Fractional Multistep Methods Research Articles

On multi-step methods for singular fractional q-integro-differential equations

Abstract The objective of this paper is to investigate, by applying the standard Caputo fractional q-derivative of order α \alpha , the existence of solutions for the singular fractional q-integro-differential equation D q α [ k ] ( t ) = Ω ( t , k 1 , k 2 , k 3 , k 4 ) {{\mathcal{D}}}_{q}^{\alpha }\left[k]\left(t)=\Omega \left(t,{k}_{1},{k}_{2},{k}_{3},{k}_{4}) , under some boundary conditions where Ω \Omega is singular at some point 0 ≤ t ≤ 1 0\le t\le 1 , on a time scale T t 0 = { t : t = t 0 q n } ∪ { 0 } {{\mathbb{T}}}_{{t}_{0}}=\left\{t:t={t}_{0}{q}^{n}\right\}\cup \left\{0\right\} , for n ∈ N n\in {\mathbb{N}} where t 0 ∈ R {t}_{0}\in {\mathbb{R}} and q ∈ ( 0 , 1 ) q\in \left(0,1) . We consider the compact map and avail the Lebesgue dominated theorem for finding solutions of the addressed problem. Besides, we prove the main results in context of completely continuous functions. Our attention is concentrated on fractional multi-step methods of both implicit and explicit type, for which sufficient existence conditions are investigated. Finally, we present some examples involving graphs, tables and algorithms to illustrate the validity of our theoretical findings.

To investigate a class of the singular fractional integro-differential quantum equations with multi-step methods

The objective of this paper is to investigate, by applying the standard Caputo fractional $q$--derivative of order $\alpha$, the existence of solutions for the singular fractional $q$--integro-differential equation $\mathcal{D}_q^\alpha [k](t) = \Omega (t , k(t), k'(t), \mathcal{D}_q^\beta [k](t), \int_0^t f(r) k(r) \, {\mathrm d}r )$, under some boundary conditions where $\Omega(t, k_1, k_2, k_3, k_4)$ is singular at some point $0 \leq t\leq 1$, on a time scale $\mathbb{T}_{ t_0} = \{ t : t =t_0q^n\}\cup \{0\}$, for $n\in \mathbb{N}$ where $t_0 \in \mathbb{R}$ and $q \in (0,1)$. We consider the compact map and avail the Lebesgue dominated theorem for finding solutions of the addressed problem. Besides, we prove the main results in context of completely continuous functions. Our attention is concentrated on fractional multi-step methods of both implicit and explicit type, for which sufficient existence conditions are investigated. Lastly, we present some examples involving graphs, tables and algorithms to illustrate the validity of our theoretical findings.

An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data

Abstract In this paper, we shall review an approach by which we can seek higher order time discretisation schemes for solving time fractional partial differential equations with nonsmooth data. The low regularity of the solutions of time fractional partial differential equations implies standard time discretisation schemes only yield first order accuracy. To obtain higher order time discretisation schemes when the solutions of time fractional partial differential equations have low regularities, one may correct the starting steps of the standard time discretisation schemes to capture the singularities of the solutions. We will consider these corrections of some higher order time discretisation schemes obtained by using Lubich’s fractional multistep methods, L1 scheme and its modification, discontinuous Galerkin methods, etc. Numerical examples are given to show that the theoretical results are consistent with the numerical results.

Discontinuous Galerkin [formula omitted]-adaptive methods for multiscale chemical reactors: Quiescent reactors

We present a class of chemical reactor systems, modeled numerically using a fractional multistep method between the reacting and diffusing modes of the system, subsequently allowing one to utilize algebraic techniques for the resulting reactive subsystems. A mixed form discontinuous Galerkin method is presented with implicit and explicit (IMEX) timestepping strategies coupled to dioristic entropy schemes for hp-adaptivity of the solution, where the h and p are adapted based on an L1-stability result. Finally we provide some numerical studies on the convergence behavior, adaptation, and asymptotics of the system applied to a pair of equilibrium problems, as well as to general three-dimensional nonlinear Lotka–Volterra chemical systems.

Numerical solution of fractional diffusion-wave equation based on fractional multistep method

This paper is devoted to application of fractional multistep method in the numerical solution of fractional diffusion-wave equation. By transforming the diffusion-wave equation into an equivalent integro-differential equation and applying Lubich’s fractional multistep method of second order we obtain a scheme of order O(τα+h2) for 1⩽α⩽1.71832 where α is the order of temporal derivative and τ and h denote temporal and spatial stepsizes. The solvability, convergence and stability properties of the algorithm are investigated and numerical experiment is carried out to verify the feasibility of the scheme.

On Multistep Methods for Differential Equations of Fractional Order

This paper concerns with numerical methods for the treatment of differential equations of fractional order. Our attention is concentrated on fractional multistep methods of both implicit and explicit type, for which order conditions and stability properties are investigated.

Fractional Multistep Methods for Weakly Singular Volterra Integral Equations of the First Kind with Perturbed Data

ABSTRACT Fractional multistep methods were introduced by C. Lubich for the quadrature of Abel integral operators and the solution of weakly singular Volterra integral equations of the first kind with exactly given right-hand sides. In the current paper, we consider the regularizing properties of these methods to solve the mentioned integral equations of the first kind for perturbed right-hand sides. Finally, numerical results are presented.