Meromorphic Solutions Research Articles

On generalized Fermat Diophantine functional and partial differential equations in $${\mathbf {C}}^2$$

In this paper, we characterize meromorphic solutions $f(z_1,z_2),g(z_1,z_2)$ to the generalized Fermat Diophantine functional equations $h(z_1,z_2)f^m+k(z_1,z_2)g^n=1$ in $\mathbf{C}^2$ for integers $m,n\geq2$ and nonzero meromorphic functions $h(z_1,z_2),k(z_1,z_2)$ in $\mathbf{C}^2$. Meromorphic solutions to associated partial differential equations are also studied.

On the transcendental entire and meromorphic solutions of certain non-linear generalized delay-differential equations

Abstract The prime intention of this paper is to study the conditions under which certain non-linear generalized delay-differential equations possess a solution. In this respect, by extending and improving recent results of [5, 15], we characterize the nature of solutions. By providing relevant examples in a remark, we also show that for the uniqueness of solutions the conditions on the Borel exceptional value can not be removed. Finally, to determine explicitly all forms of the solutions of a traditional generalized delay-differential equation, we deal with the situation under the aegis of generalized c-delay-differential equations. We exhibit some examples to show that the conclusions of the theorems actually occur.

Growth properties of meromorphic solutions of some higher-order linear differential-difference equations

Abstract This paper is devoted to the study of the growth of meromorphic solutions of homogeneous and non-homogeneous linear differential-difference equations ∑ i = 0 n ∑ j = 0 m A i ⁢ j ⁢ f ( j ) ⁢ ( z + c i ) = 0 , \displaystyle\sum_{i=0}^{n}\sum_{j=0}^{m}A_{ij}f^{(j)}(z+c_{i})=0, ∑ i = 0 n ∑ j = 0 m A i ⁢ j ⁢ f ( j ) ⁢ ( z + c i ) = F , \displaystyle\sum_{i=0}^{n}\sum_{j=0}^{m}A_{ij}f^{(j)}(z+c_{i})=F, where A i ⁢ j {A_{ij}} ( i = 0 , … , n {i=0,\ldots,n} , j = 0 , … , m {j=0,\ldots,m} ), F are meromorphic functions and c i {c_{i}} ( 0 , … , n {0,\ldots,n} ) are non-zero distinct complex numbers. Under some conditions on the coefficients, we extend early results due to Zhou and Zheng.

On meromorphic solutions of functional-differential equations

AbstractWe consider meromorphic solutions of functional-differential equations \[ f^{(k)}(z)=a(f^{n}\circ g)(z)+bf(z)+c, \]where $n,\,~k$ are two positive integers. Firstly, using an elementary method, we describe the forms of $f$ and $g$ when $f$ is rational and $a(\neq 0)$, $b$, $c$ are constants. In addition, by employing Nevanlinna theory, we show that $g$ must be linear when $f$ is transcendental and $a(\neq 0)$, $b$, $c$ are polynomials in $\mathbb {C}$.

Closed-form meromorphic solutions of some third order boundary layer ordinary differential equations

We introduce a general third order non-linear autonomous ODE which covers many ODEs coming from boundary layer problems, like the Falkner-Skan equation and the Cheng-Minkowycz equation. Using Wiman-Valiron theory and complex analytic methods recently developed, for the generic cases, it is shown that all their meromorphic solutions must be rational, or rational in one exponential, and then we find all of them explicitly. For a few non-generic cases, some solutions, which are meromorphic or singlevalued, are also obtained. Our results also explain why it is so difficult to obtain new closed-form solutions of the Falkner-Skan equation.

One advance in the proof of the conjecture on meromorphic solutions of Briot-Bouquet type equations

We study entire solutions (solutions which are entire functions) of differential equations of the form $P(y,y^{(n)})=0$, where $P$ is a polynomial with complex coefficients, $n$ is a natural number. We show that, under some constraints on $P$, all entire solutions of such equations are either polynomials, or functions of the form $e^{-L\beta z}Q(e^{\beta z})$, where $L$ is a nonnegative integer, $\beta$ is a complex number, and $Q$ is a polynomial with complex coefficients. This verifies the well-known A. E. Eremenko's conjecture on meromorphic solutions of autonomous Briot-Bouquet type equations for entire solutions in the nondegenerate case.

Meromorphic solutions of some types of q-difference differential equation and delay differential equation

Meromorphic solutions of some types of q-difference differential equation and delay differential equation

On the Meromorphic Solutions of Fermat-Type Differential Equations

In this paper, we investigate the meromorphic solutions of the Fermat-type differential equations f’(z)n + f(z+c)m = eAz+B (c ≠ 0) over the complex plane C for positive integers m, n, and A, B, c are constants. Our results improve and extend some earlier results given by Liu et al. Moreover, some examples are presented to show the preciseness of our results.

