Volume 7, Issue 6
A Novel Low-Dimensional Method for Analytically Solving Partial Differential Equations

Jie Sha, Lixiang Zhang & Chuijie Wu

Adv. Appl. Math. Mech., 7 (2015), pp. 754-779.

Published online: 2018-05

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  • Abstract

This paper is concerned with a low-dimensional dynamical system model for analytically solving partial differential equations (PDEs). The model proposed is based on a posterior optimal truncated weighted residue (POT-WR) method, by which an infinite dimensional PDE is optimally truncated and analytically solved in required condition of accuracy. To end that, a POT-WR condition for PDE under consideration is used as a dynamically optimal control criterion with the solving process. A set of bases needs to be constructed without any reference database in order to establish a space to describe low-dimensional dynamical system that is required. The Lagrangian multiplier is introduced to release the constraints due to the Galerkin projection, and a penalty function is also employed to remove the orthogonal constraints. According to the extreme principle, a set of ordinary differential equations is thus obtained by taking the variational operation of the generalized optimal function. A conjugate gradient algorithm by FORTRAN code is developed to solve the ordinary differential equations. The two examples of one-dimensional heat transfer equation and nonlinear Burgers’ equation show that the analytical results on the method proposed are good agreement with the numerical simulations and analytical solutions in references, and the dominant characteristics of the dynamics are well captured in case of few bases used only

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@Article{AAMM-7-754, author = {Jie Sha, Lixiang Zhang and Chuijie Wu}, title = {A Novel Low-Dimensional Method for Analytically Solving Partial Differential Equations}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {7}, number = {6}, pages = {754--779}, abstract = {

This paper is concerned with a low-dimensional dynamical system model for analytically solving partial differential equations (PDEs). The model proposed is based on a posterior optimal truncated weighted residue (POT-WR) method, by which an infinite dimensional PDE is optimally truncated and analytically solved in required condition of accuracy. To end that, a POT-WR condition for PDE under consideration is used as a dynamically optimal control criterion with the solving process. A set of bases needs to be constructed without any reference database in order to establish a space to describe low-dimensional dynamical system that is required. The Lagrangian multiplier is introduced to release the constraints due to the Galerkin projection, and a penalty function is also employed to remove the orthogonal constraints. According to the extreme principle, a set of ordinary differential equations is thus obtained by taking the variational operation of the generalized optimal function. A conjugate gradient algorithm by FORTRAN code is developed to solve the ordinary differential equations. The two examples of one-dimensional heat transfer equation and nonlinear Burgers’ equation show that the analytical results on the method proposed are good agreement with the numerical simulations and analytical solutions in references, and the dominant characteristics of the dynamics are well captured in case of few bases used only

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.2014.m671}, url = {http://global-sci.org/intro/article_detail/aamm/12238.html} }
TY - JOUR T1 - A Novel Low-Dimensional Method for Analytically Solving Partial Differential Equations AU - Jie Sha, Lixiang Zhang & Chuijie Wu JO - Advances in Applied Mathematics and Mechanics VL - 6 SP - 754 EP - 779 PY - 2018 DA - 2018/05 SN - 7 DO - http://dor.org/10.4208/aamm.2014.m671 UR - https://global-sci.org/intro/aamm/12238.html KW - AB -

This paper is concerned with a low-dimensional dynamical system model for analytically solving partial differential equations (PDEs). The model proposed is based on a posterior optimal truncated weighted residue (POT-WR) method, by which an infinite dimensional PDE is optimally truncated and analytically solved in required condition of accuracy. To end that, a POT-WR condition for PDE under consideration is used as a dynamically optimal control criterion with the solving process. A set of bases needs to be constructed without any reference database in order to establish a space to describe low-dimensional dynamical system that is required. The Lagrangian multiplier is introduced to release the constraints due to the Galerkin projection, and a penalty function is also employed to remove the orthogonal constraints. According to the extreme principle, a set of ordinary differential equations is thus obtained by taking the variational operation of the generalized optimal function. A conjugate gradient algorithm by FORTRAN code is developed to solve the ordinary differential equations. The two examples of one-dimensional heat transfer equation and nonlinear Burgers’ equation show that the analytical results on the method proposed are good agreement with the numerical simulations and analytical solutions in references, and the dominant characteristics of the dynamics are well captured in case of few bases used only

Jie Sha, Lixiang Zhang & Chuijie Wu. (1970). A Novel Low-Dimensional Method for Analytically Solving Partial Differential Equations. Advances in Applied Mathematics and Mechanics. 7 (6). 754-779. doi:10.4208/aamm.2014.m671
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