Partial Differential-difference Equations Research Articles

Solutions for Several Quadratic Trinomial Difference Equations and Partial Differential Difference Equations in C2

This article is to investigate the existence of entire solutions of several quadratic trinomial difference equations f(z+c)2+2αf(z)f(z+c)+f(z)2=eg(z), and the partial differential difference equations f(z+c)2+2αf(z+c)∂f(z)∂z1+∂f(z)∂z12=eg(z),f(z+c)2+2αf(z+c)∂f(z)∂z1+∂f(z)∂z2+∂f(z)∂z1+∂f(z)∂z22=eg(z). We establish some theorems about the forms of the finite order transcendental entire solutions of these functional equations. We also list a series of examples to explain the existence of the finite order transcendental entire solutions of such equations. Meantime, some examples show that there exists a very significant difference with the previous literature on the growth order of the finite order transcendental entire solutions. Our results show that some functional equations can admit the transcendental entire solutions with any positive integer order. These results make a few improvements of the previous theorems given by Xu and Cao, Liu and Yang.

A high order robust numerical scheme for the generalized Stein's model of neuronal variability

This paper continues the authors's study of robust numerical scheme for generalized Stein's model of neuronal variability, which is a singularly perturbed parabolic partial differential–difference equation with general values of shift arguments. The work on this class of problems is initiated in the papers [K. Bansal and K.K. Sharma, Parameter-robust numerical scheme for time dependent singularly perturbed reaction- diffusion problem with large delay, Numer. Funct. Anal. Optim. 39(2) (2018), pp. 127–154] for unit shift argument and in [K. Bansal and K.K. Sharma, Parameter uniform numerical scheme for time dependent singularly perturbed convection- diffusion-reaction problems with general shift arguments, Numer. Algorithms 75(1) (2017), pp. 113–145] for general shift arguments. In the present paper, this work is further extended to the development of numerical scheme based on Mickens techniques, interpolation and θ scheme. The advantage of this work over the previous research is that it deals with the problem having general values of the shift arguments with higher order of convergence and without any constraints on the number of intervals.

Entire Solutions for Complex Systems of the Second-Order Partial Differential Difference Equations of Fermat Type

This article is mainly concerned with the existence and the forms of entire solutions for several systems of the second-order partial differential difference equations of Fermat type α ∂ 2 f 1 z 1 , z 2 / ∂ z 1 2 + β ∂ 2 f 1 z 1 , z 2 / ∂ z 2 2 n 1 + f 2 z 1 + c 1 , z 2 + c 2 m 1 = 1 α ∂ 2 f 2 z 1 , z 2 / ∂ z 1 2 + β ∂ 2 f 2 z 1 , z 2 / ∂ z 2 2 n 2 + f 1 z 1 + c 1 , z 2 + c 2 m 2 = 1 and ∂ 2 f 1 z 1 , z 2 / ∂ z 1 2 2 + f 2 z 1 + c 1 , z 2 + c 2 2 = 1 ∂ 2 f 2 z 1 , z 2 / ∂ z 1 2 2 + f 1 z 1 + c 1 , z 2 + c 2 2 = 1 . Our results about the existence and the forms of solutions for these systems generalize the previous theorems given by Xu and Cao, Gao, Liu, and Yang. In addition, we give some examples to explain the existence of solutions of this system in each case.

Entire solutions for several second-order partial differential-difference equations of Fermat type with two complex variables

This paper is concerned with description of the existence and the forms of entire solutions of several second-order partial differential-difference equations with more general forms of Fermat type. By utilizing the Nevanlinna theory of meromorphic functions in several complex variables we obtain some results on the forms of entire solutions for these equations, which are some extensions and generalizations of the previous theorems given by Xu and Cao (Mediterr. J. Math. 15:1–14, 2018; Mediterr. J. Math. 17:1–4, 2020) and Liu et al. (J. Math. Anal. Appl. 359:384–393, 2009; Electron. J. Differ. Equ. 2013:59–110, 2013; Arch. Math. 99:147–155, 2012). Moreover, by some examples we show the existence of transcendental entire solutions with finite order of such equations.

