Electronic Journal of Applied Mathematics
eISSN: 2980-2474 Publisher: Mersin University Journal name: Electronic Journal of Applied Mathematics Abbreviation: Electron. J. Appl. Math. Parallel title: EJAM Frequency: Quarterly (Mar, Jun, Sep, and Dec) Electronic Journal of Applied Mathematics (Electron. J. Appl. Math.) is a peer reviewed, Open Access journal that provides an advanced forum for studies related to mathematics. All published items, including research articles, have unrestricted access and users are allowed to read, download, copy, distribute, print, search, or link to the full texts of the articles in this journal without asking prior permission from the publisher or the author. No submission or processing fees are required.
Department of Library and Documentation, Mersin University
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New solitary wave solutions of the Korteweg-de Vries (KdV) equation by new version of the trial equation method
New solitary wave solutions for the Korteweg-de Vries (KdV) equation by a new version of the trial equation method are attained. Proper transformation reduces the Korteweg-de Vries (KdV) equation to a quadratic ordinary differential equation that is fully integrated using the new version trial equation approach. The family of solitary wave solutions of the reduced equation ensures a combined expression for the Korteweg-de Vries (KdV) equation, which contains exact solutions derived in recent years using different integration methods. The analytic solution of the reduced equation permits to find exact solutions for the Korteweg-de Vries (KdV) equation, providing a variety of new solitary wave solutions that have not been reported before.
New version of the trial equation method
nonlinear partial differential equations
Copyright (c) 2023 Yusuf Pandir (Corresponding Author); Ali Ekin https://creativecommons.org/licenses/by/4.0
Blow-up solutions to the fractional solid fuel ignition model
In this study, we investigate a time fractional parabolic equation with Caputo-Fabrizio derivative and a pure exponential source term. The model can be seen as a modified version of a solid fuel ignition by considering delay effects. Because of the bad behavior of the source term, the common Banach fixed point theorem is not suitable to be applied. Therefore, we approach by an iteration method in which we find a supersolution to the problem. Then, the monotony of approximating solutions implies the existence of a mild solution. Furthermore, we show that if the given initial data is sufficiently large in term of norm measure, the corresponding solution will blow up in a finite time. Main techniques of the work are mainly based on calculations related to explicit formulas of solution operator derived from eigenpair of Helmholtz's equation and Sobolev embeddings between Hilbert scale spaces.
Copyright (c) 2023 Nguyen Dinh Huy; Anh Tuan Nguyen (Corresponding Author) https://creativecommons.org/licenses/by/4.0
Regularization of nonlocal pseudo-parabolic equation with random noise
In this paper, we consider an inverse problem for a time-fractional diffusion equation with the inhomogeneous source. These problems have many applications in engineering such as image processing, geophysics, biology. We get the result in random case as follows: • This problem is ill-posed. • We have used the nonlocal condition, instead of the final time condition. • Using the IFT regularization method, constructing the regularized solution, the a-priori choice rule for the regularization parameter is discussed and yields the corresponding convergence rate. • A numerical experiment is presented to illustrate the results in theory.
Copyright (c) 2023 Yusuf Gurefe; Le Dinh Long (Corresponding Author); Devendra Kumar https://creativecommons.org/licenses/by/4.0
On maximal solution to a degenerate parabolic equation involving a time fractional derivative
In this paper, we consider a degenerate parabolic equation associated with Caputo derivative. Our problem is studied in the unbounded domain and the nonlocal initial condition. Under some suitable conditions of the input data, we show the local existence of the mild solution. Then we show the continuation of the mild solution. We also claim the maximal mild solution. The main analysis in the current paper is based on some estimations of resolvent theory combined with many complex valuations on solutions operators in Banach spaces.
degenerate parabolic equation, fractional integrodifferential equation, mild solution, maximal solution
Copyright (c) 2023 Bui Dai Nghia; Nguyen Hoang Luc (Corresponding Author); Xiaolan Qin, Yan Wang https://creativecommons.org/licenses/by/4.0
Non-uniform trajectory tracking AILC for nonlinear time-varying parameter system with multiple unknown control directions
In this paper, a new iterative learning controller is presented for nonlinear time-varying parameter system to solve the tracking problem of different target trajectories. Based on Nussbaum function and the Lyapunov-like synthesis design the learning controller to handle system dynamics with multi-unknown control directions. Over a finite time interval, the unknown time-varying parameter is considered to be periodic, so it is expanded using a Fourier series expansion, and the remaining terms are treated with a canonical series. The controller designed in this paper can ensure that all signals of the closed-loop system are bounded in a finite time interval [0,T], and can complete non-uniform target tracking. Finally, a simulation example is given to verify the effectiveness of the designed controller.
