Article
Mathematics
Feiyun Pei, Guojiang Wu, Yong Guo
Summary: The KPI equation, originally used to describe two-dimensional shallow water waves, is a well-known nonlinear evolution equation that has recently found important applications in various fields. Obtaining the exact solutions of the KPI equation is crucial in studying these topics. In this paper, we use a general Riccati equation as an auxiliary equation and solve it through different function transformations to obtain new types of solutions. By applying this method, we construct ten sets of infinite series exact solitary wave solutions for the KPI equation, demonstrating the simplicity and effectiveness of this approach for nonlinear evolution models.
Article
Mathematics, Applied
A. Ghose-Choudhury, Sudip Garai
Summary: This article discusses the use of a comparison method to obtain exact solutions for nonlinear partial differential equations (PDEs) through their traveling wave reductions. The method, proposed by N. A. Kudryashov, is extended to include solutions expressed in terms of both the logistic function and the tanh$$ \tanh $$-class of functions. The article derives the standard set of second-order ordinary differential equations (ODEs) that have the logistic and tanh$$ \tanh $$ functions as solutions and also extends the analysis to third-order cases.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Mathematics, Interdisciplinary Applications
Guojiang Wu, Yong Guo
Summary: In this paper, abundant types of new infinite-series exact solitary wave solutions are constructed using distinct function iteration relations, which have not been reported in other documents. The numerical analysis of some solutions shows complex solitary wave phenomena, with some solutions having stable solitary wave structures and others having singularities in certain space-time positions.
FRACTAL AND FRACTIONAL
(2023)
Article
Physics, Multidisciplinary
I Area, J. J. Nieto
Summary: By using a series of fractional powers, a representation of the solution to the fractional logistic equation is presented and proven to be the exact solution in the simplest case. Numerical approximations demonstrate the good approximations obtained by truncating the fractional power series.
PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS
(2021)
Article
Mathematics, Applied
D. Ramesh Kumar
Summary: The paper aims to establish the existence and uniqueness of the common solution for nonlinear Fredholm integral equations, nonlinear Volterra integral equations, and nonlinear fractional differential equations using common fixed point results. Some common fixed point results satisfying the generalized contraction condition are proved, and an example is provided to support the usability of the result along with numerical experiments.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Jingxi Luo
Summary: A novel modified nonlinear Schrodinger equation is introduced and solved exactly and analytically using a traveling wave ansatz. The soliton solution is characterized in terms of waveform and wave speed, with different parameter settings yielding unique or degenerate waveforms. The equation is identified as a model for describing the propagation of a quantum mechanical exciton through a collectively-oscillating plane lattice, and the physical implications of the soliton solution are discussed.
STUDIES IN APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Chandrasekaran Uma Maheswari, Ramajayam Sahadevan, Munusamy Yogeshwaran
Summary: In this article, the method of separation of variables is used to derive exact solutions to a certain class of time fractional nonlinear partial differential equations and partial differential-difference equations. It is found that this method is effective in solving time fractional nonlinear differential equations with initial and boundary conditions.
FRACTIONAL CALCULUS AND APPLIED ANALYSIS
(2023)
Article
Mathematics, Applied
Juan J. Nieto
Summary: We studied the logistic differential equation of fractional order and non-singular kernel, and obtained the analytical solution.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Engineering, Electrical & Electronic
Sidika Sule Sener Kilic, Ercan Celik, Hasan Bulut
Summary: Fractional calculus is a widely used field in science and engineering, and this study investigates the conformable time-fractional (2 + 1)-dimensional quantum Zakharov-Kuznetsov and time-fractional modified Korteweg-de Vries equations using the modified exp(-O(?))-expansion function approach. New prototype solutions, including hyperbolic and trigonometric function solutions, are obtained and verified. This research is important as it provides new solutions using a strong analytical approach.
