A Least Squares Polynomial‐Based Partition of Unity Method for Tempered‐Type Integro‐Differential Equations

Source abstract

ABSTRACT This paper introduces a novel meshless method based on polynomials for solving a class of integro‐differential equations with tempered fractional singular kernels. Our method employs a least‐squares partition of unity framework based on local polynomial approximations in spatial discretization. To enhance numerical stability, the global approximation is constructed by blending local approximations over overlapping subdomains through the use of partition of unity weight functions. For temporal discretization, two convolution quadrature rules are employed for the tempered fractional operator, leading to the development of two time‐stepping schemes. The proposed method is applied to some test problems defined on rectangular, elliptical, and irregular domains. The numerical results confirm the accuracy, efficiency, and robustness of the proposed method in handling various geometries.