Solutions for systems of complex Fermat type partial differential-difference equations with two complex variables

<abstract><p>By making use of the Nevanlinna theory and difference Nevanlinna theory of several complex variables, we investigate some properties of the transcendental entire solutions for several systems of partial differential difference equations of Fermat type, and obtain some results about the existence and the forms of transcendental entire solutions of the above systems, which improve and generalize the previous results given by Cao, Gao, Liu <sup>[<xref ref-type="bibr" rid="b5">5</xref>,<xref ref-type="bibr" rid="b24">24</xref>,<xref ref-type="bibr" rid="b39">39</xref>]</sup>. Some examples are given show that there exist some significant differences in the forms of transcendental entire solutions with finite order of the systems of equations with between several complex variables and a single complex variable.</p></abstract>

On the entire solutions for several partial differential difference equations (systems) of Fermat type in $\mathbb{C}^2$

By utilizing the Nevanlinna theory of meromorphic functions in several complex variables, we will establish some theorems about the existence and the forms of entire solutions for several partial differential difference equations (systems) of Fermat type with two complex variables such as $ f(z)^2+\left[f(z+c)+\frac{\partial f}{\partial z_1}+\frac{\partial f}{\partial z_2}\right]^2 = 1 $ and $ \left\{ \begin{aligned} &amp;f_1(z)^2+\left[f_2(z+c)+\frac{\partial f_1}{\partial z_1}+\frac{\partial f_1}{\partial z_2}\right]^2 = 1, \\ &amp;f_2(z)^2+\left[f_1(z+c)+\frac{\partial f_2}{\partial z_1}+\frac{\partial f_2}{\partial z_2}\right]^2 = 1, \end{aligned}\right. $ which are some extensions and generalizations of the previous theorems given by Xu and Cao ^{[<span class="xref"><a href="#b29" ref-type="bibr">29</a></span>,<span class="xref"><a href="#b30" ref-type="bibr">30</a></span>]}, Xu, Liu and Li ^{[<span class="xref"><a href="#b28" ref-type="bibr">28</a></span>]}, and Liu, Yang ^{[<span class="xref"><a href="#b18" ref-type="bibr">18</a></span>,<span class="xref"><a href="#b19" ref-type="bibr">19</a></span>,<span class="xref"><a href="#b20" ref-type="bibr">20</a></span>]}. Moreover, we give some examples to explain that our results are precise to some extent.

Notes on the Existence of Entire Solutions for Several Partial Differential-Difference Equations

This paper is mainly devoted to discuss the existence of the finite order transcendental entire solutions for the partial differential-difference equations $$\begin{aligned}&\left( \frac{\partial f(z_1,z_2)}{\partial z_1}+\frac{\partial f(z_1,z_2)}{\partial z_2}\right) ^{n}+f(z_1+c_1,z_2+c_2)^m=1, \\&\quad \left( \frac{\partial f(z_1,z_2)}{\partial z_1}\right) ^{n}+f(z_1+c_1,z_2+c_2)^m=1. \end{aligned}$$ It is confirmed that the existence of the finite order transcendental entire solutions for the above equations in the case $$n=2$$ and $$m=1$$ . This is a very powerful supplement to the previous theorems given by Xu and Cao [L. Xu and T. B. Cao, Solutions of complex Fermat-type partial difference and differential-difference equations, Mediterr. J. Math. 15 (2018), pages, 1-14.].

Sampled-Data Fuzzy Control for Nonlinear Delayed Distributed Parameter Systems

Under spatially local averaged measurements (SLAMs), this article introduces a sampled-data fuzzy control (SDFC) for nonlinear delayed distributed parameter systems (DDPSs). First, we use a Takagi–Sugeno (T–S) fuzzy parabolic partial differential-difference equation (PDDE) to accurately describe the nonlinear DDPS. Then, on basis of the T–S fuzzy PDDE model, an SDFC design under SLAMs via space-dependent linear matrix inequalities (SDLMIs) is subsequently developed to ensure the exponential stability of the closed-loop nonlinear DDPSs by using inequality techniques and Lyapunov functional, where slow-varying and fast-varying delays are respected. Furthermore, to solve SDLMIs, the SDFC design problem for nonlinear DDPS under SLAMs is formulated as a linear matrix inequality feasibility problem. Finally, numerical simulations of two examples are presented to support the given SDFC strategy.

On Backlund transformation of Riccati equation method and its application to nonlinear partial differential equations and differential-difference equations

On Backlund transformation of Riccati equation method and its application to nonlinear partial differential equations and differential-difference equations

Necessary and Sufficient Conditions of Stability in the Quadratic Mean of Linear Stochastic Partial Differential-Difference Equations Subject to External Perturbations of the Type of Random Variables

We obtain the necessary and sufficient conditions for the stability in the quadratic mean of strong solutions to stochastic partial differential-difference equations with pairwise independent external random perturbations of the type of random variables.

