Partial Differential-difference Equations Research Articles

On solutions of certain compatible systems of quadratic trinomial Partial differential-difference equations

This paper has involved the use of a variety of variations of the Fermat-type equation $f^n(z)+g^n(z)=1$, where $n(\geq 2)\in\mathbb{N}$. Many researchers have demonstrated a keen interest to investigate the Fermat-type equations for entire and meromorphic solutions of several complex variables over the past two decades. Researchers utilize the Nevanlinna theory as the key tool for their investigations. Throughout the paper, we call the pair $(f,g)$ as a finite order entire solution for the Fermat-type compatible system $\begin{cases} f^{m_1}+g^{n_1}=1;\\ f^{m_2}+g^{n_2}=1,\end{cases}$\!\! if $f$, $g$ are finite order entire functions satisfying the system, where $m_1,m_2,n_1,n_2\in\mathbb{N}\setminus\{1\} .$\ Taking into the account the idea of the quadratic trinomial equations, a new system of quadratic trinomial equations has been constructed as follows: $\begin{cases} f^{m_1}+2\alpha f g+g^{n_1}=1;\\ f^{m_2}+2\alpha f g+g^{n_2}=1,\end{cases}$ \!\! where $\alpha\in\mathbb{C}\setminus\{0,\pm1\}.$ In this paper, we consider some earlier systems of certain Fermat-type partial differential-difference equations on $\mathbb{C}^2$, especially, those of Xu {\it{et al.}} (Entire solutions for several systems of nonlinear difference and partial differential-difference equations of Fermat-type, J. Math. Anal. Appl. 483(2), 2020) and then construct some systems of certain quadratic trinomial partial differential-difference equations with arbitrary coefficients. Our objective is to investigate the forms of the finite order transcendental entire functions of several complex variables satisfying the systems of certain quadratic trinomial partial differential-difference equations on $\mathbb{C}^n$. These results will extend the further study of this direction.

Results on solutions of several systems of the product type complex partial differential difference equations

Abstract This article is devoted to exploring the solutions of several systems of the first-order partial differential difference equations (PDDEs) with product type u ( z + c ) [ α 1 u ( z ) + β 1 u z 1 + γ 1 u z 2 + α 2 v ( z ) + β 2 v z 1 + γ 2 v z 2 ] = 1 , v ( z + c ) [ α 1 v ( z ) + β 1 v z 1 + γ 1 v z 2 + α 2 u ( z ) + β 2 u z 1 + γ 2 u z 2 ] = 1 , \left\{\begin{array}{l}u\left(z+c){[}{\alpha }_{1}u\left(z)+{\beta }_{1}{u}_{{z}_{1}}+{\gamma }_{1}{u}_{{z}_{2}}+{\alpha }_{2}v\left(z)+{\beta }_{2}{v}_{{z}_{1}}+{\gamma }_{2}{v}_{{z}_{2}}]=1,\\ v\left(z+c){[}{\alpha }_{1}v\left(z)+{\beta }_{1}{v}_{{z}_{1}}+{\gamma }_{1}{v}_{{z}_{2}}+{\alpha }_{2}u\left(z)+{\beta }_{2}{u}_{{z}_{1}}+{\gamma }_{2}{u}_{{z}_{2}}]=1,\end{array}\right. where c = ( c 1 , c 2 ) ∈ C 2 c=\left({c}_{1},{c}_{2})\in {{\mathbb{C}}}^{2} , α j , β j , γ j ∈ C , j = 1 , 2 {\alpha }_{j},{\beta }_{j},{\gamma }_{j}\in {\mathbb{C}},\hspace{0.33em}j=1,2 . Our theorems about the forms of the transcendental solutions for these systems of PDDEs are some improvements and generalization of the previous results given by Xu, Cao and Liu. Moreover, we give some examples to explain that the forms of solutions of our theorems are precise to some extent.

Characterizations of Finite Order Solutions of Circular Type Partial Differential-Difference Equations in $$ \mathbb {C}^n $$

Characterizations of Finite Order Solutions of Circular Type Partial Differential-Difference Equations in $$ \mathbb {C}^n $$

On entire solutions of Fermat type difference and kth order partial differential difference equations in several complex variables

On entire solutions of Fermat type difference and kth order partial differential difference equations in several complex variables

Entire solutions to Fermat-type difference and partial differential-difference equations in C^n

In this article, we study the existence and the form of finite order transcendental entire solutions of systems of Fermat-type difference and partial differential-difference equations in several complex variables. Our results extend previous theorems given by Xu-Cao [49], Xu et al [52], and Zheng-Xu [55]. We give some examples to illustrate the content of this article. For more information see https://ejde.math.txstate.edu/Volumes/2024/26/abstr.html

On entire solutions of system of Fermat-type difference and partial differential-difference equations in $$\mathbb {C}^n$$

