This study is devoted to constructing an approximate analytic solution of the fractional form of a strongly nonlinear boundary value problem with multi‐fractional derivatives that comes in chemical reactor theory. We construct the solution algorithm based on the generalized differential transform technique in four simple steps. The fractional derivative is defined in the sense of Caputo. We also mathematically prove the convergence of the algorithm. The applicability and effectiveness of the given scheme are justified by simulating the equation for given parameter values presented in the system and compared with existing published results in the case of standard derivatives. In addition, residual error computation is used to check the algorithm’s correctness. The results are presented in several tables and figures. The goal of this study is to justify the effects and importance of the proposed fractional derivative on the given nonlinear problem. The generalization of the adopted integer‐order problem into a fractional‐order sense which includes the memory in the system is the main novelty of this research.

Fractional derivative models involving generalized Mittag-Leffler kernels and opposing models are investigated. We first replace the classical derivative with the GMLK in order to obtain the new fractional-order models (GMLK) with the three parameters that are investigated. We utilize a spectral collocation method based on Legendre’s polynomials for evaluating the numerical solutions of the pr. We then construct a scheme for the fractional-order models by using the spectral method involving the Legendre polynomials. In the first model, we directly obtain a set of nonlinear algebraic equations, which can be approximated by the Newton-Raphson method. For the second model, we also need to use the finite differences method to obtain the set of nonlinear algebraic equations, which are also approximated as in the first model. The accuracy of the results is verified in the first model by comparing it with our analytical solution. In the second and third models, the residual error functions are calculated. In all cases, the results are found to be in agreement. The method is a powerful hybrid technique of numerical and analytical approach that is applicable for partial differential equations with multi-order of fractional derivatives involving GMLK with three parameters.

In this paper, we investigate the Sumudu transforms and homotopy analysis method (S-HAM) for solving a system of fractional partial differential equations. A general framework for solving such a kind of problems is presented. The method can also be utilized to solve systems of fractional equations of unequal orders. The algorithm is reliable and robust. Existence and convergence results concerning the proposed solution are given. Numerical examples are introduced to demonstrate the efficiency and accuracy of the algorithm.

Partial differential equations are known to be increasingly important in today’s research, and their solutions are paramount for tackling numerous real-life applications. This article extended the analytical scheme of the homotopy analysis method (HAM) to develop an approximate analytical solution for Fuzzy Partial Differential Equations (FPDEs). The scheme used its powerful tools, the auxiliary function and convergence-control parameter, in the analysis and optimization, which ensures the convergence of the approximate series solution in addition to considering all necessary concepts from fuzzy set theory to provide high precision in the fuzzy environment. Furthermore, the efficiency was shown by applying the proposed scheme to linear and nonlinear cases of Fuzzy Reaction–Diffusion Equation (FRDE) and Fuzzy Wave Equation (FWE).