OPTICAL AND QUANTUM ELECTRONICS
(2023)
Article
Mathematics, Applied
Yogeshwari F. Patel, Jayesh M. Dhodiya, Dhrien Pandit
Summary: This paper introduces a reliable semi-analytical approach, the reduced differential transform method, to solve the Newell-Whitehead-Segel equation, showing better accuracy and reliability compared to existing methods, and includes error analysis for the obtained series solution.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Review
Physics, Multidisciplinary
Nikolay K. Vitanov
Summary: Exact solutions of nonlinear differential equations are crucial for the theory and practice of complex systems. This review article discusses the Simple Equations Method (SEsM) as a specific methodology for obtaining such solutions. The article provides an overview of the literature and research related to this methodology, and presents the algorithm and applications of the SEsM.
Article
Materials Science, Multidisciplinary
Ya-Shan Guo, Wei Li, Shi-Hai Dong
Summary: Gaussian solitary wave solutions are investigated for a logarithmic nonlinear Schro dlaringer equation (NLSE) with a standard harmonic oscillator potential. It is found that the Gaussian solution does not exist in the straight line x = v t, but in the curve x = v(t). The behavior of the Gaussians and their correlations with relevant parameters are illustrated. These results will contribute to the current literature on this nonlinear equation.
RESULTS IN PHYSICS
(2023)
Article
Mathematics, Interdisciplinary Applications
Erdogan Mehmet Ozkan
Summary: In this work, the F-expansion method is applied to obtain exact solutions of the space-time fractional modified Benjamin Bona Mahony equation and the nonlinear time fractional Schrodinger equation with beta derivative. More solutions defined by the Jacobi elliptic function are obtained with the assistance of Maple.
FRACTAL AND FRACTIONAL
(2022)
Review
Mathematics
Archna Kumari, Vijay K. Kukreja
Summary: This article provides an overview of the widely used Hermite interpolating polynomials and their application in solving various types of differential equations. The use of Hermite interpolation has become an established tool in applied science.
Article
Optics
Yu-Fei Chen
Summary: This study focuses on the generalized nonlinear Schrodinger equation for the propagation of femtosecond pulses in highly-nonlinear optical medium, involving polynomial Kerr nonlinearity and higher-order non-Kerr components. The exact solutions and chirped soliton solutions of the complete types of nonlinear Schrodinger equation are obtained using the polynomial complete discriminant system. The exact solutions can be classified into three types: soliton solutions, singular solutions, and elliptic function double periodic solutions. The model is also visualized with specific parameter values, and the existence of solutions is proven through the classical two-dimensional diagrams of exact solutions and chirped soliton solutions.
Article
Engineering, Multidisciplinary
Nikolay A. Kudryashov, Mikhail Chmykhov, Michael Vigdorowitsch
Summary: This article presents a simple mathematical model for analyzing the spread of infections, focusing on SARS-Cov-2. The model takes into account the processes of infection and recovery/decease, characterized by contact rate and recovery/decease rate. The solution of the model is in the form of a quasi-logistic function, and an infection index is introduced to measure the infection level. Based on data from various sources, a threshold value of the infection index is identified, above which the spread of infection can be considered a pandemic. The article also builds a simplified (two-parameter) SIR model related to the SIS model, with general solutions that are structurally adjusted to match peak infection data.
INTERNATIONAL JOURNAL OF NONLINEAR SCIENCES AND NUMERICAL SIMULATION
(2023)
Article
Mathematics, Applied
Kristina Kan, Nikolay A. Kudryashov
Summary: A system of fourth-order nonlinear differential equations for wave propagation in optical fiber Bragg gratings is considered, taking into account arbitrary refractive index and non-local nonlinearity. The system is transformed into ordinary differential equations using traveling wave variables, and compatibility conditions for the system are defined and analyzed. Exact solutions in the form of solitary waves are found.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Optics
Nikolay A. Kudryashov, Anjan Biswas
Summary: The generalized nonlinear Schrodinger equation with two arbitrary reflective indices is studied in this paper. Transformations for dependent and independent variables are applied to find solitary wave solutions. Parametric optical solitons are discovered for describing the propagation of pulses in nonlinear optics under certain mathematical model conditions.