Copyright (c) 2023 Chunli Zhang (Corresponding Author); Yangjie Gao, Lei Yan https://creativecommons.org/licenses/by/4.0
Well-posedness results for a class of nonlinear reaction-diffusion equations with memory
In this paper, we are interested to study the initial value problem for nonlinear heat equation with memory. For our problem, we show that the local existence theory related to the finite time blow-up is also obtained for the problem with logarithmic nonlinearity. We also obtain the blowup continuation property of the mild solution. The principal techniques based on some Sobolev embeddings with \(L^p\) spaces.
Copyright (c) 2023 Ho Duy Binh; Nguyen Duc Phuong (Corresponding Author); Vo Viet Tri https://creativecommons.org/licenses/by/4.0
Fishery model influenced by global warming with optimal harvesting policy
In this article, we proposed a modified version of the Gordon-Schaefer model in which harvesting is non-linear and the mortality rate of the fish stock is a function of temperature. We showed that every solution of the system is globally bounded and that there is a single interior equilibrium point that is locally, asymptotically, and globally stable under certain conditions. We then determined the optimal levels of production and profit when the stock evolution is kept constant (MSY and MEY, respectively). Furthermore, using Pontryagin's maximum principle, we characterized the optimal harvesting policy that maximizes net present value. Finally, we performed numerical simulations to validate our theoretical results.
Copyright (c) 2025 Josué Carlos Israël OULAI (Corresponding Author); Hypolithe OKOU, Albin Tetchi N'GUESSAN https://creativecommons.org/licenses/by/4.0
A delayed eco-epidemiological prey–predator model and its discrete version under the weak Allee effect
This study looks into a delayed predator–prey model that includes competition within the hunter species and a strong Allee effect in the prey population. The time delay is used as a bifurcation measure to look at how stable the coexistence equilibrium is. Our findings show that a Hopf splitting can happen when the delay goes over a certain level. The normal for \(m\) theory and center manifold method is used to find the direction and stability of the periodic solutions that are made. Number simulations add to the evidence for the mathematical findings.
Copyright (c) 2025 Kalyan Das (Corresponding Author); G.Ranjith Kumar https://creativecommons.org/licenses/by/4.0
Blow-up solutions to logarithmic viscoelastic equations with nonlocal weak damping, delay, and acoustic boundary conditions
We in this paper consider the blow-up behavior of any nontrivial solution to the following initial boundary value problem of the logarithmic viscoelastic equations with delay and nonlocal weak damping terms under acoustic boundary conditions \[\begin{aligned} u_{tt} - \Delta u + \|u_t\|^k_2u_t &+ \int^t_0\omega(t-s)\Delta u(s)ds +c_1|u_t(t)|^{m-2}u_t(t)\\ & +c_2|u_t(t-\tau)|^{m-2}u_t(t-\tau) =|u|^{p-2}u\ln |u|. \end{aligned} \] Our results show that the local solution blows up in finite time if $p > m$ and $E(0) < 0$. More accurately, we give the upper bound of the blow-up time by the energy method. Moreover, we also give the lower bound of the blow-up time.
Copyright (c) 2025 Qianqian Luo; Jisong Duan (Corresponding Author) https://creativecommons.org/licenses/by/4.0
Some fixed point theorems of Leggett–Williams type for multivalued mappings and applications to second-order differential inclusions
This paper examines the application of fixed point index theory to set-valued mappings in ordered Banach spaces, with a specific focus on Leggett-Williams type fixed point theorems. The study provides novel conditions for the existence of one and multiple fixed points under a range of assumptions. Additionally, it applies these results to prove the existence of positive solutions for the second-order differential inclusion problem \( - x''(t) \in f\left( {t,x(t)} \right),t \in [0,1]\) with boundary conditions \(x(0)=0=x'(1)\). The obtained results contribute to expanding the scope of application of fixed point theory, providing new techniques for solving boundary value problems in different cases.
