Block Unification And Boundary Value Numerical Integration Schemes For Ordinary And Partial Differential Equations

Many physical problems can be modelled as Differential Equations (DEs). DEs require solutions, either in closed form (analytic form) or in approximate form. Closed form solutions are rare to obtain for some DEs of high order. Hence, the need for numeri- cal methods/techniques. The difficulties in obtaining closed form solutions for higher order Boundary Value Problems (BVPs) in Ordinary Differential Equation (ODEs) and Initial-Boundary Value Problems (IBVPs) in Partial Differental Equations (PDEs) are the motivations for this study. This study was aimed at developing new classes of continuous block integrators for solution of fourth and fifth order BVPs and IBVPs. The objectives were to: (i) develop new classes of continuous block integrators for step number k = 5;rn(ii) analyse the developed integrators for consistency and convergence; (iii) apply the de- rived block integrators to solve the IBVPs; and (iv) compare the efficiency of the new methods with some existing methods in literature.rnThe fourth and fifth order ODEs of the form:rny(m) = f (x, y, yJ, . . . , y(m−1)), a ≤ x ≤ b) (1)rnwhere m = 4, 5 with appropriate boundary conditions were considered. The assumed trial solution is of the formrnkrnp(x) = ρiTi∗(x) ∼= y(x) (2)rni=0rnwhere ρi’s are real constants, y ∈ Cm(a, b), and Ti∗(x) is the ith degree shifted Chebyshev polynomial of the first kind over the interval [0, k]. In order to determine the unknown coefficients ρi’s of (2) the process of collocation was adopted. This yielded the desiredrnxvrncontinuous block integrators. In order to extend the scope of application of the methodsrnto IBVPs, the Method of Lines was adopted to semi-discretize PDEsrnThe findings of this study were:rn(i) two new classes of continuous implicit five-step methods namely: Block UnificationrnMethod (BUM) and Boundary Value Method (BVM) for direct solution ofrngeneral fourth and fifth order BVPs were derived;rn(ii) the new classes of methods were proved to be consistent and convergent;rn(iii) new class of continuous implicit five-step methods were extended to solve fourthrnand fifth order IBVPs;rn(iv) the efficiency of the BUM and BVM compare favourably with some existing methodsinrnliterature in terms of accuracy;rn(v) the BUMs and BVMs are self starting, and admit easy change of step-size andrnfunction evaluation at off-step points.rnThe study concluded that the proposed two new classes of methods are efficient, and whenrncompared with some existing methods in literature are accurate. The proposed methodsrnare therefore recommended for the solution of fourth and fifth order BVPs and IBVPs.

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