Related papers: Fractional Order Runge-Kutta Methods
Implicit Runge--Kutta (IRK) methods are highly effective for solving stiff ordinary differential equations (ODEs) but can be computationally expensive for large-scale problems due to the need of solving coupled algebraic equations at each…
There has recently been considerable interest in using a nonstandard piecewise approximation to formulate fractional order differential equations as difference equations that describe the same dynamical behaviour and are more amenable to a…
Using both fractional derivatives, defined in the Riemann-Liouville and Caputo senses, and classical derivatives of the integer order we examine different numerical approaches to ordinary differential equations. Generally we formulate some…
In this work we present a new class of Runge-Kutta (RK) methods for solving systems of hyperbolic equations with a particular structure, generalization of a wave-equation. The new methods are {\it partially implicit} in the sense that a…
Finite element methods provide accurate and efficient methods for the numerical solution of partial differential equations by means of restricting variational problems to finite-dimensional approximating spaces. However, they do not…
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…
The main objective of this series of papers is to explore the entire landscape of numerical methods for fast nonlinear Fourier transformation (NFT) within the class of integrators known as the exponential integrators. In this paper, we…
The following document presents some novel numerical methods valid for one and several variables, which using the fractional derivative, allow to find solutions for some non-linear systems in the complex space using real initial conditions.…
In this work, we construct and derive a new class of exponentially fitted two-derivative diagonally implicit Runge--Kutta (EFTDDIRK) methods for the numerical solution of differential equations with oscillatory solutions. First, a general…
In this paper we present a general procedure for designing higher strong order methods for It\^o stochastic differential equations on matrix Lie groups and illustrate this strategy with two novel schemes that have a strong convergence order…
In this paper we discuss some issues that arise in the process of writing a fractional differential equation (FDE) by replacing an integer order derivative by a fractional order derivative in a given differential equation. To address these…
Exponential Runge--Kutta methods have shown to be competitive for the time integration of stiff semilinear parabolic PDEs. The current construction of stiffly accurate exponential Runge--Kutta methods, however, relies on a convergence…
Runge-Kutta methods are a popular class of numerical methods for solving ordinary differential equations. Every Runge-Kutta method is characterized by two basic parameters: its order, which measures the accuracy of the solution it produces,…
We propose a family of integrators, Flow-Composed Implicit Runge-Kutta (FCIRK) methods, for perturbations of nonlinear ordinary differential equations, consisting of the composition of flows of the unperturbed part alternated with one step…
In this article, a numerical scheme is introduced for solving the fractional partial differential equation (FPDE) arising from electromagnetic waves in dielectric media (EMWDM) by using an efficient class of finite difference methods. The…
Conventional finite-difference schemes for solving partial differential equations are based on approximating derivatives by finite-differences. In this work, an alternative theory is proposed which view finite-difference schemes as…
Based on the continuous time random walk, we derive the Fokker-Planck equations with Caputo-Fabrizio fractional derivative, which can effectively model a variety of physical phenomena, especially, the material heterogeneities and structures…
Runge-Kutta methods are the classic family of solvers for ordinary differential equations (ODEs), and the basis for the state of the art. Like most numerical methods, they return point estimates. We construct a family of probabilistic…
We propose a novel Caputo fractional derivative-based optimization algorithm. Upon defining the Caputo fractional gradient with respect to the Cartesian coordinate, we present a generic Caputo fractional gradient descent (CFGD) method. We…
In the paper, we utilize the fractional differential transformation (FDT) to solving singular initial value problem of fractional Emden-Fowler type differential equations. The solutions of our model equations are calculated in the form of…