fixed point theorem of Leggett--Williams type
Copyright (c) 2025 Nguyen Dang Quang https://creativecommons.org/licenses/by/4.0
Finite time blow-up for an inhomogeneous wave equation with Riemann--Liouville fractional integral
This paper is devoted to the study of the effect of an inhomogeneity \( \omega(x) \) on the blow-up phenomena for the wave equation \[ u_{tt} - \triangle u = I_{0^{+}}^{\alpha}(|u|^{p}) + \omega(x), \quad (t,x) \in (0,T) \times \mathbb{R}^{N}, \]where \( N \ge 1 \), \( p > 1 \), \( 0 \le \alpha < 1 \), and \( \omega(x) \) satisfies \( \int_{\mathbb{R}^{N}} \omega(x)\,dx > 0 .\) Specifically, by the method of cut-functions, this paper first proves that any non-trivial solution blows up in finite time under \( 0 < \alpha < 1 \), and later proves that any non-trivial solution blows up in finite time under \( \alpha = 0 \), \( 2 \le N \le 4 \), and \( p \) being the Strauss exponent.
Copyright (c) 2025 Changyong Xiang (Corresponding Author); Jisong Duan, Qianqian Luo https://creativecommons.org/licenses/by/4.0
Positive eigenvalues for the adjoint of an increasing operator
This paper develops a general framework for investigating the eigenvalue problem of the extended adjoint mapping \(A^\ast:E_b^\ast\to E_b^\ast\) associated with a possibly nonlinear operator \(A:E\to E\). By employing the theory of relative topological degree for condensing mappings with respect to a measure of noncompactness, we establish sufficient conditions ensuring the existence of positive eigenvalues of \(A^\ast\). The extended dual space \(E_b^\ast\) is strictly larger than the classical dual \(E^\ast\), and the two coincide only when \(E\) is finite-dimensional. In this special case, our results recover the classical eigenvalue theory for linear compact operators. The framework introduced here extends the topological degree approach to a broader nonlinear setting and provides a foundation for further applications to convex processes and control problems.
Copyright (c) 2025 Nguyen Hoang Tuan, Vo Viet Tri; Nguyen Anh Tuan (Corresponding Author) https://creativecommons.org/licenses/by/4.0
Large deviation principle for stochastic \(p\)-Laplacian reversible Selkov lattice systems
The Selkov system is a classical model for autocatalytic biochemical reactions such as glycolysis and exhibits complex spatio-temporal patterns including self-excitation, oscillations, and spatial chaos. In this paper, we consider stochastic \(p\)-Laplacian reversible Selkov lattice systems defined on the integer set \(\mathbb{Z}\), which possess two pairs of oppositely signed nonlinear terms and whose nonlinear couplings can grow polynomially with any order \(q \geq 1\), the inherent structure of the system precludes the possibility of any unidirectional dissipative influence arising from the interaction between the two coupled equations, thereby obstructing the emergence of a dominant energy-dissipation mechanism along a single directional pathway. In particular, our focus is on studying the asymptotic properties of the \((\psi^\gamma, \phi^\gamma)\) as the noise intensity \(\gamma \to 0\) for the systems. Based on the well-posedness and the convergence of the solutions of the controlled system associated with the considered system, by the weak convergence method, we establish the large deviation principle in \( C([0,T], \ell^2 \times \ell^2) \) for such infinite dimensional system. One of the advantages of the weak convergence method that does not rely on uniform exponential probability estimates of solutions. Moreover, a stopping time technique is used to prove \(\lim\limits_{\gamma \to 0} \left( \psi^\gamma_{z^\gamma} - \psi_{z^\gamma}, \phi^\gamma_{z^\gamma} - \phi_{z^\gamma} \right) = 0\) in probability in order to overcome the difficulty caused by the local monotonicity of the nonlinear and diffusion terms. Compared to the classical case of cubic nonlinearity, our results are not only more general in scope but also demonstrate a higher degree of novelty.
