Solutions Of Eighth-order Boundary-value Problems Research Articles

Least-Squares Solutions of Eighth-Order Boundary Value Problems Using the Theory of Functional Connections.

This paper shows how to obtain highly accurate solutions of eighth-order boundary-value problems of linear and nonlinear ordinary differential equations. The presented method is based on the Theory of Functional Connections, and is solved in two steps. First, the Theory of Functional Connections analytically embeds the differential equation constraints into a candidate function (called a constrained expression) containing a function that the user is free to choose. This expression always satisfies the constraints, no matter what the free function is. Second, the free-function is expanded as a linear combination of orthogonal basis functions with unknown coefficients. The constrained expression (and its derivatives) are then substituted into the eighth-order differential equation, transforming the problem into an unconstrained optimization problem where the coefficients in the linear combination of orthogonal basis functions are the optimization parameters. These parameters are then found by linear/nonlinear least-squares. The solution obtained from this method is a highly accurate analytical approximation of the true solution. Comparisons with alternative methods appearing in literature validate the proposed approach.

Numerical solutions of eighth order BVP by the Galerkin residual technique with Bernstein and Legendre polynomials

In this paper, Galerkin method with Bernstein and Legendre polynomials as basis functions is used for the numerical solutions of eighth order linear and nonlinear differential equations for two different cases of boundary conditions. In this method, the basis functions are transformed into a new set of basis functions which vanish at the boundary where the essential types of boundary conditions are defined and a matrix formulation is derived for solving the eighth order boundary value problems (BVPs). The numerical results, obtained by the proposed method, reflect that the performance of the present method is reliable, more efficient and converges monotonically to the exact solutions.

Spline approximate solution of eighth-order boundary-value problems

We use non-polynomial spline functions to develop numerical methods for the solution of the eighth-order linear boundary-value problems. End conditions of the spline are derived. We compare our results with the results produced by decomposition method and polynomial spline method. However, it is observed that our approach produce better numerical solutions in the sense that max |e i | is minimum.

Nonic spline solutions of eighth order boundary value problems

Nonic spline is used for the numerical solutions of the eighth order linear special case boundary value problems. The end conditions are derived for the definition of spline. The algorithm developed not only approximates the solutions, but their higher order derivatives as well. The method presented in this paper has also been proved to be second order convergent. Two examples compared with those considered by Inc et al. [M. Inc, D.J. Evans, An efficient approach to approximate solutions of eighth-order boundary-value problems, Int. J. Comput. Math. 81 (6) (2004) 685–692] and Siddiqi et al. [S.S. Siddiqi, E.H. Twizell, Spline solution of linear eighth-order boundary value problems, Comput. Methods Appl. Mech. Eng. 131 (1996) 309–325], show that the method developed in this paper is more efficient.