scholarly journals New approach for accelerating the nonlinear Schwarz iterations

2019 ◽  
Vol 38 (4) ◽  
pp. 51-69
Author(s):  
Nabila Nagid ◽  
Hassan Belhadj
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Convergence Analysis ◽  
Nonlinear Partial Differential Equations ◽  
Test Cases ◽  
New Approach ◽  
Additive Schwarz ◽  
Vector Sequences ◽  
Linear And Nonlinear ◽  
Epsilon Algorithm

The vector Epsilon algorithm is an effective extrapolation method used for accelerating the convergence of vector sequences. In this paper, this method is used to accelerate the convergence of Schwarz iterative methods for stationary linear and nonlinear partial differential equations (PDEs). The vector Epsilon algorithm is applied to the vector sequences produced by additive Schwarz (AS) or restricted additive Schwarz (RAS) methods after discretization. Some convergence analysis is presented, and several test-cases of analytical problems are performed in order to illustrate the interest of such algorithm. The obtained results show that the proposed algorithm yields much faster convergence than the classical Schwarz iterations.

scholarly journals ANALYSIS OF TIME-FRACTIONAL BURGERS AND DIFFUSION EQUATIONS BY USING MODIFIED q-HATM

Fractals ◽  
2021 ◽  
pp. 2240012
Author(s):  
NEHAD ALI SHAH ◽  
PRAVEEN AGARWAL ◽  
JAE DONG CHUNG ◽  
SAAD ALTHOBAITI ◽  
SAMY SAYED ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Nonlinear Partial Differential Equations ◽  
Convergent Sequence ◽  
Exact Result ◽  
Diffusion Equations ◽  
Homotopy Analysis ◽  
Linear And Nonlinear ◽  
And Diffusion

In this paper, the q-homotopy analysis transform technique is implemented to analyze the solution of fractional-order Burgers and diffusion equations with the help of Caputo operator. The results of the proposed method are shown and analyzed with the help of figures. This approach is used to determine the solution in a convergent sequence and illustrate the q-homotopy analysis transform technique solutions convergence to the exact result. Several examples showed the reliability and simplicity of the technique and highlighted the significance of this work. Therefore, the proposed method is successful in investigating other fractional-order linear and nonlinear partial differential equations.

scholarly journals Application of Conformable Sumudu Decomposition Method for Solving Conformable Fractional Coupled Burger’s Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Hassan Eltayeb ◽  
Said Mesloub
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Analytical Method ◽  
Decomposition Method ◽  
Nonlinear Partial Differential Equations ◽  
One Dimensional ◽  
Burger's Equation ◽  
Burger’S Equation ◽  
Sumudu Transform ◽  
Linear And Nonlinear

The conformable double Sumudu decomposition method (CDSDM) is a combination of decomposition method (DM) and a conformable double Sumudu transform. It is an approximate analytical method, which can be used to solve linear and nonlinear partial differential equations. In this work, one-dimensional conformable functional Burger’s equation has been solved by applying conformable double Sumudu decomposition. Two examples are used to illustrate the method.

scholarly journals Analytical Solutions for the Nonlinear Partial Differential Equations Using the Conformable Triple Laplace Transform Decomposition Method

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Shailesh A. Bhanotar ◽  
Mohammed K. A. Kaabar
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Laplace Transform ◽  
Analytical Solutions ◽  
Decomposition Method ◽  
Nonlinear Partial Differential Equations ◽  
Three Dimensions ◽  
Conformable Derivative ◽  
Partial Differential ◽  
Linear And Nonlinear

In this paper, a novel analytical method for solving nonlinear partial differential equations is studied. This method is known as triple Laplace transform decomposition method. This method is generalized in the sense of conformable derivative. Important results and theorems concerning this method are discussed. A new algorithm is proposed to solve linear and nonlinear partial differential equations in three dimensions. Moreover, some examples are provided to verify the performance of the proposed algorithm. This method presents a wide applicability to solve nonlinear partial differential equations in the sense of conformable derivative.

Continuous Solution of Partial Differential Equations on Non-Rectangular and Non-Continuous Geometry

2014 ◽  
Author(s):  
P. Venkataraman
Keyword(s):  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Continuous Solution ◽  
Finite Difference Methods ◽  
Nonlinear Partial Differential Equations ◽  
Partial Differential ◽  
Mesh Free ◽  
Numeric Calculation ◽  
Linear And Nonlinear

A high order continuous solution is obtained for partial differential equations on non-rectangular and non-continuous domain using Bézier functions. This is a mesh free alternative to finite element or finite difference methods that are normally used to solve such problems. The problem is handled without any transformation and the setup is direct, simple, and involves minimizing the error in the residuals of the differential equations along with the error in the boundary conditions over the domain. The solution can be expressed in polynomial form. The effort is same for linear and nonlinear partial differential equations. The procedure is developed as a combination of symbolic and numeric calculation. The solution is obtained through the application of standard unconstrained optimization. A constrained approach is also developed for nonlinear partial differential equations. Examples include linear and nonlinear partial differential equations. The solution for linear partial differential equations is compared to finite element solutions from COMSOL.

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