Copyright (c) 2025 Liguang Zhou, Liheng Liu, Zhenqin Nie; Jibing Leng (Corresponding Author) https://creativecommons.org/licenses/by/4.0
A priori strategy for pseudo-parabolic equations by hybrid regularization method
In this study, we address the simultaneous recovery of the source term and the initial value in a time-fractional diffusion equation, a problem inherently characterized by its ill-posed nature. To overcome this challenge, we propose a hybrid regularization method, offering robust solutions and providing precise estimations for both the source term and the initial value.
Copyright (c) 2025 Le Dinh Long, Ngo Ngoc Hung https://creativecommons.org/licenses/by/4.0
The Continuity by Order Derivative for Conformable Parabolic Equations with Exponential Nonlinearity
In this article, we examine the continuity according to the derivative order of conformable parabolic equations. We establish the existence and uniqueness of mild solutions to the problem of source function exponential non-linearity. We prove the existence and uniqueness of a mild solution based on some Sobolev embeddings, the Banach space of all continuous functions, and the Banach fixed point theorem. In addition, we will address the nonlinear model's continuity issue and demonstrate the mild solution's convergence to the nonlinear problem when \(\gamma'\to \gamma\).
Copyright (c) 2025 Ho Duy Binh, Nguyen Dinh Huy; Nguyen Van Tien (Corresponding Author); Le Dang Khoa https://creativecommons.org/licenses/by/4.0
Weak solutions of Dirichlet discrete nonlinear problems in a two-dimensional Hilbert space
In this paper we prove the existence of at least one weak solution of a discrete nonlinear Dirichlet boundary-value problem in a two-dimensional Hilbert space. The main existence results based on variational approach, specially minimization methods.
Copyright (c) 2025 A.A.K. Dianda (Corresponding Author); Yassia Ouedraogo, Malick Zoungrana https://creativecommons.org/licenses/by/4.0
Analytical Solution of the D-DimensionalKlein-Gordon Equation for Dan-Fan MolecularPotential Using AIM
In this article, the D-dimensional Klein-Gordon equation within the framework of improved Greene-Aldrich approximations scheme for Deng-Fan molecular potential is solved for s-wave and arbitrary angular momenta. The energy eigenvalues and corresponding wave functions are obtained in an exact analytical manner by using asymptotic iteration method.
improved Greene-Aldrich approximation scheme
Copyright (c) 2025 Bijon Biswas https://creativecommons.org/licenses/by/4.0
Two-weighted inequalities for maximal commutators in generalized weighted Morrey spaces on spaces of homogeneous type
In this paper we give a characterization of two-weighted inequalities for maximal commutators in generalized weighted Morrey spaces on spaces of homogeneous type \(\mathcal{M}_{\omega }^{p,\varphi }(X)\). We prove the boundedness of maximal commutators \([M,b]\) from the spaces \(\mathcal{M}_{\omega _{1}^{\delta }}^{p,\varphi _{1}}(X)\) to the spaces \(\mathcal{M}_{\omega _{2}^{\delta }}^{p,\varphi _{2}}(X)\), where \(1<p<\infty \), \(0<\delta <1\) and \((\omega _{1},\omega _{2})\in \widetilde{A}_{p}(X)\).
Copyright (c) 2023 Ayşenur Aydoğdu; Canay Aykol (Corresponding Author); Javanshir J. Hasanov https://creativecommons.org/licenses/by/4.0
Geometry of Solutions of the Geometric Curve Flows in Space
In this study, we aim at investigating the geometry of surfaces corresponding to the geometry of solutions of the geometric curve flows in Euclidean 3-space \(\mathbb R^3\) considering the Frenet frame. In particular, we express some geometric properties and some characterizations of \(u\)-parameter curves and \(t\)-parameter curves of some trajectory surfaces including the Hasimoto surface, the shortening trajectory surface, the minimal trajectory surface, the \(\sqrt{\tau}\)-normal trajectory surface in \(\mathbb R^3\).
the \(\sqrt{\tau}\)-normal trajectory surface
Copyright (c) 2023 Zehui Zhao; Shiping Zhong (Corresponding Author); Xinjie Wan https://creativecommons.org/licenses/by/4.0
Recovering solution of the reverse nonlinear time fractional diffusion equations with fluctuations data
In this study, our focus is on obtaining an estimated solution for the nonlinear fractional time diffusion equation. Specifically, we have utilized the Riemann Liouville fractional derivative. Additionally, we have concerned Gaussian white noise in the input data. As we are aware, this problem is considered ill-posed according to Hadamard's definition. To tackle this problem, we have proposed a regularized solution and demonstrated the convergence between the mild solution and the regularized solution.