Article
Optics
Nikolay A. Kudryashov
Summary: This paper considers new generalized Schrodinger equations with polynomial nonlinearities. The inverse scattering transform cannot solve the Cauchy problem for these equations, hence, traveling wave solutions are sought for optical solitons. By using transformations of dependent and independent variables, solutions for nonlinear ordinary differential equations are found. These new auxiliary equations allow for the search for optical solitons of other generalized Schrodinger equations.
Article
Optics
Nikolay A. Kudryashov
Summary: The generalized Schrödinger equation for embedded solitons is analyzed in this paper, and analytical solutions are found using the traveling wave reduction method.
Article
Optics
Nikolay A. Kudryashov
Summary: This paper considers a new generalization of the nonlinear Schrodinger equation with triple refractive index and non-local nonlinearity, and presents a new form of optical solitons. An auxiliary nonlinear differential equation of the first order with a double nonlinearity is introduced to find exact solutions of the nonlinear partial differential equation. The results show the existence of new type optical solitons of the generalized Schrodinger equation expressed via implicit functions.
Article
Optics
Nikolay A. Kudryashov
Summary: This paper considers the complex Ginzburg-Landau equation with a polynomial law of nonlinearity with four powers. The Cauchy problem for this equation cannot be solved using the inverse scattering transform. However, the equation admits the translation groups in two independent variables and the general solution is sought using the traveling wave reduction method. A first integral of the nonlinear ordinary differential equation corresponding to the complex Ginzburg-Landau equation is found and reduced to a first-order nonlinear ordinary differential equation with the general solution expressed in terms of elliptic functions. The direct method allows for exact solutions without constraints on the parameters of the mathematiocal model. Partial cases of bright and dark optical solitons of the equation are given.
Article
Optics
Nikolay A. Kudryashov
Summary: In this study, a partial differential equation in a dual-power law nonlinear medium with variable coefficients is considered. The objective is to find solitary wave solutions that can be regarded as governed optical solitons. The exact solutions of the nonlinear differential equation are found using the direct method, and constraints on the variables of the coefficients are used to simplify the equation to a nonlinear ordinary differential equation with analytical solutions. Periodic and solitary wave solutions are obtained for the nonlinear partial differential equation with dual-power law of nonlinearity and coefficients depending on variables. Two simple examples of acceleration and deceleration of solitary wave solutions are discussed.
Article
Optics
Nikolay A. Kudryashov
Summary: This paper investigates the generalized Schrodinger equation with arbitrary refractive index and unrestricted order, and proposes an algorithm for constructing optical solitons using the simplest equation method. The algorithm is applied to find optical solitons of the generalized nonlinear Schrodinger equation, and the application of the method for solving the generalized Korteweg-de Vries equation is also discussed.
Article
Optics
Nikolay A. Kudryashov
Summary: The generalized Schrodinger-Hirota equation with an arbitrary power of nonlinearity is investigated in this study. The compatibility of the overdetermined system of equations is analyzed, and the problem is reduced to the study of one of the equations. The solutions are obtained by utilizing direct calculations through variable transformations. The results demonstrate the existence of dispersive optical solitons and periodic solutions for the generalized Schrodinger-Hirota equation with arbitrary power of nonlinearity.
Article
Optics
Nikolay A. Kudryashov
Summary: This article investigates the generalized Schrodinger-Hirota equation of the fourth order and uses the traveling wave method for solution. The compatibility of the overdetermined system of equations is analyzed to determine the integrability property of the mathematical model. The simplest equation method is applied to find periodic and solitary wave solutions. The results show the optical solitons of the generalized Schrodinger-Hirota equation of the fourth order.
Article
Mathematics
Nikolay A. Kudryashov
Summary: This paper considers some types of the generalized nonlinear Schrodinger equation of the second, fourth, and sixth order. The inverse scattering transform cannot solve the Cauchy problem for equations in the general case. The main objective of this paper is to find the conservation laws of the equations by using their transformations. An algorithmic method for finding Hamiltonians of some equations is presented, allowing for the search of Hamiltonians without the derivative operator using symbolic calculation programs. The Hamiltonians of three types of the generalized nonlinear Schrodinger equation are found and examples of Hamiltonians for some equations are presented.
