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 newlinebranches of science, engineering and industries. It has numerous applications in a newlinevarious field. Computing the solutions of BVPs is a fundamental challenge to the newlineresearchers. Numerous numerical approaches are developed to approximatively solve newlinevarious differential equations. newline newlineIn this thesis, natural cubic spline method (NCS) has been developed to solve two-point newlineBVPs of second order1 differential equation. The solution of the BVPs is initially newlineapproximated by cubic splines and derived a recurrence relation with natural spline newlineconstraints. Replacing this recurrence relation and respective cubic spline newlineapproximation in BVPs, obtained a tri-diagonal system of equation. An efficient newlineThomas algorithm is used to find the solution and represents the results graphically. The famous differential equations from bessels equation, Lane-Emden equation, Thomas- newlineFermi equation, isothermal gas sphere, porous catalyst pellet and etc has been newlineconsidered to check the developed NCS method. NCS method is also applied to linear1 newlineand non-linear ODE. Quasi-linearization method (QLM) is used to linearize the nonlinear newlineODE. To know the advantage of the NCS method, its solutions are compared with newlineanalytical solutions, and also the results produced by shooting technique and spectral newlinemethod. It is observed from these examples that NCS method is more accurate for newlinesolving singular boundary value problems. NCS results are also compared with existing newlinepublished results and observed a very good agreement. Further, error analysis has been newlinedone using L norms and observed that NCS method is more efficient than other newlinenumerical techniques. newline newline