Solving a nonlinear heat equation with nonlocal boundary conditions by a method of nonlocal boundary shape functions

Abstract

When a nonlinear heat equation is subjected to nonlocal boundary conditions, the difficulty might arise because time-dependent integral conditions present in the problem. To overcome this difficulty, we derive a nonlocal boundary shape function (NLBSF) to satisfy initial condition and two nonlocal boundary conditions. Then, letting the free function in the NLBSF be the Pascal polynomials, the generated new bases automatically satisfy all the specified conditions. The solution is thus expanded in terms of these bases. After collocating points inside the space-time domain to satisfy the nonlinear heat equation and employing a novel splitting and linearizing technique to solve the resulting linear system, we can quickly find accurate solution of the nonlocal boundary conditions problem of nonlinear heat equation. Examples confirm the high accuracy and efficiency of the proposed iterative method.