Entire solutions for several systems of nonlinear difference and partial differential-difference equations of Fermat-type

The purpose of this paper is mainly to study the existence and the forms of entire solutions for the following systems of nonlinear difference and partial differential-difference equations of Fermat-type{f1(z1,z2)2+f2(z1+c1,z2+c2)2=1,f2(z1,z2)2+f1(z1+c1,z2+c2)2=1,{(∂f1(z1,z2)∂z1)n1+f2(z1+c1,z2+c2)m1=1,(∂f2(z1,z2)∂z1)n2+f1(z1+c1,z2+c2)m2=1, and{(∂f1(z1,z2)∂z1)2+[f2(z1+c1,z2+c2)−f1(z1,z2)]2=1,(∂f2(z1,z2)∂z1)2+[f1(z1+c1,z2+c2)−f2(z1,z2)]2=1. Our results about the existence and the forms of solutions for these systems are some improvements and supplements of the previous theorems given by Xu and Cao [31], Gao [4], Liu, Yang [17,18,20]. Moreover, we give some examples to utilize the correctness of our conclusions about the existence and forms of solutions for such systems.

On the traveling waves in nonlinear electrical transmission lines with intrinsic fractional-order using discrete tanh method

Abstract In this paper we investigate the solutions of the fractional partial differential-difference equations governing the dynamics of the voltage wave flowing inside two fractional (low pass and pass band) nonlinear electrical transmission lines (NETL). Through the discrete Tanh method, we derive the traveling wave solutions i.e (kink, dark, singular kink solitons) of two fractional partial differential-difference equations by using the fractional complex transform. The impact of the fractional order on the dynamical behavior of the analytical solutions in both models is highlighted. From our findings the perfect matching between the analytical and the numerical solutions is justified by the stability seen as the resulting wave propagates safely inside the fractional low pass NETL including real inductor. We get for the fractional pass band NETL a rise of nonlinear periodic waves and a train of multi periodic solitons during the numerical simulations. We show that, according to the modified Riemann–Liouville derivative properties, the findings can describe physical systems such as electrical systems and nonlinear electrical transmission lines with transient effect.

Fuzzy Control for Nonlinear Time-Delay Distributed Parameter Systems Under Spatially Point Measurements

This paper introduces a fuzzy control (FC) under spatially point measurements for nonlinear time-delay distributed parameter systems (DPSs) described by parabolic partial differential-difference equations (PDdEs). First, a Takagi–Sugeno (T–S) fuzzy PDdE model is employed to represent the nonlinear time-delay DPSs. Second, with the aid of the T–S fuzzy PDdE model, an FC design under spatially point measurements is developed in the formulation of linear matrix inequalities by constructing an appropriate Lyapunov functional, which can stabilize exponentially the time-delay DPSs. This stabilization condition can be applied to either slowing-varying time delay or fast-varying one. Finally, simulation results of a numerical example are provided to illustrate the effectiveness of the proposed method.

Fuzzy Control Under Spatially Local Averaged Measurements for Nonlinear Distributed Parameter Systems With Time-Varying Delay

This paper introduces a fuzzy control (FC) under spatially local averaged measurements (SLAMs) for nonlinear-delayed distributed parameter systems (DDPSs) represented by parabolic partial differential-difference equations (PDdEs), where the fast-varying time delay and slow-varying one are considered. A Takagi-Sugeno (T-S) fuzzy PDdE model is first derived to exactly describe the nonlinear DDPSs. Then, by virtue of the T-S fuzzy PDdE model and a Lyapunov-Krasovskii functional, an FC design under SLAMs, where the membership functions of the proposed FC law are determined by the measurement output and independent of the fuzzy PDdE plant model, is developed on basis of spatial linear matrix inequalities (SLMIs) to guarantee the exponential stability for the resulting closed-loop DDPSs. Lastly, a numerical example is offered to support the presented approach.

