A Wavelet Method for Solving Nonlinear Time-Dependent Partial Differential Equations
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TL;DR: In this paper, a wavelet method is proposed for solving a class of nonlinear time-dependent partial differential equations, and the nonlinear equations are first transfoulled into a system of ordinary differential equations by using the modified wavelet Galerkin method recently developed by the authors.
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Abstract: A wavelet method is proposed for solving a class of nonlinear time-dependent partial differential equations. Following this method, the nonlinear equations are first transfoulled into a system of ordinary differential equations by using the modified wavelet Galerkin method recently developed by the authors. Then, the classical fourth-order explicit Runge-Kutta method is employed to solve the resulting system of ordinary differential equations. To justify the present method, the coupled viscous Burgers' equations are solved as examples, results demonstrate that the proposed wavelet algorithm have a much better accuracy and efficiency than many existing numerical methods, and the order of convergence of such a wavelet method can even reach about 5.
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Citations
A HAM-based wavelet approach for nonlinear partial differential equations: Two dimensional Bratu problem as an application
Zhaochen Yang,Shijun Liao +1 more
TL;DR: A new analytic approach for boundary value problems (BVPs) governed by nonlinear partial differential equations (PDEs), which successfully combines the homotopy analysis method (HAM) and the generalized Coiflet-type wavelet and a section-based wavelet approximation for partial derivatives is proposed.
50
A HAM-based wavelet approach for nonlinear ordinary differential equations
Zhaochen Yang,Shijun Liao +1 more
TL;DR: The wavelet homotopy analysis method (wHAM) is proposed, a new analytic approximation approach for solving nonlinear boundary value problems (governed by nonlinear ordinary differential equations) that possesses high computational efficiency and much larger freedom to choose the auxiliary linear operator.
48
A space–time fully decoupled wavelet Galerkin method for solving a class of nonlinear wave problems
TL;DR: Results demonstrate that the proposed wavelet algorithm has a much better accuracy and a faster convergence rate than many existing numerical methods and is capable of capturing complex nonlinear phenomena, even those extremely sensitive to parameters.
17
A High-Order AccurateWavelet Method for SolvingThree-Dimensional Poisson Problems
TL;DR: Based on the approximation scheme for a L 2 function defined on a three-dimensional bounded space by combining techniques of boundary extension and Coiflet-type wavelet expansion, a modified wavelet Galerkin method is proposed for solving 3D Poisson problems with various boundary conditions.
1
Brief Introduction in Applications of Other Groups
You-He Zhou
- 01 Jan 2021
TL;DR: In this article, a brief introduction of those applications done by other groups using the generalized Coiflets and relevant method proposed by the author's group is given, along with a relevant method.
References
Variational iteration method for solving Burger's and coupled Burger's equations
M.A. Abdou,A.A. Soliman +1 more
TL;DR: In this article, He's variational iteration method is introduced to overcome the difficulty arising in calculating Adomian polynomials, and the solutions of Burger's equation and coupled Burger's equations are exactly obtained.
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Wavelet Analysis: The Scalable Structure of Information
Howard L. Resnikoff,Raymond O. Wells +1 more
- 27 Sep 2011
TL;DR: Part I: The Scalable Structure of Information: 1. The New Mathematical Engineering 2. Good Approximation and Algorithms 3. Wavelets: A Positional Notation for Functions
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A Chebyshev spectral collocation method for solving Burgers'-type equations
TL;DR: In this article, a spectral collocation method based on differentiated Chebyshev polynomials was proposed to obtain numerical solutions for some different kinds of nonlinear partial differential equations.
252
A review of numerical methods for nonlinear partial differential equations
TL;DR: A bird’s eye view on the development of numerical methods for solving partial differential equations with a particular emphasis on nonlinear PDEs is provided.
Nonlinear Partial Differential Equations
Abdul-Majid Wazwaz
- 01 Jan 2009
TL;DR: In this article, the authors focus on the nonlinear partial differential equations and apply the Adomina decomposition method and the variational iteration method to obtain the solutions of nonlinear wave equations.
196