Solutions of several general quadratic partial differential-difference equations in ℂ2
-
Molla Basir Ahamed
and
Sanju Mandal
Abstract
In this article, we have introduced general transformation to solving the general quadratic equations. It is of interest to know about the existence and form of the solutions of general quadratic functional equations. By utilizing Nevanlinna theory for several complex variables and the transformation, we study the existence and form of the solutions to the general quadratic partial differential or partial differential-difference equations of the form
1 Introduction
It is known that the Fermat equation
can be regarded as the Fermat-type equations over function fields. The study of equation goes back to Montel [3] and Gross [4]. In fact, the study of Fermat-type functional equations has a long history. In
The study of general quadratic equations is motivated by its importance and relevance in various branches of mathematics and science. Such an equations often arise in problems related to optimization, control theory, physics, and engineering. This type of equations is essential for the development of modern technology and science. Moreover, the general quadratic equations has been an interesting subject in the field of complex analysis in connection with extensions of Nevanlinna’s theory. As we know that, partial differential equations are occurring in various areas of applied mathematics, such as fluid mechanics, nonlinear acoustic, gas dynamics and traffic flow [10,11]. In general, it is difficult to find entire and meromorphic solutions for a nonlinear PDE. By employing Nevanlinna theory and the method of complex analysis, there were a number of literature focusing on the solutions of the PDEs and theirs many variants.
Let us also recall here the definitions of difference equations and partial differential-difference equations (PDDEs) in
Definition 1.1
[12] An equation is called a difference equation, if the equation includes shifts or difference of
Due to the development of the difference analogues of Nevanlinna theory in
must be linear, i.e., the solution takes the form
In [36], Li showed that meromorphic solutions
Recently, in [39], Xu et al. have investigated solutions of the following Fermat-type PDDEs. Our objective in this article is to explore the solutions of equations in [39] with a more general setting, specifically for general quadratic PDDEs. The study of finding solutions of general quadratic equations provide a broader perspective compared to solutions of these binomial and trinomial equations in one and several complex variables. We have seen that the equations in [39] are of the form
Therefore, to further explore Fermat-type functional equations, it is natural to pose the following question.
Question 1.1
Does there exists solutions in
In this article, our main objective is to provide answer to Question 1.1. In other words, our main interest lies in finding the solutions of the functional equation
Specifically, we will utilize Nevanlinna theory to investigate the existence and precise form of the solutions to the following general quadratic PDDEs:
Motivated by Saleeby’s articles in [37,40], our main focus in this article is to establish results for the equations in (1.2), providing a comprehensive answer to Question 1.1. In these equations, we assume that the corresponding companion matrices are nonsingular, i.e.,
2 Method for solving the general quadratic equations
The most general quadratic first order partial differential equation
where
This will represent a pair of straight lines only if the L.H.S of (2.2) can be resolve into two linear factors. By arranging the equation as a quadratic in
Therefore, if
Hence, we see that left-hand side of (2.2) can be resolve into two linear factors, if
The companion matrix of the quadratic equation (2.2) is the following
Let
Let
a straightforward computation shows that equation (2.2) reduces to
where
Hence, equation (2.5) can be written as follows:
where
To remove the
In light of this transformation, equation (2.6) is transformed to
where
It is clear that (2.7) can be expressed as follows:
Clearly, if we consider the given general equation (2.1) as a functional equation in
where
Thus, it follows from (2.4) that the solution
where
3 Main results and their consequences
In this article, we investigate the existence and form of solutions for general quadratic PDEs and PDDEs in
Theorem 3.1
Let
Then
satisfying
By providing the following examples, we show that the form of transcendental entire solutions of (3.1) in Theorem 3.1 is precise.
Example 3.1
The finite order transcendental entire solutions for the general quadratic partial differential equation
is of the form
Example 3.2
The finite order transcendental entire solutions for the general quadratic partial differential equation
is of the form
Before we state some consequences of the main result, which provide the precise form of the solutions for trinomial equations, we introduce the following notations:
From Theorem 3.1, we obtain the following result, in which the form of the solutions for trinomial quadratic PDEs is characterized.
