Abstract
Problem Statement: Many boundary value problems that arise in real life situations defy analytical solution; hence numerical techniques are desirable to find the solution of such equations. New numerical methods which are comparatively better than the existing ones in terms of efficiency, accuracy, stability, convergence and computational cost are always needed. Approach: In this study, we developed and applied three methods-standard cubic spline collocation, perturbed cubic spline collocation and perturbed cubic spline collocation tau method with exponential fitting, for solving fourth order boundary value problems. A mathematical software MATLAB was used to solve the systems of equations obtained in the illustrative examples. Results: The results obtained, from numerical examples, show that the methods are efficient and accurate with perturbed cubic spline collocation tau method with exponential fitting been the most efficient and accurate method with little computational effort involved. Conclusion: These methods are preferable to some existing methods because of their simplicity, accuracy and less computational cost involved.
Highlights
Fourth order boundary value problems occur in a number of areas of applied mathematics among which are fluid mechanics, elasticity and quantum mechanics as well as science and engineering
Small class of differential equations can be solved by analytical methods
Several authors have investigated some numerical techniques for solving boundary value problems, among which include cubic spline method, Ritz method, finite difference method, multi-derivative method and finite element method; see for instance[3,6,8,10]
Summary
Fourth order boundary value problems occur in a number of areas of applied mathematics among which are fluid mechanics, elasticity and quantum mechanics as well as science and engineering. Several authors have investigated some numerical techniques for solving boundary value problems, among which include cubic spline method, Ritz method, finite difference method, multi-derivative method and finite element method; see for instance[3,6,8,10]. Since S”(x) is continuous, collocating (17) at point x = xi-1 and moving in clockwise direction, we obtain: (7) becomes:. Collocating Eq (17) at point x = xi-1 and moving in anti-clockwise direction, we obtain: S'(xi +). By differentiating (22) once more, collocating at point x = xi-1 and moving in clockwise direction, we obtain: S′′′(xi
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