Finding analytical and semi-analytical solutions of two-dimensional nonlinear fractional-order problems arising in mathematical physics is a challenging task for research community. In this work, an innovative scheme is proposed based on the shifted Gegenbauer wavelets. For this purpose first, we introduce shifted Gegenbauer polynomials via suitable transformation. Then, we present three-dimensional shifted Gegenbauer wavelets using shifted Gegenbauer polynomials. We illustrate function approximation of three variables, for example, $u(x,y,t)$ through shifted Gegenbauer wavelets. To compute the novel operational matrices of positive integer and non-integer order derivative of shifted Gegenbauer wavelets vector in one, two and three dimensionals piecewise functions are utilized. Moreover, we describe associated theorems to validate our newly proposed scheme mathematically. The proposed algorithm is innovative because of the incorporation of Picard iterative scheme to tackle highly nonlinear problems of fractional-order. The current computational scheme coverts a mathematical model to a system of linear algebraic equations that are easier to solve. To validate the accuracy, credibility, and reliability of the present method, we analyse various fractional-order problems, including Bloch–Torrey, Burgers, Schrödinger, Rayleigh–Stokes and sine-Gordon. We also conduct a detailed comparative study, which demonstrates that the proposed computational scheme is effective to find the analytical and semi-analytical solutions of the aforementioned problems. Moreover, the proposed computational method can be utilized to analyze the solutions of other higher dimensional nonlinear fractional or variable order problems of physical nature.

对于数学物理学中出现的二维非线性分数阶问题，寻找分析和半解析解是一项艰巨的任务。在这项工作中，提出了一种基于转移的Gegenbauer小波的创新方案。为此，我们首先通过适当的变换引入移位的Gegenbauer多项式。然后，我们使用移位的Gegenbauer多项式呈现三维移位的Gegenbauer小波。我们举例说明三个变量的函数逼近，$ü（X，ÿ，Ť）$通过移位的Gegenbauer小波 为了计算一维，二维和三维分段函数中移位的Gegenbauer小波矢量的正整数和非整数阶导数的新颖运算矩阵。此外，我们描述了相关定理以数学方式验证我们新提出的方案。由于引入了Picard迭代方案来解决分数阶的高度非线性问题，因此该算法具有创新性。当前的计算方案将数学模型覆盖到了易于求解的线性代数方程组中。为了验证本方法的准确性，可信性和可靠性，我们分析了各种分数阶问题，包括布洛赫-托里，伯格斯，薛定ding，瑞利-斯托克斯和正弦高登。我们还进行了详细的比较研究，证明了所提出的计算方案对于找到上述问题的解析和半解析解是有效的。此外，所提出的计算方法可以用来分析物理性质的其他高维非线性分数阶或可变阶问题的解。