A meshless method using the Takagi–Sugeno fuzzy model

Abstract

This article proposes a new strong-form meshless method using the Takagi–Sugeno fuzzy model (MTSF) for solving differential equations (DEs). Considering the conventional fuzzy model, the fuzzy inference system (FIS) can be categorized into two architectures, a simple rule base using the Euclidean distance in a multidimensional space (Simple-FIS), and an adaptive neuro-fuzzy inference system (ANFIS). Accordingly, MTSF also can be implemented using Simple-FIS and ANFIS. Based on the two architectures, an approximation scheme for continuous functions is drawn out first. In turn, the derivation is further proposed in which the differential functions are approximated using two independent sets of points, one for the collocation point and the other for the rule point. Solving higher-order DEs becomes possible by following the derivations, and eventually numerical solutions can be obtained. Several examples of one-dimensional ordinary and two-dimensional partial DEs (ODEs and PDEs) are presented to demonstrate the performance of the MTSF method. By MTSF, solutions solved using Simple-FIS and using ANFIS are compared. Variations in boundary conditions and membership function parameters are also studied to examine the agreement among numerical solutions.