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Fractional operators (derivatives/integrals) are defined via the integration of the functions. When the function is produced by a spanning set of fractional power functions, traditional quadrature rules often need to be revised, failing to…
In this article, we deal with the efficient computation of the Wright function in the cases of interest for the expression of solutions of some fractional differential equations. The proposed algorithm is based on the inversion of the…
The coupled nonlinear space fractional Ginzburg-Landau (CNLSFGL) equations with the fractional Laplacian have been widely used to model the dynamical processes in a fractal media with fractional dispersion. Due to the existence of…
In this paper, we introduce two new non-singular kernel fractional derivatives and present a class of other fractional derivatives derived from the new formulations. We present some important results of uniformly convergent sequences of…
Nonlinear time fractional partial differential equations are widely used in modeling and simulations. In many applications, there are high contrast changes in media properties. For solving these problems, one often uses coarse spatial grid…
We propose a kernel compression method for solving Distributed-Order (DO) Fractional Partial Differential Equations (DOFPDEs) at the cost of solving corresponding local-in-time PDEs. The key concepts are (1) discretization of the integral…
This paper presents an efficient and concise double fast algorithm to solve high dimensional time-space fractional diffusion problems with spectral fractional Laplacian. We first establish semi-discrete scheme of time-space fractional…
A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of…
The PC algorithm is the state-of-the-art algorithm for causal structure discovery on observational data. It can be computationally expensive in the worst case due to the conditional independence tests are performed in an…
This paper proposes an explicit computational method for solving a three-dimensional system of nonlinear elastodynamic sine-Gordon equations subject to appropriate initial and boundary conditions. The time derivative is approximated by…
A spectral decomposition method is used to obtain solutions to a class of nonlinear differential equations. We extend this approach to the analysis of the fractional form of these equations and demonstrate the method by applying it to the…
In this article, we propose a higher order approximation to Caputo fractional (C-F) derivative using graded mesh and standard central difference approximation for space derivatives, in order to obtain the approximate solution of time…
In recent years, the use of variable-order differential operators has emerged as a powerful tool in the analysis of nonlinear fractional differential equations and chaotic systems. In finance, the accurate prediction of market trends and…
This article presents a finite element scheme with Newton's method for solving the time-fractional nonlinear diffusion equation. For time discretization, we use the fractional Crank-Nicolson scheme based on backward Euler convolution…
Fractional differential equations (FDEs) describe subdiffusion behavior of dynamical systems. Its non-local structure requires taking into account the whole evolution history during the time integration, which then possibly causes…
In arXiv:2305.03945 [math.NA], a first-order optimization algorithm has been introduced to solve time-implicit schemes of reaction-diffusion equations. In this research, we conduct theoretical studies on this first-order algorithm equipped…
We propose two efficient numerical approaches for solving variable-order fractional optimal control-affine problems. The variable-order fractional derivative is considered in the Caputo sense, which together with the Riemann-Liouville…
A scheme for approximating the kernel $w$ of the fractional $\alpha$-integral by a linear combination of exponentials is proposed and studied. The scheme is based on the application of a composite Gauss-Jacobi quadrature rule to an integral…
In this work, we develop variational formulations of Petrov-Galerkin type for one-dimensional fractional boundary value problems involving either a Riemann-Liouville or Caputo derivative of order $\alpha\in(3/2, 2)$ in the leading term and…
In this paper, the boundary element method is combined with Chebyshev operational matrix technique to solve two-dimensional multi-order time-fractional partial differential equations; nonlinear and linear in respect to spatial and temporal…