Copyright (c) 2023 Doan Thi Thanh Xuan; Vo Thi Thanh Ha (Corresponding Author) https://creativecommons.org/licenses/by/4.0
Optical soliton solutions of the generalized sine-Gordon equation
In this study, the extended trial equation method based on the general form of the nonlinear elliptic ordinary differential equation is employed to solve the nonlinear generalized sine-Gordon equations. By using this method, we achieve, unlike new types of exact wave solutions such as Elliptic-F, Elliptic-E, and Elliptic-Π functions that are known as elliptic integrals.
Copyright (c) 2023 Yusuf Pandir; Humaira Yasmin (Corresponding Author) https://creativecommons.org/licenses/by/4.0
The Traveling Wave Solutions to a Variant of the Boussinesq Equation
In this study, we study a variant of the Boussinesq equation called as \(B( n+1,\,1,\,n )\) equation, and construct some traveling wave solutions by using an effective approach called the extended trial equation method. Thus, the soliton solutions, rational function solutions, elliptic function solutions and Jacobi elliptic function solutions, which show the existence of various mathematical and physical structures and events in the fundamental equation considered, have been constructured. In order to make a more detailed examination of the physical behavior of these solutions, two- and three-dimensional graphs of some solution functions were drawn with the help of the Mathematica package program. In the section of Discussion, we suggest a more general version of the trial equation method for nonlinear differential equations.
Copyright (c) 2023 Muhammad Nadeem (Corresponding Author); Loredana Florentina Iambor https://creativecommons.org/licenses/by/4.0
Weak Mean Random Attractor of Reversible Selkov Lattice Systems Driven by Locally Lipschitz Lévy Noises
This paper is concerned with weak pullback mean random attractor of reversible Selkov lattice systems defined on the entire integer set \(\mathbb{Z}\) driven by locally Lipschitz Lévy noises. Firstly, we formulate the stochastic lattice equations to an abstract system defined in the non-concrete space \(\ell^2\times\ell^2\) of square-summable sequences. Secondly, we establish the global well-posedness of the systems with locally Lipschitz diffusion terms. Under certain conditions, we show that the long-time dynamics can be captured by a weakly compact and weakly attracting mean random attractor in the Bochner space \(L^2(\Omega,\ell^2\times\ell^2)\). To overcome the difficulty caused by the drift and diffusion terms, we adopt a stopping time technique to prove the convergence of solutions in probability. The mean random dynamical systems theory proposed by Wang (J. Differ. Equ., 31:2177-2204, 2019) is used to deal with the difficulty caused by the nonlinear noise.
Copyright (c) 2024 Guofu Li, Xiulan Yang, Liguang Zhou; Yan Wang (Corresponding Author) https://creativecommons.org/licenses/by/4.0
Global Attractors of the Delay 2D Navier-Stokes Equations on Unbounded Channel-Like Domains
This paper studies the global attractors of 2D Navier-Stokes equations with delay defined in unbounded Channel-like domains. We establish the uniform tail-ends estimates of the solutions by establishing all the solutions are uniformly small for overcome the non-compactness of the solutions.
Copyright (c) 2024 Zhang Zhang, Xiaobin Yao (Corresponding Author) https://creativecommons.org/licenses/by/4.0
Dynamics of a Diffusive Two Predators-One Prey System
This paper analyses a diffusive predator-prey model consisting of a single prey species and two predator species with modified Leslie-Gower term Holling type II functional response subject to the homogeneous Neumann boundary condition. Local stability condition is derived by the application of Routh-Hurwitz criterion. Global asymptotic stability of the unique positive steady state is shown by constructing a suitable Lyapunov function when self diffusion is allowed where as non-constant positive steady states can exist due to the presence of cross-diffusion, that means, cross-diffusion can induce stationary pattern. Taking the cross diffusion as a bifurcation parameter, one can show the existence of positive non-constant solutions with the help of bifurcation theory. A brief conclusion completes the paper.
