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Well-posedness of a kind of nonlinear coupled system of fractional differential equations

Science China Mathematics 2016年第6期59卷 1209-1220页

作者： ZHOU XiaoJun XU ChuanJu School of Mathematical Sciences Xiamen University School of Mathematics and Computer Science Guizhou Normal University

We study the boundary value problem of a coupled differential system of fractional order, and prove the existence and uniqueness of solutions to the considered problem. The underlying differential system is featured by a fractional differential operator, which is defined in the Riemann-Liouville sense, and a nonlinear term in which different solution components are coupled. The analysis is based on the reduction of the given system to an equivalent system of integral equations. By means of the nonlinear alternative of Leray-Schauder,the existence of solutions of the factional differential system is obtained. The uniqueness is established by using the Banach contraction principle.

关键词： fractional differential equations nonlinear coupled system existence and uniqueness of solutions

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SUPERCONVERGENCE ANALYSIS OF FINITE ELEMENT METHODS FOR OPTIMAL CONTROL PROBLEMS OF THE STATIONARY B(?)NARD TYPE

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Journal of Computational Mathematics 2008年第5期26卷 660-676页

作者： Yanzhen Chang Danping Yang School of Mathematics and system Science Shandong University Jinan 250100 China Department of Mathematics East China Normal University Shanghai 200062 China

In this paper, we consider the finite element approximation of the distributed optimal control problems of the stationary Benard type under the pointwise control constraint. The states and the co-states are approximated by polynomial functions of lowest-order mixed finite element space or piecewise linear functions and the control is approximated by piecewise constant functions. We give the superconvergence analysis for the control; it is proved that the approximation has a second-order rate of convergence. We further give the superconvergence analysis for the states and the co-states. Then we derive error estimates in L^∞-norm and optimal error estimates in L^2-norm.

关键词： Optimal control problem The stationary Benard problem nonlinear coupled system Finite element approximation Superconvergence.

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An attractive analytical technique for coupled system of fractional partial differential equations in shallow water waves with conformable derivative

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Communications in Theoretical Physics 2020年第8期72卷 1-17页

作者： Mohammed Al-Smadi Omar Abu Arqub Samir Hadid Department of Applied Science Ajloun CollegeAl-Balqa Applied UniversityAjloun 26816Jordan Department of Mathematics Faculty of ScienceAl-Balqa Applied UniversitySalt 19117Jordan Department of Mathematics and Sciences College of Humanities and SciencesAjman UniversityAjmanUnited Arab Emirates

Mathematical simulation of nonlinear physical and abstract systems is a very vital process for predicting the solution behavior of fractional partial differential equations(FPDEs)corresponding to different applications in science and engineering. In this paper, an attractive reliable analytical technique, the conformable residual power series, is implemented for constructing approximate series solutions for a class of nonlinear coupled FPDEs arising in fluid mechanics and fluid flow, which are often designed to demonstrate the behavior of weakly nonlinear and long waves and describe the interaction of shallow water waves. In the proposed technique the n-truncated representation is substituted into the original system and it is assumed the(n-1) conformable derivative of the residuum is zero. This allows us to estimate coefficients of truncation and successively add the subordinate terms in the multiple fractional power series with a rapidly convergent form. The influence, capacity, and feasibility of the presented approach are verified by testing some real-world applications. Finally, highlights and some closing comments are attached.

关键词： nonlinear coupled system fractional partial differential equations residual power series method conformable fractional derivative

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