This article investigates the RBF partition of unity method for solving the two-dimensional Sobolev equation with a Burgers-type nonlinearity on irregular domains. To the best of our knowledge, there are limited results in solving the Sobolev equation with a Burgers-type nonlinearity on complex irregular domains. Two variants of the radial basis functions partition of unity method, namely the classical and direct methods with smooth and discontinuous weight functions, are employed for spatial discretization, and their results are compared in terms of accuracy, execution time, and sparsity of their computational matrices. We Utilized the polyharmonic radial kernels due to their remarkable properties, including their ability to approximate and scalability. The scalability attribute of these radial kernels is used to regulate the instability of the method. To ensure the accuracy and reliability of our proposed approach, we carried out numerical simulations and provided several examples for comparison. Our comparisons of the radial basis functions partition of unity procedures reveal that the direct method is computationally more efficient than the classical approach, while both versions have admissible accuracy.