On entire solutions of system of Fermat-type difference and partial differential-difference equations in $$\mathbb {C}^n$$

The study of solutions of several systems of the product type nonlinear partial differential difference equations

The study of solutions of several systems of the product type nonlinear partial differential difference equations

Characterizations of entire solutions for the system of Fermat-type binomial and trinomial shift equations in ℂ n#

Abstract In this article, we investigate the existence and the precise form of finite-order transcendental entire solutions of some system of Fermat-type quadratic binomial and trinomial shift equations in C n {{\mathbb{C}}}^{n} . Our results are the generalizations of the results of [H. Y. Xu, S. Y. Liu, and Q. P. Li, Entire solutions for several systems of nonlinear difference and partial differential-difference equations of Fermat-type, J. Math. Anal. Appl. 483 (2020), 123641, 1–22, DOI: https://doi.org/10.1016/j.jmaa.2019.123641.] and [H. Y. Xu and Y. Y. Jiang, Results on entire and meromorphic solutions for several systems of quadratic trinomial functional equations with two complex variables, RACSAM 116 (2022), 8, DOI: https://doi.org/10.1007/s13398-021-01154-9.] to a large extent. Most interestingly, as a consequence of our main result, we have shown that the system of quadratic trinomial shift equation has no solution when it reduces to a system of quadratic trinomial difference equation. In addition, some examples relevant to the content of the article have been exhibited.

Results on Solutions for Several Systems of Partial Differential-Difference Equations with Two Complex Variables

This article is to describe the entire solutions of some partial differential-difference equations and systems. Some theorems about the forms of transcendental entire solutions with finite order for several high-order partial differential-difference equations (or systems) of the Fermat type with two complex variables are obtained. Moreover, some examples are provided to explain that our results are precise to some extent.

The study of solutions for several systems of PDDEs with two complex variables

Abstract The purpose of this article is to describe the properties of the pair of solutions of several systems of Fermat-type partial differential difference equations. Our theorems exhibit the forms of finite order transcendental entire solutions for these systems, which are some extensions and improvement of the previous theorems given by Xu, Cao, Liu, etc. Furthermore, we give a series of examples to show that the existence conditions and the forms of transcendental entire solutions with finite order of such systems are precise.

Fuzzy Boundary Control for Nonlinear Delayed DPSs Under Boundary Measurements.

For nonlinear delayed distributed parameter systems (DDPSs), this article considers a fuzzy boundary control (FBC) under boundary measurements (BMs). Initially, we accurately describe the nonlinear DDPS through a Takagi-Sugeno (T-S) fuzzy partial differential-difference equation (PDDE). Then, in accordance with the T-S fuzzy PDDE model, an FBC design under BMs ensuring the exponential stability for closed-loop DDPS is subsequently presented by spatial linear matrix inequalities (SLMIs) via using Wirtinger's inequality, Halanay's inequality, and the Lyapunov direct method, which respects the fast-varying and slow-varying delays. Moreover, we formulate SLMIs as LMIs for solving the fuzzy boundary controller design of nonlinear DDPSs under BMs. Finally, the effectiveness of the proposed FBC strategy is presented via simulation examples.

A Systematic Review on the Solution Methodology of Singularly Perturbed Differential Difference Equations

This review paper contains computational methods or solution methodologies for singularly perturbed differential difference equations with negative and/or positive shifts in a spatial variable. This survey limits its coverage to singular perturbation equations arising in the modeling of neuronal activity and the methods developed by numerous researchers between 2012 and 2022. The review covered singularly perturbed ordinary delay differential equations with small or large negative shift(s), singularly perturbed ordinary differential–differential equations with mixed shift(s), singularly perturbed delay partial differential equations with small or large negative shift(s) and singularly perturbed partial differential–difference equations of the mixed type. The main aim of this review is to find out what numerical and asymptotic methods were developed in the last ten years to solve such problems. Further, it aims to stimulate researchers to develop new robust methods for solving families of the problems under consideration.

Existence of meromorphic solutions of systems of partial differential–difference equations

Existence of meromorphic solutions of systems of partial differential–difference equations

The Exact Solutions for Several Partial Differential-Difference Equations with Constant Coefficients

This article is concerned with the description of the entire solutions of several Fermat type partial differential-difference equations (PDDEs) μf(z)+λfz1(z)2+[αf(z+c)−βf(z)]2=1, and μf(z)+λ1fz1(z)+λ2fz2(z)2+[αf(z+c)−βf(z)]2=1, where fz1(z)=∂f∂z1 and fz2(z)=∂f∂z2, c=(c1,c2)∈C2, α,β,μ,λ,λ1,λ2,c1,c2 are constants in C. Our theorems in this paper give some descriptions of the forms of transcendental entire solutions for the above equations, which are some extensions and improvement of the previous theorems given by Xu, Cao, Liu, and Yang. In particular, we exhibit a series of examples to explain that the existence conditions and the forms of transcendental entire solutions with a finite order of such equations are precise.