Article
Physics, Multidisciplinary
Nikolay A. Kudryashov
Summary: The complex Ginzburg-Landau equation is studied, and it is found that there are three conservation laws associated with this equation. These conservation laws are obtained through a direct transformation of the equation. The first integral of the ordinary differential equation is derived by reducing it to traveling wave variables. The conservative quantities corresponding to the power, momentum, and energy of the optical soliton are calculated.
Article
Mathematics
Nikolay A. Kudryashov, Sofia F. Lavrova
Summary: The Chavy-Waddy-Kolokolnikov model for bacterial colonies is studied. The Painleve test is used to determine if the mathematical model is integrable, providing restrictions on the parameters. The inverse scattering transform method is found ineffective for solving the Cauchy problem due to the requirement of stationary solutions. The stability of stationary points and construction of periodic and solitary solutions are also explored.
Article
Optics
Nikolay A. Kudryashov, Anjan Biswas, Agniya G. Borodina, Yakup Yildirim, Hashim M. Alshehri
Summary: This paper applies Painleve analysis to a concatenated model consisting of the well-known nonlinear Schrodinger's equation, Lakshmanan-Porsezian-Daniel model, and Sasa-Satsuma equation. A bright 1-soliton solution is then obtained using the intermediary Jacobi's elliptic function.
Article
Mathematics, Applied
Peter Frolkovic, Nikola Gajdosova
Summary: This paper presents compact semi-implicit finite difference schemes for solving advection problems using level set methods. Through numerical tests and stability analysis, the accuracy and stability of the proposed schemes are verified.
APPLIED MATHEMATICS AND COMPUTATION
(2024)
Article
Mathematics, Applied
Md. Rajib Arefin, Jun Tanimoto
Summary: Human behaviors are strongly influenced by social norms, and this study shows that injunctive social norms can lead to bi-stability in evolutionary games. Different games exhibit different outcomes, with some showing the possibility of coexistence or a stable equilibrium.
APPLIED MATHEMATICS AND COMPUTATION
(2024)
Article
Mathematics, Applied
Dingyi Du, Chunhong Fu, Qingxiang Xu
Summary: A correction and improvement are made on a recent joint work by the second and third authors. An optimal perturbation bound is also clarified for certain 2 x 2 Hermitian matrices.
APPLIED MATHEMATICS AND COMPUTATION
(2024)
Article
Mathematics, Applied
Pingrui Zhang, Xiaoyun Jiang, Junqing Jia
Summary: In this study, improved uniform error bounds are developed for the long-time dynamics of the nonlinear space fractional Dirac equation in two dimensions. The equation is discretized in time using the Strang splitting method and in space using the Fourier pseudospectral method. The major local truncation error of the numerical methods is established, and improved uniform error estimates are rigorously demonstrated for the semi-discrete scheme and full-discretization. Numerical investigations are presented to verify the error bounds and illustrate the long-time dynamical behaviors of the equation with honeycomb lattice potentials.
APPLIED MATHEMATICS AND COMPUTATION
(2024)
Article
Mathematics, Applied
Kuan Zou, Wenchen Han, Lan Zhang, Changwei Huang
Summary: This research extends the spatial PGG on hypergraphs and allows cooperators to allocate investments unevenly. The results show that allocating more resources to profitable groups can effectively promote cooperation. Additionally, a moderate negative value of investment preference leads to the lowest level of cooperation.
APPLIED MATHEMATICS AND COMPUTATION
(2024)
Article
Mathematics, Applied
Kui Du
Summary: This article introduces two new regularized randomized iterative algorithms for finding solutions with certain structures of a linear system ABx = b. Compared to other randomized iterative algorithms, these new algorithms can find sparse solutions and have better performance.
APPLIED MATHEMATICS AND COMPUTATION
(2024)
Article
Mathematics, Applied
Shadi Malek Bagomghaleh, Saeed Pishbin, Gholamhossein Gholami
Summary: This study combines the concept of vanishing delay arguments with a linear system of integral-algebraic equations (IAEs) for the first time. The piecewise collocation scheme is used to numerically solve the Hessenberg type IAEs system with vanishing delays. Well-established results regarding regularity, existence, uniqueness, and convergence of the solution are presented. Two test problems are studied to verify the theoretical achievements in practice.
