In this paper, we present a novel approach based on shifted Gegenbauer wavelets to attain approximate solutions of some classed of time‐fractional nonlinear problems. First, we present the approximation of a function of two variables u(x,t) with help of shifted Gegenbauer wavelets and then some novel operational matrices are proposed with the help of piecewise functions to investigate the positive integer derivative (D_x and D_t), fractional‐order derivative ( and ), fractional‐order integration ( and ) and delay terms ( and ) of approximated function u(x,t). In order to transform the discussed nonlinear problem into linear problem Picard iterative scheme has been adopt. The current scheme converts the discussed highly nonlinear time‐fractional problem into system of linear algebraic equation the help of developed operational matrices and Picard idea. Analysis on the error bound and convergence to authenticate the mathematical formulation of the computational algorithm. We solve various test problems, such as the van der Pol oscillator model, generalized Burger–Huxley, neutral delay parabolic differential equations, sine‐Gordon, parabolic integro‐differential equation and nonlinear Schrödinger equations to show the efficiency and accuracy of linearized shifted Gegenbauer wavelets method. A comprehensive comparative examination shows the credibility, accuracy, and reliability of the presently proposed computational approach. Also, this scheme can be extended conveniently to other multi‐dimensional physical problems of highly nonlinear fractional or variable order of complex nature.

中文翻译：

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基于新颖运算矩阵的鲁棒方案，用于解决力学和数学物理中出现的某些类型的时间分数非线性问题

在本文中，我们提出了一种基于移位的Gengenbauer小波的新颖方法，以获得一类时分非线性问题的近似解。首先，我们借助移位的Gegenbauer小波给出了两个变量u（x，t）函数的逼近，然后借助分段函数提出了一些新颖的运算矩阵，以研究正整数导数（D _x和D _t），分数阶导数（和），分数阶积分（和）和延迟术语（和近似函数的）ü（X，Ť）。为了将讨论的非线性问题转化为线性问题，采用了Picard迭代方案。目前的方案是在发达的运算矩阵和Picard思想的帮助下，将讨论的高度非线性时间分数问题转换为线性代数方程组。分析误差范围和收敛性，以验证计算算法的数学公式。我们解决了各种测试问题，例如van der Pol振荡器模型，广义Burger-Huxley，中性时滞抛物线微分方程，sine-Gordon，抛物线积分微分方程和非线性Schrödinger方程，以证明线性化移位Gegenbauer小波的效率和精度方法。全面的比较检查显示出可信度，准确性，和目前提出的计算方法的可靠性。同样，该方案可以方便地扩展到其他具有复杂性的高度非线性分数或可变阶数的多维物理问题。