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Modified approach for linear and non-linear IBVPs with fractional dynamics

Qasim Khan, Anthony Suen

TL;DR

The work addresses initial-boundary value problems for fractional PDEs with Caputo time derivatives $^{C}D^{\alpha}_{t}$. It introduces a Modified Laplace Adomian Decomposition Method MLDM that combines ADM, the Laplace transform, and Daftardar-Jafari polynomials, with boundary conditions enforced by corrected iterates and B_n^* corrections. Numerical experiments on linear and nonlinear FPDE IBVPs in 1D and 2D show rapid convergence and higher accuracy with few terms, outperforming classical LADM in many cases. The results provide a practical framework for solving FPDE IBVPs and point to extensions to broader nonlinear FPDEs and related PDEs.

Abstract

Analytical and numerical techniques have been developed for solving fractional partial differential equations (FPDEs) and their systems with initial conditions. However, it is much more challenging to develop analytical or numerical techniques for FPDEs with boundary conditions, although some methods do exist to address such problems. In this paper, a modified technique based on the Adomian decomposition method with Laplace transformation is presented, which effectively treats initial-boundary value problems. The non-linear term has been controlled by Daftardar-Jafari polynomials. Our proposed technique is applied to several initial and boundary value problems and the obtained results are presented through graphs. The differing behavior of the solutions for the suggested problems is observed by using various fractional orders. It is found that our proposed technique has a high rate of convergence towards the exact solutions of the problems. Moreover, while implementing this modification, higher accuracy is achieved with a small number of calculations, which is the main novelty of the proposed technique. The present method requires a new approximate solution in each iteration that adds further accuracy to the solution. It demonstrates that our suggested technique can be used effectively to solve initial-boundary value problems of FPDEs.

Modified approach for linear and non-linear IBVPs with fractional dynamics

TL;DR

The work addresses initial-boundary value problems for fractional PDEs with Caputo time derivatives . It introduces a Modified Laplace Adomian Decomposition Method MLDM that combines ADM, the Laplace transform, and Daftardar-Jafari polynomials, with boundary conditions enforced by corrected iterates and B_n^* corrections. Numerical experiments on linear and nonlinear FPDE IBVPs in 1D and 2D show rapid convergence and higher accuracy with few terms, outperforming classical LADM in many cases. The results provide a practical framework for solving FPDE IBVPs and point to extensions to broader nonlinear FPDEs and related PDEs.

Abstract

Analytical and numerical techniques have been developed for solving fractional partial differential equations (FPDEs) and their systems with initial conditions. However, it is much more challenging to develop analytical or numerical techniques for FPDEs with boundary conditions, although some methods do exist to address such problems. In this paper, a modified technique based on the Adomian decomposition method with Laplace transformation is presented, which effectively treats initial-boundary value problems. The non-linear term has been controlled by Daftardar-Jafari polynomials. Our proposed technique is applied to several initial and boundary value problems and the obtained results are presented through graphs. The differing behavior of the solutions for the suggested problems is observed by using various fractional orders. It is found that our proposed technique has a high rate of convergence towards the exact solutions of the problems. Moreover, while implementing this modification, higher accuracy is achieved with a small number of calculations, which is the main novelty of the proposed technique. The present method requires a new approximate solution in each iteration that adds further accuracy to the solution. It demonstrates that our suggested technique can be used effectively to solve initial-boundary value problems of FPDEs.
Paper Structure (10 sections, 1 theorem, 66 equations, 14 figures)

This paper contains 10 sections, 1 theorem, 66 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Research Methodology
  3. Adomian Decomposition Method
  4. Modified ADM for Linear IBVPs
  5. Classical LADM for IVPs
  6. Modified LADM for non-linear IBVPs
  7. Numerical Experiments
  8. Result and Discussion
  9. Conclusion
  10. Declaration

Key Result

Lemma 1

For $j-1 < \alpha \leq j$ with $j \in \mathbb{N}$ and $h : \mathbb{R}^+\to\mathbb{C}$, the operator $I^{\alpha}$ satisfies the following identities

Figures (14)

  • Figure 1: Comparison plots of Exact, Classical, and Modified LADM solutions, along with their corresponding absolute errors for Problem \ref{['problem1']}.
  • Figure 2: 3D comparison plot presenting the Exact, Classical, and Modified LADM solutions for problem \ref{['problem1']}.
  • Figure 3: Comparison plots of Exact, Classical, and Modified LADM solutions at different fractional order $\alpha$ for Problem \ref{['problem2']}.
  • Figure 4: 3D comparison plot presenting the Exact, Classical, and Modified LADM solutions for problem \ref{['problem2']}.
  • Figure 5: Comparison plots of Exact, Classical, and Modified LADM solutions at different fractional order $\alpha$ for Problem \ref{['problem32']}.
  • ...and 9 more figures

Theorems & Definitions (5)

  • Definition 1
  • Lemma 1
  • Definition 2
  • Definition 3
  • Definition 4