Artificial Boundary Method For Nonhomogeneous Parabolic Pdes On Unbounded Domains

Parabolic PDEs are the basic models of heat,gas,electromagnetic fields and so on.A large number of practical problems in the field of science and engineering,such as stress analysis with infinite elastic foundation dams,flow problems in infinitely long pipelines,are described by the parabolic partial differential equations on the unbounded domain.Due to the complexity of the practical problem and the unboundedness of the physical region,especially for the nonhomogeneous problem,it is not easy to get its exact solution in theory,or its exact solution is not conducive to the calculation,so to find a stable,high-precision method has important theoretical and practical value.The heat equation and the Burgers' equation are two most classical parabolic PDEs.Through the unremitting efforts of many scientific scholars,the study of the heat conduction equation and the Burgers' equation in the bounded region has made many valuable achievements.However,there are few studies on nonhomogeneous heat and Burgers' equations on unbounded domains.In this paper,the artificial boundary method and the Crank-Nicolson difference scheme are used to solve the nonhomogeneous heat and Burgers' Equation on the unbounded domain.The main contents and innovations are stated as follows:In the first part,similar to the study of nonhomogeneous heat equations on semi-bounded regions,we consider a finite difference scheme for the one-dimensional nonhomogeneous Burgers' equation on unbounded domains.Two exact artificial boundary conditions are introduced to transform the original problems into problems on finite domains.The finite difference scheme is developed by using reduced order method and the Crank-Nicolson difference scheme for the obtained equation and artificial conditions.The stability and error estimate of the discretization schemes are analyzed.Nonhomogeneous numerical examples demonstrate the accuracy and the efficiency of the algorithms.In the second part,we investigate a new finite difference scheme for one dimensional nonhomogeneous Burgers' equation on the unbounded domain.The exact nonlinear artificial boundary conditions are used on two artificial boundaries to limit the original problem onto a bounded computational domain.A function transformation makes both the equation and the boundary conditions linear.The novel finite difference scheme is developed by using the method of reduction of order for the obtained equation and artificial boundary conditions.The stability and the convergence with order 3/2 in time and 2 in space in an energy norm are proved for this method for Burgers' equation.The efficiency and accuracy of the proposed method are illustrated by different examples.Compared with the previous research results,this method not only avoids the difficulty of solving the nonlinear problem,but also eliminates the intermediate variable,which can greatly save the computation time and reduce the calculation cost.