Corollary 3.1
Let
Then
where
where
Extending the scope of [39, Theorem 2.2], this article presents its second primary result, which is applicable to a broader range of quadratic PDEs.
Theorem 3.2
Let
does not admit any transcendental entire solution with finite order.
Remark 3.1
In view of the proof of Theorem 3.2, it can be easily shown that the trinomial quadratic PDE
does not admit any transcendental entire solutions. Consequently, we present a more general framework for the equation.
The following statement constitutes the third significant outcome of this article, involving the expansion of [39, Theorem 2.3] to address general quadratic partial differential equations.
Theorem 3.3
Let
Then
Example 3.3
The finite order transcendental entire solutions for the general quadratic partial differential equation
of the form
In case of trinomial quadratic PDEs, we obtain the following result as a consequence of Theorem 3.3.
Corollary 3.2
Let
Then
We obtain the next result as a generalization of [39, Theorem 2.4] for quadratic PDEs.
Theorem 3.4
Let
Then
where
and
and also
The following examples are exhibited to ensure the form of the transcendental entire solution of (3.4) in Theorem 3.4.
Example 3.4
Let
takes the form
where
Example 3.5
Let
The finite order transcendental entire solutions for the general quadratic partial differential equation
is of the form
where
In the case of trinomial quadratic PDEs, the following result can be obtained as a corollary of Theorem 3.4.
Corollary 3.3
Let
The function
where
and
and, also
We obtain the following result as a generalization [39, Theorem 2.5] for quadratic partial differential equations PDEs.
Theorem 3.5
Let
Then
where
and
To support the accuracy of the solutions’ form in (3.5) of Theorem 3.5, we provide the following two examples.
Example 3.6
Let
of the form
Example 3.7
Let
and
The finite order transcendental entire solutions for the general quadratic partial differential equation
is of the form
As a consequence of Theorem 3.5, we establish the existence and form of trinomial quadratic PDEs and obtain the following result.
Corollary 3.4
Let
Then
where
and
4 Proof of the main results
Let
First, we recall here some necessary lemmas, which will play a key role in proving the main results.
Lemma 4.1
[45,46] For any entire function F on
Remark 4.1
Here, denote
Lemma 4.2
[47] Let
for all
Remark 4.2
Here,
Lemma 4.3
[48] If g and h are entire functions on the complex plane
the internal function h is a polynomial and the external function g is of finite order or
the internal function h is not a polynomial but a function of finite order, and the external function g is of zero order.
In the following section, we will discuss proof of the main results.
Proof of Theorem 3.1
Assuming that
The proof is divided into two distinct cases for clarity.
Case 1: If
in view of (4.1), it is easy to see that
Considering (4.2) and (4.3), it is evident that
The characteristic equations of (4.4) are as follows:
Using the initial conditions
where
Since
where
Consequently, equation (4.5) is transformed into
By substituting (4.6) into (4.3), we obtain
Thus considering (4.6) and (4.7), we can conclude that the solution takes the precise form of
Case 2: Assuming that
By subtracting (4.9) from (4.8), we obtain
Using the fact that
From (4.10) and (4.11), we obtain
Sub-case 2.1: Suppose that
It follows from (4.12) that
By considering the Nevanlinna characteristic function and applying the “First Fundamental Theorem of Nevanlinna,” we derive
which shows that
Given that
Therefore, from (4.13), we see that
outside possibly a set of finite Lebesgue measure in view of the results (see [18,46]) that
when
Sub-case 2.2: Suppose that
Thus, (4.12) can be written as follows:
Given that
where
Sub-case 2.3: Let us assume
By the similar argument used in sub-case
Sub-case 2.4: Suppose that
By employing the same reasoning as in sub-case 1, we reach a contradictory conclusion.