Entire solutions of several quadratic binomial and trinomial partial differential-difference equations in $$ {\mathbb {C}}^2 $$

Entire solutions of several quadratic binomial and trinomial partial differential-difference equations in $$ {\mathbb {C}}^2 $$

The Study of Solutions of Several Systems of Nonlinear Partial Differential Difference Equations

The Study of Solutions of Several Systems of Nonlinear Partial Differential Difference Equations

Exotical solitons for an intrinsic fractional circuit using the sine-cosine method

This work focuses on seeking soliton solutions for an intrinsic fractional discrete nonlinear electrical transmission network obtained through the simplest sine-cosine method. The studied model is governed by a fractional nonlinear partial differential-difference equation in (2 + 1) spatio-temporal dimensions, and the method used to get exact solutions is simple, concise and well-known. We achieve for the model studied here that, the sought solutions specifically by means of the sine-cosine method are functions of all the capacitor's nonlinearities (quadratic and cubic) if and only if, we use the fourth-order spatial dispersion (FOSD)during the continuous media approximation. In contrast, in the absence of the FOSD term, the solutions only exist if, either the quadratic or the cubic nonlinearity is considered separately. In addition, the obtained solutions shapes are exotical, unexpected and novel. These findings (singular bright solitary waves, pulse, U-shaped and M-shaped waves trains) get many applications; for instance, codifying data in the allowed or forbidden band for the signal's transmission in the waveguides.

Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2

Abstract By utilizing the Nevanlinna theory of meromorphic functions in several complex variables, we mainly investigate the existence and the forms of entire solutions for the partial differential-difference equation of Fermat type α ∂ f ( z 1 , z 2 ) ∂ z 1 + β ∂ f ( z 1 , z 2 ) ∂ z 2 m + f ( z 1 + c 1 , z 2 + c 2 ) n = 1 {\left(\alpha \frac{\partial f\left({z}_{1},{z}_{2})}{\partial {z}_{1}}+\beta \frac{\partial f\left({z}_{1},{z}_{2})}{\partial {z}_{2}}\right)}^{m}+f{\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})}^{n}=1 and α ∂ f ( z 1 , z 2 ) ∂ z 1 + β ∂ f ( z 1 , z 2 ) ∂ z 2 2 + [ γ 1 f ( z 1 + c 1 , z 2 + c 2 ) − γ 2 f ( z 1 , z 2 ) ] 2 = 1 , {\left(\alpha \frac{\partial f\left({z}_{1},{z}_{2})}{\partial {z}_{1}}+\beta \frac{\partial f\left({z}_{1},{z}_{2})}{\partial {z}_{2}}\right)}^{2}+{\left[{\gamma }_{1}f\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})-{\gamma }_{2}f\left({z}_{1},{z}_{2})]}^{2}=1, where m , n m,n are positive integers and α , β , γ 1 , γ 2 \alpha ,\beta ,{\gamma }_{1},{\gamma }_{2} are constants in C {\mathbb{C}} . We give some results about the forms of solutions for these equations, which are great improvements of the previous theorems given by Xu and Cao et al. Moreover, it is very satisfactory that we give the corresponding examples to explain the conclusions of our theorems in each case.

Existence of the Solution to the Cauchy Problem for Nonlinear Stochastic Partial Differential-Difference Equations of Neutral Type

Existence of the Solution to the Cauchy Problem for Nonlinear Stochastic Partial Differential-Difference Equations of Neutral Type

Notes on Solutions for Some Systems of Complex Functional Equations in ℂ 2

The purpose of this article is to give the details of finding the transcendental entire solutions with finite order for the systems of nonlinear partial differential-difference equations ∂ f 1 z 1 , z 2 / ∂ z 1 + ∂ f 1 z 1 , z 2 / ∂ z 2 n 1 + P 1 z f 2 z 1 + c 1 , z 2 + c 2 m 1 = Q 1 z , ∂ f 1 z 1 , z 2 / ∂ z 1 + ∂ f 1 z 1 , z 2 / ∂ z 2 n 2 + P 2 z f 1 z 1 + c 1 , z 2 + c 2 m 2 = Q 2 z , where P 1 z , P 2 z , Q 1 z , and Q 2 z are polynomials in ℂ 2 ; n 1 , n 2 , m 1 , and m 2 are positive integers, and c = c 1 , c 2 ∈ ℂ 2 . We obtain that there exist some pairs of the transcendental entire solutions of finite order for the above system, which is a very powerful supplement to the previous theorems given by Xu and Cao and Xu and Yang.