Abdul Hamid Ganie, Saurav Mallik, Mashael M. AlBaidani, Adnan Khan, Mohd Asif Shah
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引用次数: 0
摘要
在这项工作中,我们采用同调扰动变换法和杨变换分解法这两种独特的方法来求解分数非线性七阶 Kaup-Kupershmidt (KK) 问题。化学、物理学和工程学中出现的物理现象都可以用这个方程进行数学解释,特别是非线性光学、量子力学、等离子体物理学、流体动力学等。所提供的方法用于解决分数非线性七阶 KK 问题以及杨变换和分数卡普托导数。这些结果对于探索一系列物理过程具有重要意义和必要性。本文利用现代方法和分数算子对所提供的问题进行了令人满意的近似。为了求解分数 KK 方程,我们首先使用了杨变换和分数卡普托导数。He's 和 Adomian 多项式可用于管理非线性项。结果表明,建议的近似解收敛于精确解。在这些方法中,计算结果都是收敛级数。推荐方法的主要优点是只需很少的计算量就能得到非常精确的结果。建议方法的结果会与精确解进行比较。通过使用图形和表格将结果与精确解进行比较,我们可以验证所提供策略的有效性。此外,我们还研究了所建议方法在不同小数阶的结果,结果表明,随着数值从小数阶移动到整数阶,结果会变得更加精确。此外,所提供的方法新颖、简单且相当精确,表明它们在解决微分方程方面非常有效。
Novel analysis of nonlinear seventh-order fractional Kaup–Kupershmidt equation via the Caputo operator
In this work, we use two unique methodologies, the homotopy perturbation transform method and Yang transform decomposition method, to solve the fractional nonlinear seventh-order Kaup–Kupershmidt (KK) problem. The physical phenomena that arise in chemistry, physics, and engineering are mathematically explained in this equation, in particular, nonlinear optics, quantum mechanics, plasma physics, fluid dynamics, and so on. The provided methods are used to solve the fractional nonlinear seventh-order KK problem along with the Yang transform and fractional Caputo derivative. The results are significant and necessary for exploring a range of physical processes. This paper uses modern approaches and the fractional operator to develop satisfactory approximations to the offered problem. To solve the fractional KK equation, we first use the Yang transform and fractional Caputo derivative. He’s and Adomian polynomials are useful to manage nonlinear terms. It is shown that the suggested approximate solution converges to the exact one. In these approaches, the results are calculated as convergent series. The key advantage of the recommended approaches is that they provide highly precise results with little computational work. The suggested approach results are compared to the precise solution. By comparing the outcomes with the precise solution using graphs and tables we can verify the efficacy of the offered strategies. Also, the outcomes of the suggested methods at various fractional orders are examined, demonstrating that the findings get more accurate as the value moves from fractional order to integer order. Moreover, the offered methods are innovative, simple, and quite accurate, demonstrating that they are effective for resolving differential equations.
期刊介绍:
The main aim of Boundary Value Problems is to provide a forum to promote, encourage, and bring together various disciplines which use the theory, methods, and applications of boundary value problems. Boundary Value Problems will publish very high quality research articles on boundary value problems for ordinary, functional, difference, elliptic, parabolic, and hyperbolic differential equations. Articles on singular, free, and ill-posed boundary value problems, and other areas of abstract and concrete analysis are welcome. In addition to regular research articles, Boundary Value Problems will publish review articles.