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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A strategy for inverse source term and the initial data for the time-fractional diffusion equation for the Hyper Bessel operator
In this paper, we investigate the inverse problem of simultaneously identifying the source term and the initial value for a time-fractional diffusion equation involving a Caputo-like hyper-Bessel operator. We first establish the ill-posedness of the problem and derive a conditional stability estimate. To overcome this difficulty, we apply the modified Fractional Landweber regularization method and provide error estimates under a priori parameter choice rules.
Copyright (c) 2025 Vo Viet Tri https://creativecommons.org/licenses/by/4.0
Convergence of physics-informed neural networks for time-fractional diffusion equations via stability and generalization bounds
Physics-informed neural networks (PINNs) have emerged as a flexible framework for solving partial differential equations (PDEs), yet rigorous convergence results remain limited for models with nonlocal time dependence. This paper studies PINNs for the time-fractional diffusion equation with the Caputo derivative of order \(\alpha\in(0,1)\). We propose a residual-based PINN formulation and establish a stability-driven error bound: the solution error is controlled by the PDE residual together with the boundary and initial condition residuals. By combining this PDE stability with sampling-based generalization bounds for the empirical PINN loss, we prove convergence of empirical minimizers to the true solution as the number of collocation points increases and the approximation and optimization errors vanish. Numerical experiments with manufactured solutions support the theory and demonstrate accuracy.
Copyright (c) 2025 Trinh Phuoc Toan; Huynh Huuu Dinh (Corresponding Author) https://creativecommons.org/licenses/by/4.0
Fishery model influenced by pollution with optimal harvesting policy
In this work, we proposed a simplified version of the Gordon-Schaefer model in which the mortality rate of the fish stock biomass and the unit cost of effort are due to aquatic pollution, and harvesting is non-linear. Our dynamic study shows that our system is globally bounded and asymptotically and globally stable under certain conditions around the single interior equilibrium point. We then performed a numerical simulation to validate our theoretical results. Next, we determined the maximum sustainable yield (MSY) and maximum equilibrium yield (MEY), and presented numerical results. Finally, using Pontryagin's maximum principle, we determined the optimal control and performed a numerical simulation.
Copyright (c) 2025 Josué Carlos Israël OULAI (Corresponding Author); Hypolithe OKOU, David Paul TUO https://creativecommons.org/licenses/by/4.0
The effect of fear on two predator-prey pairs linked by competition
This paper presents a mathematical model examining the dynamics of a linked predator-prey system that incorporates fear. Here each predator consumes one prey only. and the preys are in competition. The nature of predation is of Holling type II. The principal results explore various aspects , including positivity, boundedness, local and global stability of the coexistence equilibrium point, uniform persistence and Hopf bifurcation. The numerical simulations support the theoretical findings, offering practical insights into the model's behaviour.
Copyright (c) 2025 Debasis Mukherjee 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
Efficient coupling technique for approximate and exact solutions of nonlinear Caputo fractional differential equations
Complex phenomena such as anomalous diffusion, fractal structures, viscoelasticity, and chaotic dynamics are effectively described using fractional-order nonlinear models. Solving these nonlinear fractional models provides deeper insight into the mechanisms and behaviors underlying such processes. In this work, we develop a hybrid analytical technique for solving nonlinear fractional partial differential equations by coupling the residual function with the Laplace transform, referred to as the Laplace residual power series method (LRPSM). This method is constructed upon a modified fractional-order power series formulated within the framework of the Caputo derivative. The accuracy of the proposed approach is confirmed through absolute, relative, and residual error analyses. LRPSM efficiently determines the coefficients of series solutions using a simple limit principle at infinity, in contrast to well-established methods such as the variational iteration method, Adomian decomposition method, and homotopy perturbation method, which typically require complex integrations. Likewise, the classical residual power series method (RPSM) depends on higher-order derivatives, which become increasingly challenging in fractional settings. Since LRPSM avoids the use of Adomian's or He's polynomials for handling nonlinearities, it proves to be a more straightforward and computationally efficient alternative for obtaining accurate series solutions to nonlinear fractional problems.
Copyright (c) 2025 Muhammad Imran Liaqat (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
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
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
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
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
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
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