Solving a parabolic PDE with nonlocal boundary conditions using the sinc method
The authors use the sinc method to determine the solution \(u\) of the heat equation \(u_t-u_{xx}=q(x,t)\) for \(x\in(0,1)\) and \(t\in(0,T]\) with initial condition \(u(x,0)=f(x)\). They consider two problems for \(0<t\leq T\). Problem I: \(u(0,t)=\int_0^1 k_1(x,t)u(x,t)\,dx+g_1(t)\) and \(u(1,t)=\int_0^1 k_2(x,t)u(x,t)\,dx+g_2(t)\). Problem II: \(\alpha_1(t)u_x(0,t)+\beta(t)u(0,t)=g_1(t)\) and \(\alpha_2(t)u_x(1,t)+\gamma(t)u(1,t)=g_2(t)\). Problem I appears in the study of the quasi-static flexure of a thermoelastic bar. Problem II can be regarded as a heat transfer process where \(u\) represents the temperature distribution. In Section 2, both problems are transformed into systems of Volterra integral equations. In the Sections 3 and 4, convolution integrals are approximated by the sinc collocation method. Convergence analysis is discussed in Section 5. In Section 6, the performance of the algorithm in solving parabolic partial differential equations (PDEs) with nonlocal boundary conditions and the accuracy and efficiency of the method are illustrated with two examples.
scientific article; zbMATH DE number 6148996
scientific article