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Highlights • System of fractional differential equations involving non-singular Mittag-Leffler kernel is solved by collocation methods on discontinuous piecewise polynomial space. • Existence and regularity of the solutions are investigated. • The convergence and super-convergence properties of the introduced methods are derived on graded meshes. • Non-singular fractional diffusion equation is solved by transforming to the system of fractional differential equations. Abstract A system of fractional differential equations involving non-singular Mittag-Leffler kernel is considered. This system is transformed to a type of weakly singular integral equations in which the weak singular kernel is involved with both the unknown and known functions. The regularity and existence of its solution is studied. The collocation methods on discontinuous piecewise polynomial space are considered. The convergence and superconvergence properties of the introduced methods are derived on graded meshes. Numerical results provided to show that our theoretical convergence bounds are often sharp and the introduced methods are efficient. Some comparisons and applications are discussed. [ABSTRACT FROM AUTHOR]

In this paper we consider multi variable orders differential equations (MVODEs) with non-local and no-singular kernel. The derivative is described in Atangana and Baleanu sense of variable order. We use the fifth-kind Chebyshev polynomials as basic functions to obtain operational matrices. We transfer the original equations to a system of algebraic equations using operational matrices and collocation method. The convergence analysis of the presented method is discussed. Few examples are presented to show the efficiency of the presented method. [ABSTRACT FROM AUTHOR]

This paper focuses on providing a new scheme to find the fuzzy approximate solution of fractional differential equations (FDEs) under uncertainty. The Caputo-type derivative base on the generalized Hukuhara differentiability is approximated by a linearization formula to reduce the corresponding uncertain FDE to an ODE under fuzzy concept. This new approach may positively affect on the computational cost and easily apply for the other types of uncertain fractional-order differential equation. The performed numerical simulations verify the proficiency of the presented scheme. [ABSTRACT FROM AUTHOR]

This article presents a variable order nonlinear mathematical model and its optimal control for a Tumor under immune suppression. The formulation generalizes the one proposed by Kim et. al. consisting of eleven integer order differential equations. The new approach adopts a variable-order fractional model with the derivatives defined in the Caputo sense. Two control variables, one for immunotherapy and one for Chemotherapy, are proposed to eliminate or reduce the Tumor cells. A simple numerical technique called the nonstandard generalized Euler method is developed to solve the proposed optimal control problem. Moreover, the stability analysis and the truncation error are studied. Numerical simulations and comparative studies are implemented. Our findings disclose that the proposed scheme used here has two main advantages: it is faster than the generalized Euler scheme and it can reduce the number of Tumor cells in a proper process better than the second scheme. [ABSTRACT FROM AUTHOR]