On meromorphic solutions of certain type of nonlinear differential-difference equations

On meromorphic solutions of certain type of nonlinear differential-difference equations

On the absence of logarithmic singularities in the solutions of Lamé-type equations

The object of this research is linear differential equations of the second order with regular singularities. We extend the concept of a regular singularity to linear partial differential equations. The general solution of a linear differential equation with a regular singularity is a linear combination of two linearly independent solutions, one of which in the general case contains a logarithmic singularity. The well-known Lamé equation, where the Weierstrass elliptic function is one of the coefficients, has only meromorphic solutions. We consider such linear differential equations of the second order with regular singularities, for which as a coefficient instead of the Weierstrass elliptic function we use functions that are the solutions to the first Painlevé or Korteweg – de Vries equations. These equations will be called Lamé-type equations. The question arises under what conditions the general solution of Lamé-type equations contains no logarithms. For this purpose, in the present paper, the solutions of Lamé-type equations are investigated and the conditions are found that make it possible to judge the presence or absence of logarithmic singularities in the solutions of the equations under study. An example of an equation with an irregular singularity having a solution with an logarithmic singularity is given, since the equation, defining it, has a multiple root.

On meromorphic solutions of nonlinear delay-differential equations

Using Cartan's second main theorem and Nevanlinna's theorem concerning a group of meromorphic functions, we obtain the growth and zero distribution of meromorphic solutions of the nonlinear delay-differential equation fn(z)+P(z)f(k)(z+η)=H0(z)+H1(z)eω1zq+⋯+Hm(z)eωmzq, where n,k,q,m are positive integers, η,ω1,⋯,ωm are complex numbers with ω1⋯ωm≠0, and P,H0,H1,⋯,Hm are entire functions of order less than q with PH1⋯Hm≢0. Especially for η=0, some sufficient conditions are given to guarantee the above equation has no meromorphic solutions of few poles.

New exact solutions of the sixth-order thin-film equation with complex method

By the Wiman–Valiron theory and the complex method, we prove that all meromorphic solutions of the fifth-order ODE u n u ′ ′ ′ ′ ′ − c u = 0 belong to W , and obtain several new solutions of the sixth-order thin-film equation comparing to the opening literature. At last we propose two unsolved problems for the readers. • New exact solutions of the sixth-order thin-film equation have been investigated. • The complex method has been used first time for PDEs with arbitrary degree n. • The obtained results are different from the founded ones in literature. • Two open questions are given for further research.

Lower order for meromorphic solutions to linear delay-differential equations

In this article, we study the order of growth for solutions of the non-homogeneous linear delay-differential equation $$ \sum_{i=0}^n\sum_{j=0}^{m}A_{ij}f^{(j)} (z+c_i)=F(z), $$ where \(A_{ij}(z)\) \((i=0,\dots ,n;j=0,\dots ,m)\), \(F(z)$\)are entire or meromorphic functions and \(c_i\) \((0,1,\dots ,n)\) are non-zero distinct complex numbers. Under the condition that there exists one coefficient having the maximal lower order, or having the maximal lower type, strictly greater than the order, or the type, of the other coefficients, we obtain estimates of the lower bound of the order of meromorphic solutions of the above equation. For more information see https://ejde.math.txstate.edu/Volumes/2021/92/abstr.html

Results on entire and meromorphic solutions for several systems of quadratic trinomial functional equations with two complex variables

Results on entire and meromorphic solutions for several systems of quadratic trinomial functional equations with two complex variables

On the existence of meromorphic solutions of certain nonlinear difference equations

By Cartan’s version of Nevanlinna’s theory, we prove the following result: Let m and n be two positive integers satisfying n≥m+2 and m≥1, let p be a polynomial such that p≢0, let η be a finite complex number such that η≠0 and Δηf≢0, let α1,α2,…,αm be m distinct finite nonzero complex numbers, and let Hj be either an exponential polynomial of degree less than q or an ordinary polynomial in z such that Hj≢0 for 1≤j≤m. Suppose that f is a nonconstant meromorphic solution of the difference equation fn(z)+p(z)Δηf(z)=H1(z)eα1zq+H2(z)eα2zq+⋯+Hm(z)eαmzq. Then m≥2 and f is reduced to a transcendental entire function such that its order ρ(f)=∞ or its order satisfies ρ(f)=q with m=2, while f can be either expressed as f(z)=A1(z)eα1zq, with A1(z)=H1fpΔηf and nα1−α2=0, or f(z)=A2(z)eα2zq, with A2(z)=H2fpΔηf and nα2−α1=0, where A1 and A2 are entire function such that their characteristic functions satisfy T(r,Aj)=o(T(r,f)),1≤j≤2, as r∉E and r→∞. Here, E⊂(0,+∞) is a subset of finite linear measure. An example is provided to show that the assumption n≥m+2, in a sense, is the best possible. This result improves and extends the corresponding results from Yang–Laine (2010), Liu–Lü–Shen–Yang (2014), and Latreuch (2017).

