Abstract
A hybrid convergent method of tenth-order is presented in this work for directly solving fifth-order boundary value problems in ordinary differential equations. A unique direct block approach is obtained by combining multiple Finite Difference Formulas which are derived via the collocation technique. The proposed method is fully analyzed and the existence and uniqueness of the discrete solution is established. Different numerical examples are considered and the results are compared with those provided by existing works in the literature. The comparison shows the good performance of the present method over some cited works in the literature, confirming the competitiveness and superiority of the new numerical integrator.
Highlights
This paper considers the direct numerical solution of fifth-order BVPs of the form y(5) = f x, y, y, y y, y(4)
Momoh where α0, α1, α2, β0 and β1 are real constants, and f is assumed to be a continuous function on a prescribed domain of interest. Problems of this nature usually arise in the mathematical modelling of viscoelastic flows, induction motor and different aspects of mathematical, physical and engineering sciences
The theorems that provide the conditions for existence and uniqueness of solutions of the boundary value problems of type (1.1) are extensively discussed in [1]
Summary
Caglar et al [3] presented a method for solving fifth-order boundary value problems where they adopted an approximation by a sixthdegree B-spline function and exhibited a first order convergence. A tenth-order block method for directly solving a fifth-order boundary value problem which is assumed to have a unique solution within the integration interval is presented. To further improve the order and the stability of the method, we have considered sixth derivative terms This resulted into a method that is of uniform tenth theoretical order which is capable of handling directly the solution of equations of the type in (1.1). Let us consider a generic two-block subinterval [xn, xn+2] and assume that the theoretical solution to (1.1) is approximated here by a polynomial of the form: y(x) ≈ p(x) = arxr, r=0. We do not present here these coefficients as they are cumbersome expressions, and can be obtained with a CAS
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