating the relevant definitions and numerical algorithms, it outlines the theory and concepts of the B-spline functions. After a brief introduction on B-spline it describes how the B-spline functions can be implemented to solve initial and boundary value problems. Various degree of B-spline function definition is extracted by using the recursive function. Some of the important properties of the B-spline basis functions are also discussed. Chapter 2 deals with the numerical solution of Fisher's equation = vu + p f (u) ,x E (-00 , 00) , t > 0 with f (u) = u(1— u), Here, v is the (constant) diffusion coefficient, p is the reaction factor and f is the nonlinear reaction term. Fisher's equation is a second order nonlinear partial differential equation which is having applications in population growth model. Fisher's equation represents the evolution of the population due to the two competing physical processes, diffusion and nonlinear local multiplication. Solutions to the equation are obtained by using cubic B-spline collocation method. For discretization, the weighted average approximation is used along the time direction. Experiments are done to obtain the solutions by selecting different values of . To establish the applicability and accuracy of the proposed method, the derived method has been applied on three practical problems available in the literature. From 111 experiments it is concluded that the minimum value of the .L,0 errors are obtained when B = Y2 .The derived scheme is shown to have second order of convergence. A part of this Chapter has been published in International Journal of Computer Mathematics, (2010). Chapter 3 discusses the application of collocation method with cubic B-spline basis functions to find solution of a coupled system of viscous Burgers' equation of the form u uu x+ a (uv)x = 0, x E [a, b], t E [0, 11 vt—v,+rivvx+ fl(uv)x = 0, x e[a,b], t e [0, 71 where 77 is a real constant, a and 13 are arbitrary constants depending on the system parameters such as Peclet number, Stokes velocity of particles due to gravity and Brownian diffusivity. The coupled equation is used to model poly dispersive sedimentation. In order to solve the system of equations cubic B-spline collocation method is used. By applying method, equations get reduced to a bi-tridiagonal system of equation which is solved by a modified form of Thomas algorithm. The adaptability and efficiency of the scheme is shown by applying it on three numerical examples. The scheme is also analyzed for stability and convergence. It is shown that the schemeis unconditionally stable. The order of convergence of the scheme is found to be approaching two. A part of this Chapter has been published in Commun Nonlinear Sci Numer Sim ulat. (2011). Chapter 4 deals with the numerical solution of the Hyperbolic Telegraph equation using a collocation method with the cubic B-spline functions. The linear form of one-'dimensional Hyperbolic Telegraph equation utt + 2a u + feu = u + f (x, t), x e[a,b], t 0 iv where a , /3 are real constants, arises in the study of propagation of electrical signals in a cable of transmission line and wave phenomena. With a > 0 , f3 = 0 the equation represents a damped wave equation and for a > 13 > 0 it represents the telegraph equation. The Telegraph equation is used to model pulsate blood flow in arteries and in one-dimensional random motion of bugs along a hedge. The method is applied on four examples and computed results are compared with those available in literature. The stability of the scheme is also discussed and found to be unconditionally stable. Chapter 5 addresses the collocation method with quartic B-spline basis functions to find numerical solutions of Benjamin-Bona-Mahony-Burgers (BBMB) equation. The equation describes the mathematical model of propagation of small-amplitude long waves in nonlinear dispersive media. Benjamin-Bona-Mahony-Burgers equation is given by —u,c,1— au + tat, + flux = 0 , x E [a, b ] ,t e [0,T] where u(x,t) is a real-valued function of the two variables x and t that represents the fluid velocity in the horizontal direction and a, , 13 are positive constants. By applying the method, the equation gets reduced to a four diagonal system of matrix. This matrix system is solved by applying modified form of Thomas algorithm. The efficiency of the scheme is shown by applying it on four numerical examples. The order of convergence of the scheme is found to be approaching three. Chapter 6 is devoted to find the numerical solutions of Kuramoto-Sivashinsky (KS) equation using quintic B-spline collocation method. The KS equation 11, +wt., + + = 0 ,x E [a,b] , t E [0,T] was originally derived in the context of plasma instabilities, flame front propagation, and phase turbulence in reaction-diffusion system. This equation is useful to model solitary pulses in a falling thin film. By applying the method the equation get reduced to a system of penta diagonal system of matrix. This matrix system can be solved by applying modified form of Thomas algorithm. Five numerical examples are solved to illustrate the accuracy and efficiency of the scheme. In order to estimate the accuracy of the scheme the global relative error is evaluated for examples and the stability of the scheme is also discussed. The scheme is found to be conditionally stable with the condition a When v = 0 the KS equation gets reduced to the Burgers' equation. Therefore, the method is applied on one example of Burgers' equation. A part of this Chapter has been published in Commun Nonlinear Sci Numer Simulat. (2010). Chapter 7 is concerned with the numerical solution of the extended Fisher-Kolmogorov equation and Swift-Hohenberg equation. Both equations are fourth-order semi-linear equations. The general equation can be written as ut + y u +riur, +u3 + (1— ote)u = 0,x c[xo,xj, t > 0 For 77= —1 and a =2 equation reduced to extended Fisher-Kolmogorov equation ut +y +u3 —u = 0 while for y =1 and 77 = 2 equation reduced to Swift-Hohenberg equation u, +u +2u, +u3 + (1— cz)u = 0, The equations are used for the study of various issues in pattern formation in bi-stable systems. We have solved both the equations by applying quintic B-spline collocation method. The scheme is checked for stability and found to be unconditionally stable for both the equations. The order of convergence of the schemes is found to be more than two. Graphs are plotted to describe the characteristics of the solutions obtained. The numerical results are plotted for different values of parameters in Swift-Hohenberg equation as different patterns occur for a slight change in the value of parameters. 12v h2 vi A part of this Chapter has been published in Inter. J. of Appl. Math and Mech. (2010). Finally, in Chapter 8, based on the present study, conclusions are drawn and future research work in this direction is suggested