The Four-Dimensional Natural Transform Adomian Decomposition Method and (3+1)-Dimensional Fractional Coupled Burgers’ Equation
Abstract
:1. Introduction
2. Basic Definitions of the Natural Transform Method
3. Analysis of the Four-Dimensional Natural Adomian Decomposition Method
- Step 1: By implementing the four-dimensional natural transform to Equation (9), we obtain
- Step 2: Now, by using the differentiation property of the natural transform, we have
- Step 3: By employing the inverse four-dimensional natural transform for Equation (12), we obtain
- Step 4: The four-dimensional natural Adomian decomposition method assumes series solutions of the functions , , and , which are determined by
- Step 5: After applying the four-dimensional natural Adomian decomposition method, we introduce the recursive relations as follows:
4. Four-Dimensional Natural Adomian Decomposition Method and Singular (3+1)-Dimensional Fractional Coupled Burgers’ Equation
- Step 1: Multiply both sides of Equation (41) by to obtain
- Step 2: Operating the four-dimensional natural transform for both sides of Equation (42) yields
- Step 3: By integrating both sides of Equation (46) from 0 to to and 0 to with respect to , and , respectively, we have
- Step 6: After applying the four-dimensional natural Adomian decomposition method, we introduce the recursive relations as follows:
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Alsaud, H.; Eltayeb, H. The Four-Dimensional Natural Transform Adomian Decomposition Method and (3+1)-Dimensional Fractional Coupled Burgers’ Equation. Fractal Fract. 2024, 8, 227. https://doi.org/10.3390/fractalfract8040227
Alsaud H, Eltayeb H. The Four-Dimensional Natural Transform Adomian Decomposition Method and (3+1)-Dimensional Fractional Coupled Burgers’ Equation. Fractal and Fractional. 2024; 8(4):227. https://doi.org/10.3390/fractalfract8040227
Chicago/Turabian StyleAlsaud, Huda, and Hassan Eltayeb. 2024. "The Four-Dimensional Natural Transform Adomian Decomposition Method and (3+1)-Dimensional Fractional Coupled Burgers’ Equation" Fractal and Fractional 8, no. 4: 227. https://doi.org/10.3390/fractalfract8040227