Studies on Natural Cubic splines to boundary value problems of differential equation

Abstract

Boundary value problems (BVPs) of differential equations can be found in many branches of science, engineering and industries. It has numerous applications in a various field. Computing the solutions of BVPs is a fundamental challenge to the researchers. Numerous numerical approaches are developed to approximatively solve various differential equations. newlineIn this thesis, natural cubic spline method (NCS) has been developed to solve two-point BVPs of second order1 differential equation. The solution of the BVPs is initially approximated by cubic splines and derived a recurrence relation with natural spline constraints. Replacing this recurrence relation and respective cubic spline approximation in BVPs, obtained a tri-diagonal system of equation. An efficient Thomas algorithm is used to find the solution and represents the results graphically. The famous differential equations from Bessel s equation, Lane-Emden equation, Thomas-Fermi equation, isothermal gas sphere, porous catalyst pellet and etc has been considered to check the developed NCS method. NCS method is also applied to linear1 and non-linear ODE. Quasi-linearization method (QLM) is used to linearize the non-linear ODE. To know the advantage of the NCS method, its solutions are compared with analytical solutions, and also the results produced by shooting technique and spectral method. It is observed from these examples that NCS method is more accurate for solving singular boundary value problems. NCS results are also compared with existing published results and observed a very good agreement. Further, error analysis has been done using Land#8734; norms and observed that NCS method is more efficient than other numerical techniques. newlineThe procedure for NCS method is also developed to solve PDEs. In this thesis, it is restricted to consider parabolic and hyperbolic partial differential equations for the brevity of the thesis. For this time derivatives are replaced by forward finite difference operator and space derivatives are replaced by derivatives of cubic spline po