We introduce the fractional integral corresponding to the new concept of fractional derivative recently introduced by Caputo and Fabrizio and we study some related fractional differential equations.

/pdf/properties-of-a-new-fractional-derivative-without-singular-4xwsnzcmp4.pdf

Properties of a New Fractional Derivative without Singular Kernel

This chapter mainly introduces theories of one-dimensional linear viscoelasticity based on Boltzmann superposition and the concept of linear response. The second section deals with measurement of linear viscoelasticity (experimental aspects), and the third section deals with phenomenological models such as spring–dashpot and parsimonious models. The last section is devoted to molecular theories of polymer viscoelasticity in linear regime.

Theory of Linear Viscoelasticity

Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel

https://www.kau.edu.sa/Files/130/Researches/61664_32628.pdf

Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations

Fractional differentials provide more accurate models of systems under consideration. In this paper, approximation techniques based on the shifted Legendre-tau idea are presented to solve a class of initial-boundary value problems for the fractional diffusion equations with variable coefficients on a finite domain. The fractional derivatives are described in the Caputo sense. The technique is derived by expanding the required approximate solution as the elements of shifted Legendre polynomials. Using the operational matrix of the fractional derivative the problem can be reduced to a set of linear algebraic equations. From the computational point of view, the solution obtained by this method is in excellent agreement with those obtained by previous work in the literature and also it is efficient to use.

A tau approach for solution of the space fractional diffusion equation

Applications of fractional calculus

By defining an order and lattice in the finite dimensional σ-commutative G-graded algebra A (the space of supernumbers) we see that it is Riesz space. Also by considering some order continuous norms on A we prove that it is Banach lattice. Some of definitions and results of Riesz space theory are investigated. We extend the Riesz space structure on Kobayashi and Nagamachi’s superspace[4]. We will investigate some other properties of Riesz space theory on it.

/pdf/generalized-riesz-superspace-43tmink231.pdf

Generalized Riesz Superspace

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Applications of fractional calculus

Generalized Riesz Superspace