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Integration and differentiation of non-integer orders for N-dimensional physical lattices with long-range particle interactions are suggested. The proposed lattice fractional derivatives and integrals are represented by kernels of lattice long-range interactions, such that their Fourier series transformations have a power-law form with respect to components of wave vector. Continuous limits for these lattice fractional derivatives and integrals give the continuum derivatives and integrals of non-integer orders with respect to coordinates. Lattice analogs of fractional differential equations that include suggested lattice differential and integral operators can serve as an important element of microscopic approach to nonlocal continuum models in mechanics and physics. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new refinements and similar extensions of Hermite–Hadamard inequality for fractional integrals and present several applications in the Beta function. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a systematic study of some extended special functions like B b ; ρ , λ α , β x , y , 2 F 1 α , β ; ρ , λ z ; b and p F q α , β ; ρ , λ z ; b . We obtain various properties of these extended functions and establish their some connections with the Laguerre polynomial and Fox’s H -function. Furthermore, we also establish the extended Riemann–Liouville type fractional integral operator and extended Kober type fractional integral operators. [ABSTRACT FROM AUTHOR]

Abstract: It is shown that, if the problem is defined in the setting of fractional calculus via fractional difference on non-differentiable functions, then the solution of the functional equation f(x + y)= f(x)f(y) is exactly defined as the solution of a linear fractional differential equation. The dual or counterpart problem, that is the fractional solution of the equation g(xy)= g(x)+ g(y), is also considered, and it is shown that the corresponding solution is the logarithm of fractional order defined as the inverse of a generalized Mittag–Leffler function which is nowhere differentiable. This framework suggests a definition of fractional Laplace’s transform expressed in terms of generalized Mittag–Leffler function, and its main properties are outlined: mainly inverse function and convolution. One takes this opportunity to display a (new) fractional Taylor’s series for functions f(x,y) of two variables x and y. Many open problems are stated, which are directly related to the non-differentiability of the functions so involved, therefore the title “on the fractional solution…”. [Copyright &y& Elsevier]

Abstract: An algorithm has been developed to convert the multi-order fractional differential equation:where m < α ⩽ m +1, 0< β 1 < β 2 <⋯< β n < α and denotes Caputo fractional derivative of order α into a system of fractional differential equations. Further Adomian decomposition method has been employed to solve the system of fractional differential equations. Some illustrative examples are presented. [Copyright &y& Elsevier]

Abstract: In this paper, a new generalized Taylor’s formula of the kindwhere a j ∈ R , x > a, 0< α ⩽1, is established. Such an expression is precisely the classical Taylor’s formula in case of α =1. In addition, detailed expressions for and a j (α), that involving Caputo fractional derivatives, are well constructed. Some applications including approximation of functions and solutions of fractional differential equations are also given. [Copyright &y& Elsevier]

Abstract: In this paper, we present a powerful modification of the rectangular decomposition method that is used to solve fractional diffusion-wave equations. The new modification extends the application of the rectangular decomposition method to solve a wide class of space–time fractional partial differential equations. The proposed modification gives an analytical solution in the form of a convergent series with easily computable components. Some numerical examples are given to illustrate the efficiency and reliability of the method. The new modification introduces a promising tool for many partial differential equations. [Copyright &y& Elsevier]

Abstract: In this paper, a new modification of the Adomian decomposition method is effectively implemented for solving fractional diffusion-wave equations that will facilitate the calculations, where the fractional derivative is based on Caputo definition. The proposed algorithm gives an analytical solution in the form of a convergent series with easily computable components. Numerical examples are given to show the application of the present algorithm. The new modification introduces a promising tool for many partial differential equations. [Copyright &y& Elsevier]

Abstract: In this paper we propose two algorithms for numerical fractional integration and Caputo fractional differentiation. We present a modification of trapezoidal rule that is used to approximate finite integrals, the new modification extends the application of the rule to approximate integrals of arbitrary order α >0. We then, using the new modification derive an algorithm to approximate fractional derivatives of arbitrary order α >0, where the fractional derivative based on Caputo definition, for a given function by a weighted sum of function and its ordinary derivatives values at specified points. The study is conducted through illustrative examples and error analysis. [Copyright &y& Elsevier]

Abstract: The existence of solutions to an integral equation of fractional order with hysteresis has been established. A fixed point theorem due to Schauder is the main tool in carrying out our proof. [Copyright &y& Elsevier]

Abstract: The aim of the present analysis is to apply Adomian decomposition method for the solution of a nonlinear fractional differential equation. Finally, the solution obtained by the decomposition method has been numerically evaluated and presented in the form of tables and then compared with those obtained by truncated series method. A good agreement of the results is observed. [Copyright &y& Elsevier]