Purpose

The purpose of this paper is to apply an efficient hybrid computational numerical technique, namely, q-homotopy analysis Sumudu transform method (q-HASTM) and residual power series method (RPSM) for finding the analytical solution of the non-linear time-fractional Hirota–Satsuma coupled KdV (HS-cKdV) equations.

Design/methodology/approach

The proposed technique q-HASTM is the graceful amalgamations of q-homotopy analysis method with Sumudu transform via Caputo fractional derivative, whereas RPSM depend on generalized formula of Taylors series along with residual error function.

Findings

To illustrate and validate the efficiency of the proposed technique, the authors analyzed the projected non-linear coupled equations in terms of fractional order. Moreover, the physical behavior of the attained solution has been captured in terms of plots and by examining the L₂ and L_∞ error norm for diverse value of fractional order.

Originality/value

The authors implemented two technique, q-HASTM and RPSM to obtain the solution of non-linear time-fractional HS-cKdV equations. The obtained results and comparison between q-HASTM and RPSM, shows that the proposed methods provide the solution of non-linear models in form of a convergent series, without using any restrictive assumption. Also, the proposed algorithm is easy to implement and highly efficient to analyze the behavior of non-linear coupled fractional differential equation arisen in various area of science and engineering.

中文翻译：

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分数阶Hirota-Satsuma耦合KdV方程的两种有效计算技术

目的

本文的目的是应用一种高效的混合计算数值技术，即q同伦分析Sumudu变换方法（q-HASTM）和残差幂级数方法（RPSM）来寻找非线性时间分数阶的解析解广田-萨摩耦合KdV（HS-cKdV）方程。

设计/方法/方法

提出的技术q-HASTM是通过Caputo分数导数进行Sumudu变换的q-同伦分析方法的完美融合，而RPSM依赖于Taylors级数的广义公式以及残差函数。

发现

为了说明和验证所提出技术的效率，作者按照分数阶分析了投影的非线性耦合方程。此外，实现溶液的物理行为已经在情节等方面并通过检查所捕获的大号₂和大号_∞误差范为分数阶的多样值。

创意/价值

作者实施了q-HASTM和RPSM这两种技术来获得非线性时间分数HS-cKdV方程的解。所得结果和q-HASTM与RPSM的比较表明，所提出的方法以收敛序列的形式提供了非线性模型的解，而没有使用任何限制性假设。而且，所提出的算法易于实现并且对于分析在科学和工程学的各个领域中出现的非线性耦合分数阶微分方程的行为是高效的。