Sub-case 2.5: Suppose that
In view of (4.12), we see that
Sub-case 2.6: Assume that
By employing equation (4.12), we can establish that
Sub-case 2.7: Let
By observing equation (4.12), it becomes clear that
Sub-case 2.8: Suppose that
Since
where
By substituting this
The characteristic equations of (4.16) are as follows:
and hence,
By using the initial conditions:
where
is a transcendental entire function with finite order in
Substituting (4.17) into (4.9) gives that
where
It follows easily that
Consequently, in view of (4.17) and (4.18), we obtain form of the solution
This completes the proof.□
Proof of Theorem 3.2
Let us assume that
We discuss the following two possible cases.
Case 1: If
In view of (4.19), it is easy to see that
By subtracting (4.21) from (4.20), we see that
By differentiating (4.20) with respect to
Consequently, it can be deduced from equations (4.21) and (4.23) that
By differentiating (4.24) with respect to
In view of (4.22) and (4.25), it is clear that
where
Consequently, from (4.25), it follows that
where
Case 2: If
By the similar reasoning applied previously, we obtain
Differentiating (4.27) with respect to
By considering equations (4.28) and (4.30), it becomes evident that
By differentiating (4.31) with respect to
Since
In view of (4.32) and (4.33), we see that
where
is a transcendental, which is a contradiction as
Proof of Theorem 3.3
Suppose that
Now we will discuss two cases below.
Case 1: If
In view of (4.36), we have
An easy computation using (4.37) and (4.38) shows that
By differentiating (4.37) with respect to
From (4.38) and (4.40), it follows that
By differentiating (4.41) with respect to
In view of (4.40) and (4.42), it is easy to see that
where
Through the combination of equations (4.40) and (4.42), it follows that
which shows that
Case 2: If
Considering equations (4.44) and (4.45), a direct calculation shows that
By differentiating (4.44) with respect to
From (4.45) and (4.47), it follows that
Differentiating (4.48) with respect to
However, since
In view of (4.50) and (4.51), a simple computation shows that
which implies that
is a transcendental, which is a contradiction as
Proof of Theorem 3.4
Suppose that
To complete the proof, it is sufficient to discuss the following cases.
Case 1: If
From (4.53), it is clear that
From (4.54) and (4.55), we obtain
The characteristic equations of (4.56) are as follows:
Using the initial conditions:
where
In view of
From (4.57) and using
Case 2: If
Subtracting (4.59) from (4.58), we see that
Since
Therefore, it follows from (4.60) and (4.61) that
The equation (4.62) can be written as
Otherwise, we obtain
and this shows that the polynomial
If
then we have
Otherwise, we have
which implies that
where
By the similar argument being used in [39], using second fundamental theorem of Nevanlinna for several complex variables, we obtain
Similarly, it can be shown that
That is,
is satisfied for all
or,
If
then we have
Since
Now, we claim that
It follows that
Next, we aim to show that
and
In view of (4.66) and (4.67), it follows that
which implies that
and
By substituting
The characteristic equations of (4.68) are
By using the initial conditions:
where
Consequently, we have
which can be written as follows:
Substituting (4.70) into (4.58), it is easy to verify that
This completes the proof.□
Proof of Theorem 3.5
Suppose that
Below, we will analyze two cases.
