Linear stability of numerical methods for systems of functional differential equations with a proportional delay

2003 ◽  
Vol 13 (5) ◽  
pp. 329
Author(s):  
Chengming HUANG
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Linear Stability ◽  
Functional Differential Equations ◽  
Proportional Delay ◽  
Functional Differential

scholarly journals Implicit difference schemes for quasilinear parabolic functional equations

2012 ◽  
Vol 45 (4) ◽  
Author(s):  
Milena Matusik
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Functional Differential Equations ◽  
Initial Boundary ◽  
Difference Schemes ◽  
New Class ◽  
Implicit Difference Schemes ◽  
Functional Differential ◽  
The Stability ◽  
Quasilinear Parabolic

AbstractWe present a new class of numerical methods for quasilinear parabolic functional differential equations with initial boundary conditions of the Robin type. The numerical methods are difference schemes which are implicit with respect to time variable. We give a complete convergence analysis for the methods and we show that the new methods are considerable better than the explicit schemes. The proof of the stability is based on a comparison technique with nonlinear estimates of the Perron type for given functions with respect to functional variables. Results obtained in the paper can be applied to differential equations with deviated variables and to differential integral problems.

scholarly journals Numerical Solution of Pantograph-Type Delay Differential Equations Using Perturbation-Iteration Algorithms

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
M. Mustafa Bahşi ◽  
Mehmet Çevik
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Functional Differential Equations ◽  
Taylor Series Expansion ◽  
Perturbation Parameter ◽  
Iteration Algorithm ◽  
Proportional Delay ◽  
Pantograph Equation ◽  
Functional Differential ◽  
Delay Differential

The pantograph equation is a special type of functional differential equations with proportional delay. The present study introduces a compound technique incorporating the perturbation method with an iteration algorithm to solve numerically the delay differential equations of pantograph type. We put forward two types of algorithms, depending upon the order of derivatives in the Taylor series expansion. The crucial convenience of this method when compared with other perturbation methods is that this method does not require a small perturbation parameter. Furthermore, a relatively fast convergence of the iterations to the exact solutions and more accurate results can be achieved. Several illustrative examples are given to demonstrate the efficiency and reliability of the technique, even for nonlinear cases.

On application of the i-smooth analysis methodology to elaboration of numerical methods for solving functional differential equations

2021 ◽  
pp. 26-34
Author(s):  
Arkadii V. Kim
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Order Of Convergence ◽  
Functional Differential Equations ◽  
Convergence Conditions ◽  
Neutral Differential Equations ◽  
Infinite Dimensional ◽  
Analysis Methodology ◽  
Different Types ◽  
Functional Differential

The article discusses a number of aspects of the application of i -smooth analysis in the development of numerical methods for solving functional differential equations (FDE). The principle of separating finite- and infinite-dimensional components in the structure of numerical schemes for FDE is demonstrated with concrete examples, as well as the usage of different types of prehistory interpolation, those by Lagrange and Hermite. A general approach to constructing Runge–Kutta-like numerical methods for nonlinear neutral differential equations is presented. Convergence conditions are obtained and the order of convergence of such methods is established.

scholarly journals Numerical Solution of a Class of Functional-Differential Equations Using Jacobi Pseudospectral Method

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
A. H. Bhrawy ◽  
M. A. Alghamdi ◽  
D. Baleanu
Keyword(s):  
Differential Equations ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Functional Differential Equations ◽  
Variational Iteration Method ◽  
Proportional Delay ◽  
Neutral Functional Differential Equations ◽  
Special Cases ◽  
Functional Differential ◽  
Pseudo Spectral

The shifted Jacobi-Gauss-Lobatto pseudospectral (SJGLP) method is applied to neutral functional-differential equations (NFDEs) with proportional delays. The proposed approximation is based on shifted Jacobi collocation approximation with the nodes of Gauss-Lobatto quadrature. The shifted Legendre-Gauss-Lobatto Pseudo-spectral and Chebyshev-Gauss-Lobatto Pseudo-spectral methods can be obtained as special cases of the underlying method. Moreover, the SJGLP method is extended to numerically approximate the nonlinear high-order NFDE with proportional delay. Some examples are displayed for implicit and explicit forms of NFDEs to demonstrate the computation accuracy of the proposed method. We also compare the performance of the method with variational iteration method, one-legθ-method, continuous Runge-Kutta method, and reproducing kernel Hilbert space method.

scholarly journals Nonlinear Pantograph-Type Diffusion PDEs: Exact Solutions and the Principle of Analogy

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 511
Author(s):  
Andrei D. Polyanin ◽  
Vsevolod G. Sorokin
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Functional Differential Equations ◽  
Reaction Diffusion ◽  
Proportional Delay ◽  
Type Reaction ◽  
Self Similar ◽  
Functional Differential ◽  
Similar Solutions ◽  
Varying Delay

We study nonlinear pantograph-type reaction–diffusion PDEs, which, in addition to the unknown u=u(x,t), also contain the same functions with dilated or contracted arguments of the form w=u(px,t), w=u(x,qt), and w=u(px,qt), where p and q are the free scaling parameters (for equations with proportional delay we have 0<p<1, 0<q<1). A brief review of publications on pantograph-type ODEs and PDEs and their applications is given. Exact solutions of various types of such nonlinear partial functional differential equations are described for the first time. We present examples of nonlinear pantograph-type PDEs with proportional delay, which admit traveling-wave and self-similar solutions (note that PDEs with constant delay do not have self-similar solutions). Additive, multiplicative and functional separable solutions, as well as some other exact solutions are also obtained. Special attention is paid to nonlinear pantograph-type PDEs of a rather general form, which contain one or two arbitrary functions. In total, more than forty nonlinear pantograph-type reaction–diffusion PDEs with dilated or contracted arguments, admitting exact solutions, have been considered. Multi-pantograph nonlinear PDEs are also discussed. The principle of analogy is formulated, which makes it possible to efficiently construct exact solutions of nonlinear pantograph-type PDEs. A number of exact solutions of more complex nonlinear functional differential equations with varying delay, which arbitrarily depends on time or spatial coordinate, are also described. The presented equations and their exact solutions can be used to formulate test problems designed to evaluate the accuracy of numerical and approximate analytical methods for solving the corresponding nonlinear initial-boundary value problems for PDEs with varying delay. The principle of analogy allows finding solutions to other nonlinear pantograph-type PDEs (including nonlinear wave-type PDEs and higher-order equations).

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