A Novel Low-Dimensional Method for Analytically Solving Partial Differential Equations (Advances in Applied Mathematics and Mechanics, 2021)
This paper is concerned with a low-dimensional dynamical system modelfor analytically solving partial differential equations (PDEs). The model proposed isbased on a posterior optimal truncated weighted residue (POT-WR) method, by whichan infinite dimensional PDE is optimally truncated and analytically solved in requiredcondition of accuracy. To end that, a POT-WR condition for PDE under considerationis used as a dynamically optimal control criterion with the solving process. A set ofbases needs to be constructed without any reference database in order to establish aspace to describe low-dimensional dynamical system that is required. The Lagrangianmultiplier is introduced to release the constraints due to the Galerkin projection, anda penalty function is also employed to remove the orthogonal constraints. Accordingto the extreme principle, a set of ordinary differential equations is thus obtainedby taking the variational operation of the generalized optimal function. A conjugategradient algorithm by FORTRAN code is developed to solve the ordinary differentialequations. The two examples of one-dimensional heat transfer equation and nonlinearBurgers’ equation show that the analytical results on the method proposed are goodagreement with the numerical simulations and analytical solutions in references, andthe dominant characteristics of the dynamics are well captured in case of few basesused only.