Case 1: If
In view of (4.71), it is easy to see that
A simple computation using (4.72) and (4.73) shows that
Differentiating (4.72) with respect to
Thus, it follows from (4.73) and (4.75) that
By differentiating (4.76) with respect to
In view of (4.74) and (4.77), it is easy to see that
In view of (4.74), using
where
Case 2: If
In view of (4.79) and (4.80), we obtain
By differentiating (4.79) with respect to
Thus, it follows from (4.80) and (4.82) that
By differentiating (4.83) with respect to
This can be formulated as follows:
Equation (4.84) can be written as
Otherwise, we obtain
This shows that
If
then from (4.84) it is easy to see that
Otherwise, it follows that
and a simple computation shows that
where
By the similar argument being used previously, in view of the second fundamental theorem of Nevanlinna for several complex variables, we obtain
Similarly, we obtain
That is,
for all
or
If
then it is easy to see that
Since
It follows that the polynomial
We claim that
and
Combining with the fact that
and
From (4.90) and (4.91), a simple computation shows that
and
Furthermore, we observe that equation (4.83) can be formulated as follows:
that is,
Therefore, a simple computation shows that
where
Through the substitution of (4.93) into (4.79), a simple computation leads to
By comparing the coefficients of
This completes the proof.□
Acknowledgement
The authors express their gratitude to the referee for the helpful suggestions and insightful comments aimed at improving the exposition of the article. The first named author is supported by the DST FIST (SR/FST/MS-II/2021/101(C)), Department of Mathematics, Jadavpur University. The second named author is supported by CSIR-SRF (File No: 09/0096(12546)/2021-EMR-I, dated: 08/10/2024), Government of India, New Delhi.
-
Funding information: Authors state no funding involved.
-
Author contributions: Both authors have contributed equally to the manuscript.
-
Conflict of interest: The authors declare that there is no conflict of interest regarding the publication of this article.
-
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
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- Nonlinear heat equation with viscoelastic term: Global existence and blowup in finite time
- New Jensen's bounds for HA-convex mappings with applications to Shannon entropy
- Special Issue on Approximation Theory and Special Functions 2024 conference
- Ulam-type stability for Caputo-type fractional delay differential equations
- Faster approximation to multivariate functions by combined Bernstein-Taylor operators
- (λ, ψ)-Bernstein-Kantorovich operators
- Some special functions and cylindrical diffusion equation on α-time scale
- (q, p)-Mixing Bloch maps
- Orthogonalizing q-Bernoulli polynomials
- On better approximation order for the max-product Meyer-König and Zeller operator
- Moment-based approximation for a renewal reward process with generalized gamma-distributed interference of chance
- Special Issue on Variational Methods and Nonlinear PDEs
- A note on mean field type equations
- Ground states for fractional Kirchhoff double-phase problem with logarithmic nonlinearity
- Solution of nonlinear Langevin equations involving Hilfer-Hadamard fractional order derivatives and variable coefficients
Articles in the same Issue
- Research Articles
- On approximation by Stancu variant of Bernstein-Durrmeyer-type operators in movable compact disks
- Circular n,m-rung orthopair fuzzy sets and their applications in multicriteria decision-making
- Grand Triebel-Lizorkin-Morrey spaces
- Coefficient estimates and Fekete-Szegö problem for some classes of univalent functions generalized to a complex order
- Proofs of two conjectures involving sums of normalized Narayana numbers
- On the Laguerre polynomial approximation errors and lower type of entire functions of irregular growth
- New convolutions for the Hartley integral transform imbedded in the Banach algebras and convolution-type integral equations
- Some inequalities for rational function with prescribed poles and restricted zeros
- Lucas difference sequence spaces defined by Orlicz function in 2-normed spaces
- Evaluating the efficacy of fuzzy Bayesian networks for financial risk assessment
- Fixed point results for contractions of polynomial type
- Estimation for spatial semi-functional partial linear regression model with missing response at random
- Investigating the controllability of differential systems with nonlinear fractional delays, characterized by the order 0 < η ≤ 1 < ζ ≤ 2
- New forms of bilateral inequalities for K-g-frames
- Rate of pole detection using Padé approximants to polynomial expansions
- Existence results for nonhomogeneous Choquard equation involving p-biharmonic operator and critical growth
- Note on the shape-preservation of a new class of Kantorovich-type operators via divided differences
- Geršhgorin-type theorems for Z1-eigenvalues of tensors with applications
- New topologies derived from the old one via operators
