scholarly journals Approximate Solution of Fractional Nonlinear Partial Differential Equations by the Legendre Multiwavelet Galerkin Method

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
M. A. Mohamed ◽  
M. Sh. Torky
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Galerkin Method ◽  
Nonlinear Partial Differential Equations ◽  
Algebraic Equations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
The Galerkin Method

The Legendre multiwavelet Galerkin method is adopted to give the approximate solution for the nonlinear fractional partial differential equations (NFPDEs). The Legendre multiwavelet properties are presented. The main characteristic of this approach is using these properties together with the Galerkin method to reduce the NFPDEs to the solution of nonlinear system of algebraic equations. We presented the numerical results and a comparison with the exact solution in the cases when we have an exact solution to demonstrate the applicability and efficiency of the method. The fractional derivative is described in the Caputo sense.

scholarly journals Technique to Solve Linear Fractional Differential Equations Using B-Polynomials Bases

2021 ◽  
Vol 5 (4) ◽  
pp. 208
Author(s):  
Muhammad I. Bhatti ◽  
Md. Habibur Rahman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Galerkin Method ◽  
Fractional Order ◽  
Initial Conditions ◽  
Operational Matrix ◽  
Basis Sets ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
The Galerkin Method

A multidimensional, modified, fractional-order B-polys technique was implemented for finding solutions of linear fractional-order partial differential equations. To calculate the results of the linear Fractional Partial Differential Equations (FPDE), the sum of the product of fractional B-polys and the coefficients was employed. Moreover, minimization of error in the coefficients was found by employing the Galerkin method. Before the Galerkin method was applied, the linear FPDE was transformed into an operational matrix equation that was inverted to provide the values of the unknown coefficients in the approximate solution. A valid multidimensional solution was determined when an appropriate number of basis sets and fractional-order of B-polys were chosen. In addition, initial conditions were applied to the operational matrix to seek proper solutions in multidimensions. The technique was applied to four examples of linear FPDEs and the agreements between exact and approximate solutions were found to be excellent. The current technique can be expanded to find multidimensional fractional partial differential equations in other areas, such as physics and engineering fields.

Wavelet Galerkin method for fourth-order multi-dimensional elliptic partial differential equations

2018 ◽  
Vol 16 (05) ◽  
pp. 1850045 ◽  
Author(s):  
B. V. Rathish Kumar ◽  
Gopal Priyadarshi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Galerkin Method ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Computational Cost ◽  
Stability Estimates ◽  
Partial Differential ◽  
Wavelet Galerkin Method

We describe a wavelet Galerkin method for numerical solutions of fourth-order linear and nonlinear partial differential equations (PDEs) in 2D and 3D based on the use of Daubechies compactly supported wavelets. Two-term connection coefficients have been used to compute higher-order derivatives accurately and economically. Localization and orthogonality properties of wavelets make the global matrix sparse. In particular, these properties reduce the computational cost significantly. Linear system of equations obtained from discretized equations have been solved using GMRES iterative solver. Quasi-linearization technique has been effectively used to handle nonlinear terms arising in nonlinear biharmonic equation. To reduce the computational cost of our method, we have proposed an efficient compression algorithm. Error and stability estimates have been derived. Accuracy of the proposed method is demonstrated through various examples.

Solution Non Linear Partial Differential Equations By ZMA Decomposition Method

2021 ◽  
Vol 20 ◽  
pp. 712-716
Author(s):  
Zainab Mohammed Alwan
Keyword(s):  
Exact Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Decomposition Method ◽  
Integral Transformation ◽  
Nonlinear Partial Differential Equations ◽  
Adomian Decomposition Method ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Adomian Decomposition

In this survey, viewed integral transformation (IT) combined with Adomian decomposition method (ADM) as ZMA- transform (ZMAT) coupled with (ADM) in which said ZMA decomposition method has been utilized to solve nonlinear partial differential equations (NPDE's).This work is very useful for finding the exact solution of (NPDE's) and this result is more accurate obtained with compared the exact solution obtained in the literature.

Numerical Solution of Fractional Partial Differential Equations by Discrete Adomian Decomposition Method

2014 ◽  
Vol 6 (01) ◽  
pp. 107-119 ◽  
Author(s):  
D. B. Dhaigude ◽  
Gunvant A. Birajdar
Keyword(s):  
Exact Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Diffusion Equation ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Discrete Schrödinger Equation ◽  
Adomian Decomposition

AbstractIn this paper we find the solution of linear as well as nonlinear fractional partial differential equations using discrete Adomian decomposition method. Here we develop the discrete Adomian decomposition method to find the solution of fractional discrete diffusion equation, nonlinear fractional discrete Schrodinger equation, fractional discrete Ablowitz-Ladik equation and nonlinear fractional discrete Burger’s equation. The obtained solution is verified by comparison with exact solution whenα= 1.

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