Про існування розв’язку задачі Коші одного класу стохастичних диференціально-різницевих рівнянь в частинних похідних із зовнішними випадковими збуреннями

Про існування розв’язку задачі Коші одного класу стохастичних диференціально-різницевих рівнянь в частинних похідних із зовнішними випадковими збуреннями

Finite Difference Solution to the Space-Time Fractional Partial Differential-Difference Toda Lattice Equation

This paper deals with the numerical solution of space-time fractional partial differential-difference Toda lattice equation $\frac{\partial^{2\alpha} u_n}{\partial x^{\alpha}\partial t^{\alpha}}=(1+\frac{\partial^\alpha u_n}{\partial t^{\alpha}})(u_{n-1}-2u_n+u_{n+1})$, $\alpha \in (0,1)$. The finite differences method (FD-method) is used for numerical solution of this problem. According to the method, we approximate the unknown values $u_n$ of the desired function by finite differences approximation. As an application we demonstrate the capabilities of this method for identification of various values of order of fractional derivative $\alpha$. Numerical results show that the proposed version of FD-method allows to obtain all data from the initial and boundary conditions with enough high accuracy.

Delay-Dependent Exponential Stabilization for Linear Distributed Parameter Systems With Time-Varying Delay

This paper extends the framework of Lyapunov–Krasovskii functional to address the problem of exponential stabilization for a class of linearly distributed parameter systems (DPSs) with continuous differentiable time-varying delay and a spatiotemporal control input, where the system model is described by parabolic partial differential-difference equations (PDdEs) subject to homogeneous Neumann or Dirichlet boundary conditions. By constructing an appropriate Lyapunov–Krasovskii functional candidate and using some inequality techniques (e.g., spatial integral form of Jensen's inequalities and vector-valued Wirtinger's inequalities), some delay-dependent exponential stabilization conditions are derived, and presented in terms of standard linear matrix inequalities (LMIs). These stabilization conditions are applicable to both slow-varying and fast-varying time delay cases. The detailed and rigorous proof of the closed-loop exponential stability is also provided in this paper. Moreover, the main results of this paper are reduced to the constant time delay case and extended to the stochastic time-varying delay case, and also extended to address the problem of exponential stabilization for linear parabolic PDdE systems with a temporal control input. The numerical simulation results of two examples show the effectiveness and merit of the main results.

Parameter-Robust Numerical Scheme for Time-Dependent Singularly Perturbed Reaction–Diffusion Problem with Large Delay

ABSTRACTThis work presents the development of numerical scheme for second-order time-dependent singularly perturbed reaction-diffusion problem with large delay in the undifferentiated term. These types of problems arise frequently in many areas of science and engineering that take into consideration the effect of present situation as well as the past history of the physical system. As the characteristics of the reduced problem (𝜀 = 0) corresponding to the original singularly perturbed problem considered here are parallel to the boundary of the domain this implies, parabolic layers exhibit in the solution. In this paper, we initiate the study of parabolic layers together with interior layers in the solution of singularly perturbed parabolic partial differential-difference equations due to propagation of singularity. Proposed numerical scheme comprised of finite difference scheme and piecewise uniform Shishkin mesh. The method is shown to be accurate of order , where M and N are the number of mesh elements in time and spatial direction, respectively. Proposed numerical scheme is proved to be parameter uniform convergent in the maximum norm. Numerical experiments have been performed to show the existence of interior layer due to large state-dependent delay argument in the reaction term and to confirm the predicted theory.

Symmetries of the Hirota Difference Equation

Continuous symmetries of the Hirota difference equation, commuting with shifts of independent variables, are derived by means of the dressing procedure. Action of these symmetries on the dependent variables of the equation is presented. Commutativity of these symmetries enables interpretation of their parameters as "times" of the nonlinear integrable partial differential-difference and differential equations. Examples of equations resulting in such procedure and their Lax pairs are given. Besides these, ordinary, symmetries the additional ones are introduced and their action on the Scattering data is presented.

On existence of solution of the Cauchy problem for nonlinear diffusion stochastic partial differential-difference equations of neutral type with random external perturbations

The question related to the existence of the Cauchy problem solution in the class of nonlinear diffusion stochastic partial differential-difference equations of a neutral type with random external disturbances which are independent from the Wiener process is considered. Sufficient conditions are obtained for the diffusion coefficients of nonlinear stochastic differential-difference equations of a neutral type (NDSDRRNT) that guarantee the existence of the solution with the probability of 1. The method of the proof is based on the results of O.M. Stanzhitsky and A.O. Tsukanova on the existence and uniqueness of the Cauchy problem solution for the stochastic differential reaction-diffusion equation of a neutral type. In this paper, we prove the existence of a "mild solution" of NDSDRRNT. In some cases, such equations are mathematical models of real processes, the consideration of which is planned in the future work.