Some Properties on The [p,q]-Order of Meromorphic Solutions of Homogeneous and Non-homogeneous Linear Differential Equations With Meromorphic Coefficients

In the present paper, we investigate the $\left[p,q\right] $-order of solutions of higher order linear differential equations \begin{equation*} A_{k}\left(z\right) f^{\left( k\right) }+A_{k-1}\left( z\right) f^{\left(k-1\right)}+\cdots +A_{1}\left( z\right) f^{\prime }+A_{0}\left( z\right) f=0 \end{equation*} and \begin{equation*} A_{k}\left( z\right) f^{\left( k\right) }+A_{k-1}\left( z\right) f^{\left(k-1\right) }+\cdots +A_{1}\left( z\right) f^{\prime }+A_{0}\left( z\right) f=F\left( z\right), \end{equation*} where $A_{0}\left( z\right) ,$ $A_{1}\left( z\right) ,...,A_{k}\left(z\right) \not\equiv 0$ and $F\left( z\right) \not\equiv 0$ are meromorphic functions of finite $\left[ p,q\right] $-order. We improve and extend some results of the authors by using the concept $\left[ p,q\right] $-order.

Growth of solutions of linear differential-difference equations with coefficients having the same logarithmic order

In this paper, we investigate the relations between the growth of meromorphic coefficients and that of meromorphic solutions of complex linear differential-difference equations with meromorphic coefficients of finite logarithmic order. Our results can be viewed as the generalization for both the cases of complex linear differential equations and complex linear difference equations.

Exact meromorphic solutions of Schwarzian differential equations

This paper studies exact meromorphic solutions of the autonomous Schwarzian differential equations. All transcendental meromorphic solutions of five canonical types (among six) of the autonomous Schwarzian differential equations are constructed explicitly. In particular, the solutions of four types are shown to be elliptic functions. Also, all transcendental meromorphic solutions that are locally injective or possess a Picard exceptional value are characterized for the remaining canonical type.

Meromorphic solutions of delay differential equations related to logistic type and generalizations

Let {bj}j=1k be meromorphic functions, and let w be admissible meromorphic solutions of delay differential equationw′(z)=w(z)[P(z,w(z))Q(z,w(z))+∑j=1kbj(z)w(z−cj)] with distinct delays c1,…,ck∈C∖{0}, where the two nonzero polynomials P(z,w(z)) and Q(z,w(z)) in w with meromorphic coefficients are prime each other. We obtain that if limsupr→∞log⁡T(r,w)r=0, thendegw⁡(P/Q)≤k+2. Furthermore, if Q(z,w(z)) has at least one nonzero root, then degw⁡(P)=degw⁡(Q)+1≤k+2; if all roots of Q(z,w(z)) are nonzero, then degw⁡(P)=degw⁡(Q)+1≤k+1; if degw⁡(Q)=0, then degw⁡(P)≤1.In particular, whenever degw⁡(Q)=0 and degw⁡(P)≤1 and without the growth condition, any admissible meromorphic solution of the above delay differential equation (called Lenhart-Travis' type logistic delay differential equation) with reduced form can not be an entire function w satisfying N‾(r,1w)=O(N(r,1w)); while if all coefficients are rational functions, then the condition N‾(r,1w)=O(N(r,1w)) can be omitted. Furthermore, any admissible meromorphic solution of the logistic delay differential equation (that is, for the simplest special case where k=1 and degw⁡(P/Q)=0) satisfies that N(r,w) and T(r,w) have the same growth category. Some examples support our results.

Three results on transcendental meromorphic solutions of certain nonlinear differential equations

In this paper, we study the transcendental meromorphic solutions for the nonlinear differential equations: $f^{n}+P(f)=R(z)e^{\alpha(z)}$ and $f^{n}+P_{*}(f)=p_{1}(z)e^{\alpha_{1}(z)}+p_{2}(z)e^{\alpha_{2}(z)}$ in the complex plane, where $P(f)$ and $P_{*}(f)$ are differential polynomials in $f$ of degree $n-1$ with coefficients being small functions and rational functions respectively, $R$ is a non-vanishing small function of $f$, $\alpha$ is a nonconstant entire function, $p_{1}, p_{2}$ are non-vanishing rational functions, and $\alpha_{1}, \alpha_{2}$ are nonconstant polynomials. Particularly, we consider the solutions of the second equation when $p_{1}, p_{2}$ are nonzero constants, and $°\alpha_{1}=°\alpha_{2}=1$. Our results are improvements and complements of Liao (Complex Var. Elliptic Equ. 2015, 60(6): 748--756), and Rong-Xu (Mathematics 2019, 7, 539), etc., which partially answer a question proposed by Li (J. Math. Anal. Appl. 2011, 375: 310--319).