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).

Existence, uniqueness and stability of impulsive stochastic neutral functional differential equations driven by Rosenblatt process with varying-time delays

2019 ◽  
Vol 27 (4) ◽  
pp. 213-223
Author(s):  
El Hassan Lakhel ◽  
Abdelmonaim Tlidi
Keyword(s):  
Differential Equations ◽  
Fixed Point Theory ◽  
Functional Differential Equations ◽  
Point Theory ◽  
Uniqueness Result ◽  
Similar Process ◽  
Neutral Functional Differential Equations ◽  
Rosenblatt Process ◽  
Self Similar ◽  
Functional Differential

Abstract Hermite processes are self-similar processes with stationary increments; the Hermite process of order 1 is fractional Brownian motion (fBm) and the Hermite process of order 2 is the Rosenblatt process. In this paper we consider a class of impulsive neutral stochastic functional differential equations with variable delays driven by Rosenblatt process with index {H\in(\frac{1}{2},1)} , which is a special case of a self-similar process with long-range dependence. More precisely, we prove an existence and uniqueness result, and we establish some conditions, ensuring the exponential decay to zero in mean square for the mild solution by means of the Banach fixed point theory. Finally, an illustrative example is given to demonstrate the effectiveness of the obtained result.

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.

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.

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