Related papers: Jacobi-Predictor-Corrector Approach for the Fracti…
In this work, we present a collocation method based on the Legendre wavelet combined with the Gauss--Jacobi quadrature formula for solving a class of fractional delay-type integro-differential equations. The problem is considered with…
We look for differential equations satisfied by the generalized Jacobi polynomials which are orthogonal on the interval [-1,1] with respect to the classical weight function for the Jacobi polynomials together with point masses at both…
This paper presents a numerical method to solve a time-fractional Burgers equation, achieving order of convergence $(2-\alpha)$ in time, here $\alpha$ represents the order of the time derivative. The fractional derivative is modeled by…
For terminal value problems of fractional differential equations of order $\alpha \in (0,1)$ that use Caputo derivatives, shooting methods are a well developed and investigated approach. Based on recently established analytic properties of…
The application of the approximation-operational approach to solving linear differential equations of fractional order with variable coefficients is considered. It is shown that the method can also be applied to solving differential…
We propose a spectral collocation method, based on the generalized Jacobi wavelets along with the Gauss-Jacobi quadrature formula, for solving a class of third-kind Volterra integral equations. To do this, the interval of integration is…
The differentiation by integration method with Jacobi polynomials was originally introduced by Mboup, Join and Fliess. This paper generalizes this method from the integer order to the fractional order for estimating the fractional order…
The aim of this paper is to study differential properties of orthogonal polynomials with respect to a discrete Jacobi-Sobolev bilinear form with mass point at $-1$ and/or $+1$. In particular, we construct the orthogonal polynomials using…
We introduce a new class of fractional backward orthogonal functions designed for the spectral approximation of weakly singular adjoint Volterra integral equations. These basis functions generate an approximation space that naturally…
This paper presents a new predictor-corrector numerical scheme suitable for fractional differential equations. An improved explicit Atangana-Seda formula is obtained by considering the neglected terms and used as the predictor stage of the…
In this paper, the authors propose the utilization of Fibonacci Neural Networks (FNN) for solving arbitrary order differential equations. The FNN architecture comprises input, middle, and output layers, with various degrees of Fibonacci…
This paper presents a Jacobi-type iteration for computing a given specified eigenpair of a symmetric matrix. For a certain class of diagonally dominant matrices, the procedure is shown to converge at a linear rate depending on how the…
The fractional Laplacian $(-\Delta)^{\alpha/2}$ is a non-local operator which depends on the parameter $\alpha$ and recovers the usual Laplacian as $\alpha \to 2$. A numerical method for the fractional Laplacian is proposed, based on the…
The work in this paper is four-fold. Firstly, we introduce an alternative approach to solve fractional ordinary differential equations as an expected value of a random time process. Using the latter, we present an interesting numerical…
This paper investigates, a new class of fractional order Runge-Kutta (FORK) methods for numerical approximation to the solution of fractional differential equations (FDEs). By using the Caputo generalizedTaylor formula and the total…
We describe a general operational method that can be used in the analysis of fractional initial and boundary value problems with additional analytic conditions. As an example, we derive analytic solutions of some fractional generalisation…
A novel efficient and high accuracy numerical method for the time-fractional differential equations (TFDEs) is proposed in this work. We show the equivalence between TFDEs and the integer-order extended parametric differential equations…
An effective numerical method is presented for optimizing model parameters that can be applied to any type of system of non-linear equations and any number of data-points, which does not require explicit formulation of the objective…
We use the well-known observation that the solutions of Jacobi's differential equation can be represented via non-oscillatory phase and amplitude functions to develop a fast algorithm for computing multi-dimensional Jacobi polynomial…
Solving complex optimization problems in engineering and the physical sciences requires repetitive computation of multi-dimensional function derivatives. Commonly, this requires computationally-demanding numerical differentiation such as…