- Blow up solutions for two-dimensional semilinear elliptic problem of Liouville type with nonlinear gradient terms
- Infinitely many normalized solutions for Schrödinger equations with local sublinear nonlinearity
- Nonparametric expectile shortfall regression for functional data
- Advancing analytical solutions: Novel wave insights and methodologies for beta fractional Kuralay-II equations
- A generalized p-Laplacian problem with parameters
- A study of solutions for several classes of systems of complex nonlinear partial differential difference equations in ℂ2
- Towards finding equalities involving mixed products of the Moore-Penrose and group inverses by matrix rank methodology
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- Coefficient functionals for Sakaguchi-type-Starlike functions subordinated to the three-leaf function
- Solutions of several general quadratic partial differential-difference equations in ℂ2
- Inequalities for the generalized trigonometric functions with respect to weighted power mean
- Optimization of Lagrange problem with higher-order differential inclusion and special boundary-value conditions
- Hankel determinants for q-starlike functions connected with q-sine function
- System of partial differential hemivariational inequalities involving nonlocal boundary conditions
- A new family of multivalent functions defined by certain forms of the quantum integral operator
- A matrix approach to compare BLUEs under a linear regression model and its two competing restricted models with applications
- Weighted composition operators on bicomplex Lorentz spaces with their characterization and properties
- Behavior of spatial curves under different transformations in Euclidean 4-space
- Commutators for the maximal and sharp functions with weighted Lipschitz functions on weighted Morrey spaces
- A new kind of Durrmeyer-Stancu-type operators
- A study of generalized Mittag-Leffler-type function of arbitrary order
- On the approximation of Kantorovich-type Szàsz-Charlier operators
- Split quaternion Fourier transforms for two-dimensional real invariant field
- Review Article
- Characterization generalized derivations of tensor products of nonassociative algebras
- Special Issue on Differential Equations and Numerical Analysis - Part II
- Existence and optimal control of Hilfer fractional evolution equations
- Persistence of a unique periodic wave train in convecting shallow water fluid
- Existence results for critical growth Kohn-Laplace equations with jumping nonlinearities
- Monotonicity and oscillation for fractional differential equations with Riemann-Liouville derivatives
- Nontrivial solutions for a generalized poly-Laplacian system on finite graphs
- Stability and bifurcation analysis of a modified chemostat model
- Special Issue on Nonlinear Evolution Equations and Their Applications - Part II
- Analytic solutions of a generalized complex multi-dimensional system with fractional order
- Extraction of soliton solutions and Painlevé test for fractional Peyrard-Bishop DNA model
- Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part II
- Some fixed point results with the vector degree of nondensifiability in generalized Banach spaces and application on coupled Caputo fractional delay differential equations
- On the sum form functional equation related to diversity index
- Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part II
- Simpson, midpoint, and trapezoid-type inequalities for multiplicatively s-convex functions
- Converses of nabla Pachpatte-type dynamic inequalities on arbitrary time scales
- Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part II
- Energy decay of a coupled system involving a biharmonic Schrödinger equation with an internal fractional damping
- Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part II
- Nonlinear heat equation with viscoelastic term: Global existence and blowup in finite time
- New Jensen's bounds for HA-convex mappings with applications to Shannon entropy
- Special Issue on Approximation Theory and Special Functions 2024 conference
- Ulam-type stability for Caputo-type fractional delay differential equations
- Faster approximation to multivariate functions by combined Bernstein-Taylor operators
- (λ, ψ)-Bernstein-Kantorovich operators
- Some special functions and cylindrical diffusion equation on α-time scale
- (q, p)-Mixing Bloch maps
- Orthogonalizing q-Bernoulli polynomials
- On better approximation order for the max-product Meyer-König and Zeller operator
- Moment-based approximation for a renewal reward process with generalized gamma-distributed interference of chance
- Special Issue on Variational Methods and Nonlinear PDEs
- A note on mean field type equations
- Ground states for fractional Kirchhoff double-phase problem with logarithmic nonlinearity
- Solution of nonlinear Langevin equations involving Hilfer-Hadamard fractional order derivatives and variable coefficients
