Solutions of Time-Space Fractional PDEs Using Picard's Iterative Method

Manoj Kumar, Aman Jhinga, J. T. Majithia
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Abstract

In this paper, we present Picard's iterative method for solving time-space fractional partial differential equations, where the derivatives are considered in the Caputo sense. We prove the existence and uniqueness of solutions. Additionally, we demonstrate the versatility of our proposed approach by obtaining exact solutions for a diverse set of equations. This method is user-friendly and directly applicable to any computer algebra system. The present method avoids intricate computations associated with the Adomian decomposition method, such as calculating Adomian polynomials, or the requirements of other methods like choosing a homotopy in the homotopy perturbation method, identification and manipulation of the invariant subspace in invariant subspace method or constructing a variational function in the variational iteration method. Thus, the proposed method is a versatile and efficient tool for exploring systems that involve both temporal and spatial fractional derivatives.
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使用皮卡尔迭代法求解时空分式 PDEs
在本文中,我们介绍了 Picard 解决时空分数偏微分方程的迭代法,其中导数是在 Caputo 意义上考虑的。我们证明了解的存在性和唯一性。此外,我们还通过获得不同方程组的精确解,证明了我们提出的方法的通用性。这种方法对用户友好,可直接应用于任何计算机代数系统。本方法避免了与阿多米分解法相关的复杂计算,如计算阿多米多项式,或其他方法的要求,如在同调扰动法中选择同调,在不变子空间法中识别和操作不变子空间,或在变分迭代法中构建变分函数。因此,所提出的方法是探索涉及时间和空间分数导数的系统的多功能高效工具。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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