APPLIED MATHEMATICS AND COMPUTATION
(2024)
Article
Mathematics, Applied
Qi Hu, Tao Jin, Yulian Jiang, Xingwen Liu
Summary: Public supervision plays an important role in guiding and influencing individual behavior. This study proposes a reputation incentives mechanism with public supervision, where each player has the authority to evaluate others. Numerical simulations show that reputation provides positive incentives for cooperation.
APPLIED MATHEMATICS AND COMPUTATION
(2024)
Article
Mathematics, Applied
Werner M. Seiler, Matthias Seiss
Summary: This article proposes a geometric approach for the numerical integration of (systems of) quasi-linear differential equations with singular initial and boundary value problems. It transforms the original problem into computing the unstable manifold at a stationary point of an associated vector field, allowing efficient and robust solutions. Additionally, the shooting method is employed for boundary value problems. Examples of (generalized) Lane-Emden equations and the Thomas-Fermi equation are discussed.
APPLIED MATHEMATICS AND COMPUTATION
(2024)
Article
Mathematics, Applied
Lisandro A. Raviola, Mariano F. De Leo
Summary: We evaluated the performance of novel numerical methods for solving one-dimensional nonlinear fractional dispersive and dissipative evolution equations and showed that the proposed methods are effective in terms of accuracy and computational cost. They can be applied to both irreversible models and dissipative solitons, offering a promising alternative for solving a wide range of evolutionary partial differential equations.
APPLIED MATHEMATICS AND COMPUTATION
(2024)
Article
Mathematics, Applied
Yong Wang, Jie Zhong, Qinyao Pan, Ning Li
Summary: This paper studies the set stability of Boolean networks using the semi-tensor product of matrices. It introduces an index-vector and an algorithm to verify and achieve set stability, and proposes a hybrid pinning control technique to reduce computational complexity. The issue of synchronization is also discussed, and simulations are presented to demonstrate the effectiveness of the results obtained.
APPLIED MATHEMATICS AND COMPUTATION
(2024)
Article
Mathematics, Applied
Ling Cheng, Sirui Zhang, Yingchun Wang
Summary: This paper considers the optimal capacity allocation problem of integrated energy systems (IESs) with power-gas systems for clean energy consumption. It establishes power-gas network models with equality and inequality constraints, and designs a novel full distributed cooperative optimal regulation scheme to tackle this problem. A distributed projection operator is developed to handle the inequality constraints in IESs. The simulation demonstrates the effectiveness of the distributed optimization approach.
APPLIED MATHEMATICS AND COMPUTATION
(2024)
Article
Mathematics, Applied
Abdurrahim Toktas, Ugur Erkan, Suo Gao, Chanil Pak
Summary: This study proposes a novel image encryption scheme based on the Bessel map, which ensures the security and randomness of the ciphered images through the chaotic characteristics and complexity of the Bessel map.
APPLIED MATHEMATICS AND COMPUTATION
(2024)
Article
Mathematics, Applied
Xinjie Fu, Jinrong Wang
Summary: In this paper, we establish an SAIQR epidemic network model and explore the global stability of the disease in both disease-free and endemic equilibria. We also consider the control of epidemic transmission through non-instantaneous impulsive vaccination and demonstrate the sustainability of the model. Finally, we validate the results through numerical simulations using a scale-free network.
APPLIED MATHEMATICS AND COMPUTATION
(2024)
Article
Mathematics, Applied
Maria Han Veiga, Lorenzo Micalizzi, Davide Torlo
Summary: The paper focuses on the iterative discretization of weak formulations in the context of ODE problems. Several strategies to improve the accuracy of the method are proposed, and the method is combined with a Deferred Correction framework to introduce efficient p-adaptive modifications. Analytical and numerical results demonstrate the stability and computational efficiency of the modified methods.
APPLIED MATHEMATICS AND COMPUTATION
(2024)
