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We investigate a local modification of a variable-order fractional wave equation, which describes the propagation of diffusive wave in viscoelastic media with evolving physical property. We incorporate an equivalent formulation to prove the…
This paper introduces a fast and numerically stable algorithm for the solution of fourth-order linear boundary value problems on an interval. This type of equation arises in a variety of settings in physics and signal processing. Our method…
We introduce a novel technique for constructing higher-order variational integrators for Hamiltonian systems of ODEs. In particular, we are concerned with generating globally smooth approximations to solutions of a Hamiltonian system. Our…
This article describes an approximation technique based on fractional order Bernstein wavelets for the numerical simulations of fractional oscillation equations under variable order, and the fractional order Bernstein wavelets are derived…
We discuss the derivation and the solutions of integro-differential equations (variable-order time-fractional diffusion equations) following as continuous limits for lattice continuous time random walk schemes with power-law waiting-time…
System of semilinear ordinary differential equation and fractional differential equation of distributed order is investigated and solved in a mild and classical sense. Such a system arises as a distributed derivative model of…
In this paper we present three types of Caputo-Hadamard derivatives of variable fractional order, and study the relations between them. An approximation formula for each fractional operator, using integer-order derivatives only, is…
In this paper, two kinds of high-order compact finite difference schemes for second-order derivative are developed. Then a second-order numerical scheme for Riemann-Liouvile derivative is established based on fractional center difference…
The modulating functions method has been used for the identification of linear and nonlinear systems. In this paper, we generalize this method to the on-line identification of fractional order systems based on the Riemann-Liouville…
In this paper we introduce a new mathematical tool to solve fractional equations representing models of fractional systems : The Ultradistributions. Ultradistributions permit us to unify the notion of integral and derivative in one only…
We introduce two ordinary second-order linear differential equations of the Laguerre- and Jacobi-type. Solutions are written as infinite series of square integrable functions in terms of the Laguerre and Jacobi polynomials, respectively.…
The relations between solutions of the three types of totally linear partial differential equations of first order are presented. The approach is based on factorization of a non-homogeneous first order differential operator to products…
In this article, we introduce a new class of coupled fractional Lane-Emden boundary value problems. We employ a novel approach, the fractional Haar wavelet collocation method with the Newton-Raphson method. We analyze the conditions in two…
In this paper we consider a class of partial integro-differential equations of fractional order, motivated by an equation which arises as a result of modeling surface-volume reactions in optical biosensors. We solve these equations by…
We develop a sparse spectral method for a class of fractional differential equations, posed on $\mathbb{R}$, in one dimension. These equations can include sqrt-Laplacian, Hilbert, derivative and identity terms. The numerical method utilizes…
In this paper, we investigate a spectral Petrov-Galerkin method for fractional initial value problems. Singularities of the solution at the origin inherited from the weakly singular kernel of the fractional derivative are considered, and…
A numerical scheme is presented for approximating fractional order Poisson problems in two and three dimensions. The scheme is based on reformulating the original problem posed over $\Omega$ on the extruded domain…
The finite Laguerre transform is applied to solve Differential Equations Problems of order higher than two and a one-dimensional steady-state Schr\"{o}dinger equation, by using elementary Linear Algebra methods.
In this work we study the solutions to some fractional higher-order equations. Special cases in which time-fractional derivatives take integer values are also examined and the explicit solutions are presented. Such solutions can be…
This paper deals with fractional differential equations, with dependence on a Caputo fractional derivative of real order. The goal is to show, based on concrete examples and experimental data from several experiments, that fractional…