141 research outputs found
The Asymptotic Behaviours of a Class of Neutral Delay Fractional-Order Pantograph Differential Equations
By using fractional calculus and the summation by parts formula in this paper, the asymptotic behaviours of solutions of nonlinear neutral fractional delay pantograph equations with continuous arguments are investigated. The asymptotic estimates of solutions for the equation are obtained, which may imply asymptotic stability of solutions. In the end, a particular case is provided to illustrate the main result and the speed of the convergence of the obtained solutions
A Shifted Jacobi-Gauss Collocation Scheme for Solving Fractional Neutral Functional-Differential Equations
The shifted Jacobi-Gauss collocation (SJGC) scheme is proposed and implemented to solve the fractional neutral functional-differential equations with proportional delays. The technique we have proposed is based upon shifted Jacobi polynomials with the Gauss quadrature integration technique. The main advantage of the shifted Jacobi-Gauss scheme is to reduce solving the generalized fractional neutral functional-differential equations to a system of algebraic equations in the unknown expansion. Reasonable numerical results are achieved by choosing few shifted Jacobi-Gauss collocation nodes. Numerical results demonstrate the accuracy, and versatility of the proposed algorithm
A Modified Generalized Laguerre-Gauss Collocation Method for Fractional Neutral Functional-Differential Equations on the Half-Line
The modified generalized Laguerre-Gauss collocation (MGLC) method is applied to obtain an approximate solution of fractional neutral functional-differential equations with proportional delays on the half-line. The proposed technique is based on modified generalized Laguerre polynomials and Gauss quadrature integration of such polynomials. The main advantage of the present method is to reduce the solution of fractional neutral functional-differential equations into a system of algebraic equations. Reasonable numerical results are achieved by choosing few modified generalized Laguerre-Gauss collocation points. Numerical results demonstrate the accuracy, efficiency, and versatility of the proposed method on the half-line
Numerical Algorithm for Nonlinear Delayed Differential Systems of th Order
The purpose of this paper is to propose a semi-analytical technique
convenient for numerical approximation of solutions of the initial value
problem for -dimensional delayed and neutral differential systems with
constant, proportional and time varying delays. The algorithm is based on
combination of the method of steps and the differential transformation.
Convergence analysis of the presented method is given as well. Applicability of
the presented approach is demonstrated in two examples: A system of pantograph
type differential equations and a system of neutral functional differential
equations with all three types of delays considered. Accuracy of the results is
compared to results obtained by the Laplace decomposition algorithm, the
residual power series method and Matlab package DDENSD. Comparison of computing
time is done too, showing reliability and efficiency of the proposed technique.Comment: arXiv admin note: text overlap with arXiv:1501.00411 Author's reply:
the text overlap may be caused by the fact that this article is concerning
systems of equations, while the other paper was about single equation
Well-posedness and Ulam-Hyers stability results of solutions to pantograph fractional stochastic differential equations in the sense of conformable derivatives
One kind of stochastic delay differential equation in which the delay term is dependent on a proportion of the current time is the pantograph stochastic differential equation. Electric current collection, nonlinear dynamics, quantum mechanics, and electrodynamics are among the phenomena modeled using this equation. A key idea in physics and mathematics is the well-posedness of a differential equation, which guarantees that the solution to the problem exists and is a unique and meaningful solution that relies continuously on the initial condition and the value of the fractional derivative. Ulam-Hyers stability is a property of equations that states that if a function is approximately satisfying the equation, then there exists an exact solution that is close to the function. Inspired by these findings, in this research work, we established the Ulam-Hyers stability and well-posedness of solutions of pantograph fractional stochastic differential equations (PFSDEs) in the framework of conformable derivatives. In addition, we provided examples to analyze the theoretical results
Vieta-Lucas Wavelet based schemes for the numerical solution of the singular models
In this paper, numerical methods based on Vieta-Lucas wavelets are proposed
for solving a class of singular differential equations. The operational matrix
of the derivative for Vieta-Lucas wavelets is derived. It is employed to reduce
the differential equations into the system of algebraic equations by applying
the ideas of the collocation scheme, Tau scheme, and Galerkin scheme
respectively. Furthermore, the convergence analysis and error estimates for
Vieta-Lucas wavelets are performed. In the numerical section, the comparative
analysis is presented among the different versions of the proposed Vieta-Lucas
wavelet methods, and the accuracy of the approaches is evaluated by computing
the errors and comparing them to the existing findings.Comment: 23 pages, 4 figures, 2 Table
Analytical study of ABC-fractional pantograph implicit differential equation with respect to another function
This article aims to establish sufficient conditions for qualitative properties of the solutions for a new class of a pantograph implicit system in the framework of Atangana-Baleanu-Caputo () fractional derivatives with respect to another function under integral boundary conditions. The Schaefer and Banach fixed point theorems (FPTs) are utilized to investigate the existence and uniqueness results for this pantograph implicit system. Moreover, some stability types such as the Ulam-Hyers , generalized , Ulam-Hyers-Rassias and generalized are discussed. Finally, interpretation mathematical examples are given in order to guarantee the validity of the main findings. Moreover, the fractional operator used in this study is more generalized and supports our results to be more extensive and covers several new and existing problems in the literature
Analytical Approximate Solutions for a General Class of Nonlinear Delay Differential Equations
We use the polynomial least squares method (PLSM), which allows us to compute analytical approximate polynomial solutions for a very general class of strongly nonlinear delay differential equations. The method is tested by computing approximate solutions for several applications including the pantograph equations and a nonlinear time-delay model from biology. The accuracy of the method is illustrated by a comparison with approximate solutions previously computed using other methods
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