Copyright (c) 2023 Debasis Mukherjee https://creativecommons.org/licenses/by/4.0
Analytical Solutions of the D-Dimensional Klein-Gordon Equation with Q-Deformed Modified Pöschl-Teller Potential
In this article, the D-dimensional Klein-Gordon equation within the framework of Greene-Aldrich approximations scheme for q-deformed modified P¨oschl-Teller Potential is solved for s-wave and arbitrary angular momenta. The energy eigenvalues and corresponding wave functions are obtained in an exact analytical manner via the Nikiforov-Uvarov (N-U) method. Further, it is shown that in the non-relativistic limit, the energy eigenvalues reduce to that of Schrodinger equations for the potential. It is also shown that, the obtained results lead to the solutions of the same problem for modified P¨oschl-Teller potential for \(q = 1\).
q-deformed modified P¨oschl-Teller potential, Greene-Aldrich approxi15 mation, Nikiforov-Uvarov (N-U) method
Copyright (c) 2024 Bijon Biswas https://creativecommons.org/licenses/by/4.0
On the Existence of Flux as a Function of the Surface Elevation for Long Wave Solution of Shallow Water Equations
The shallow water equations in mechanics of fluids, govern the motion of a shallow layer of water over a fixed impervious bed. In this paper, the bed form is assumed to be rough and horizontal, and the motion of water is assumed to be of the long wave type (Lamb [1], pp. 254-256) such that the free surface has a gradually varying propagating profile. Gravitation permits such motion but is resisted by the turbulence generated by the bed friction. A model of the governing equations based on the Reynolds averaged Navier-Stokes equations has recently been given by Bose [2], which is highly nonlinear. A heuristic approach of numerically solving the equations for the modified long waves is also presented in that article, by assuming that the horizontal flux across a section of flow is some function of the free surface elevation alone. This key assertion is analysed in this article and proved to hold provided some boundedness criteria are satisfied by the flux gradients. The theory is apparently applicable to find appropriate boundedness conditions on the flux of flow for numerically solving long wave equations in the case of other models for long wave propagation as well.
Copyright (c) 2024 Sujit K.Bose; Ganesh C. Gorain (Corresponding Author) https://creativecommons.org/licenses/by/4.0
Dynamics of a Two Competing Prey-One Predator System with Fear Effect and Anti-Predator Behaviour
This paper examines the effect of fear and anti-predator behaviour in a two competing prey and one predator system. Here, one of the competing prey species is affected by the fear of predators while the other competing prey species shows anti-predator behaviour. Basic results on positivity, the boundedness of the solutions, local and global stability of the coexistence equilibrium point, uniform persistence, Hopf bifurcation and the nature of the limit cycle emerging through Hopf bifurcation are investigated. Numerical examples are provided to support our obtained results.
Copyright (c) 2024 Debasis Mukherjee https://creativecommons.org/licenses/by/4.0
Existence of Solutions for a Class of Kirchhoff-Type Equations with Indefinite Potential
This study explores the existence of solutions to the following Kirchhoff-type problem \[ \left\{ \begin{array} [c]{ll}% -\left(a+b\int_{\mathbb{R}^3}|\nabla u|^2dx\right)\Delta u+ V(x)u=f(x,u),~{\rm{in}}~ \mathbb{R}^{3},\\ u\in H^1(\mathbb{R}^3),% \end{array} % \right. \] where a and b are postive constants, and the potential \(V(x)\) is continuous and indefinite in sign. With suitable assumptions on \(V(x)\) and \(f\), we establish the existence of solutions using the Symmetric Mountain Pass Theorem.
Copyright (c) 2024 Linlian Xiao, Jiaqian Yuan; Jian Zhou (Corresponding Author); Yunshun Wu https://creativecommons.org/licenses/by/4.0
Exponential Stability of Viscoelastic Structure with Second Sound
This manuscript deals with a thermo-viscoelastic system describing the vibrations of a flexible structure. We study this structure's stabilization problem when subjected to Kelvin-Voigt damping and the second sound. Cattaneo-Vernotte's law governs the thermal effect, eliminating the physical paradox of equal speed of waves in the classical thermoelastic theory. Semigroup theory proves the solution's existence and uniqueness. Exponential stability is proved by the energy method.
Copyright (c) 2024 Ganesh C. Gorain; Carlos Raposo (Corresponding Author); Octavio Vera, Srikanta Ram https://creativecommons.org/licenses/by/4.0