On the numerical solution of some differential equations with nonlocal integral boundary conditions via Haar wavelet

Abstract

Differential equations with nonlocal boundary conditions are used to model a number of physical phenomena encountered in situations where data on the boundary cannot be measured directly. This study explores numerical solutions to elliptic, parabolic and hyperbolic equations with two different types of nonlocal integral boundary conditions. The numerical solutions are obtained using the Haar wavelet collocation method with the aid of Finite Differences for time derivatives. The method is applicable to both linear and nonlinear problems. To obtain the numerical solutions, Gauss elimination method is used for linear and Newton's method for nonlinear differential equations. The validity of the proposed method is demonstrated by solving several benchmark test problems from the literature: two elliptic linear and two nonlinear samples covering both types of nonlocal integral boundary conditions; one nonlinear and two linear test problems for parabolic partial differential equations; two linear samples for hyperbolic partial differential equations. The accuracy of the method is verified by comparing the numerical results with the analytical solutions. The numerical results confirm that the method is simple and effective.