In this article, the existence of solution for the first-order nonlinear coupled system of ordinary differential equations with nonlinear coupled boundary condition (CBC for short) is studied using a coupled lower and upper solution approach. Our method for a nonlinear coupled system with nonlinear CBC is new and it unifies the treatment of many different first-order problems. Examples are included to ensure the validity of the results.

Many conventional physical and engineering phenomena have been identified to be well expressed by making use of the fractional order partial differential equations (FOPDEs). For that reason, a proficient and stable numerical method is needed to find the approximate solution of FOPDEs. This article is designed to develop the numerical scheme able to find the approximate solution of generalized fractional order coupled systems (FOCSs) with mixed partial derivative terms of fractional order. Our main objective in this article is the development of a new operational matrix for fractional mixed partial derivatives based on the orthogonal shifted Legendre polynomials (SLPs). The fractional derivatives are considered herein in the sense of Caputo. The proposed method has the advantage to reduce the considered problems to a system of algebraic equations which are simple in handling by any computational software. Being easily solvable, the associated algebraic system leads to finding the solution of the problem. Some examples are included to demonstrate the accuracy and validity of the proposed method. ARTICLE HISTORY

In this paper, the adapted (G’/G)-expansion scheme is executed to obtain exact solutions to the fractional Clannish Random Walker’s Parabolic (FCRWP) equation. Some innovative results of the FCRWP equation are gained via the scheme. A diverse variety of exact outcomes are obtained. The proposed procedure could also be used to acquire exact solutions for other nonlinear fractional mathematical models (NLFMMs).

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The present article investigates the existence of solutions of the following nonlinear second order coupled system with nonlinear coupled boundary conditions (CBCs)where f 1 , f 2 : [0, 1] × R → R, µ : R 6 → R 2 and ν : R 2 → R 2 are continuous functions. The results presented in [7,11] are extended in our article. Coupled lower and upper solutions, Arzela-Ascoli theorem and Schauder's fixed point theorem play an important role in establishing the arguments. Some examples are taken to ensure the validity of the theoretical results.