Computational Methods for Differential Equations
Volume:13 Issue: 2, Spring 2025

The predator-prey model is a pair of first-order nonlinear differential equations which are used to explain the dynamics of biological systems. These systems contain two species interacting, one as a predator and the other as prey. This work proposes a meshless local Petrov-Galerkin (MLPG) method based upon the interpolating moving least squares (IMLS) approximation, for the numerical solution of the predator-prey systems. With this aim, the space derivative is discretized by the MLPG technique in which the test and trial functions are chosen from the shape functions of IMLS approximation. Next, a semi-implicit finite difference approach is utilized to discretize the time derivative. The main aim of this work is to bring forward a flexible numerical procedure to solve predator-prey systems on complicated geometries.

This paper employs Colombeau algebra as a mathematical framework to establish both the existence and uniqueness of solutions for the fractional Schrödinger equation when subjected to singular potentials. A noteworthy contribution lies in the introduction of the concept of a generalized conformable semigroup, marking the first instance of its application. This innovative approach plays a pivotal role in demonstrating the sought-after results within the context of the fractional Schrödinger equation. The utilization of Colombeau algebra, coupled with the introduction of the generalized conformable semigroup, represents a novel and effective strategy for addressing challenges posed by singular potentials in the study of this particular type of Schrödinger equation.

This research introduces an algorithmically efficient framework for analyzing the fractional impulsive system, which can be seen as specific instances of the broader fractional Lorenz impulsive system. Notably, these systems find pertinent applications within the financial domain. To this end, the utilization of cubic splines is embraced to effectively approximate the fractional integral within the context of the system. The outcomes derived from this method are subsequently compared with those yielded by alternative techniques documented in existing literature, all pertaining to the integration of functions.Furthermore, the proposed methodology is not only applied to the resolution of the fractional impulsive system, but also extended to encompass scenarios involving the fractional Lorenz system with impulsive characteristics. The discernible effects stemming from the selection of disparate impulse patterns are meticulously demonstrated. In synthesis, this paper endeavors to present a pragmatic and proficient resolution to the intricate challenges posed by impulsive systems.

In this paper, we obtained the Poincare return maps for the planar piecewise linear differential systems of the type focus-focus. Normal forms for planar piecewise smooth systems with two zones of the type focus-focus and saddle-saddle, separated by a straight line and with a center at the origin, are obtained. Upper bounds for the number of limit cycles bifurcated from the period annulus of these normal forms due to perturbation by polynomial functions of any degree are established.

This article introduces a new numerical hybrid approach based on an operational matrix and spectral technique tosolve Caputo fractional sub-diffusion equations. This method transforms the model into a set of nonlinear algebraicequation systems. Chebyshev polynomials are used as basis functions. The study includes theoretical analysisto demonstrate the convergence and error bounds of the proposed method. Two test problems are conducted toillustrate the method’s accuracy. The results indicate the efficiency of the proposed method.

The applications of a Reaction-Diffusion boundary value problems are found in science, biochemical applications, and chemical applications. The Ananthaswamy-Sivasankari method (ASM) is employed to solve the considered specific models like non-linear reaction-diffusion model in porous catalysts, spherical catalysts pellet, and catalytic reaction-diffusion process in a catalyst slab. An accurate semi-analytical expression for the concentrations and effectiveness factors are given in the explicit form. Graphical representations are used to display the impacts of several parameters, including the Thiele modulus, characteristic reaction rate, concentration of half-saturation, reaction order and dimensionless constant in Langmuir-Hinshelwood kinetics. The impact of numerous parameters namely the Langmuir-Hinshelwood kinetics and Thiele modulus on effectiveness factors are displayed graphically. Our semi-analytical findings shows good match in all parameters when compared to numerical simulation using MATLAB. Many non-linear problems in chemical science especially, the Reaction-Diffusion equations, Michaelis-Menten kinetic equation, can be resolved with the aid of the new approximate analytical technique, ASM.

This manuscript provides an efficient technique for solving time-fractional diffusion-wave equations using general Lagrange scaling functions (GLSFs). In GLSFs, by selecting various nodes of Lagrange polynomials, we get various kinds of orthogonal or non-orthogonal Lagrange scaling functions. The general Riemann-Liouville fractional integral operator (GRLFIO) of GLSFs is obtained generally. General Riemann-Liouville fractional integral operator of the general Lagrange scaling function is calculated exactly using the Hypergeometric functions. The operator extraction method is precisely calculated and this has a direct impact on the accuracy of our method. The operator and optimization method are implemented to convert the problem to a set of algebraic equations. Also, error analysis is discussed. To demonstrate the efficiency of the numerical scheme, some numerical examples are examined.

A hybrid computational procedure of Newton Raphson method and orthogonal collocation method have been applied to study the behavior of nonlinear astrophysics equations. The nonlinear Lane Emden equation has been discretized using the orthogonal collocation method using $n^{th}$-order Bessel polynomial as $J_n(\xi)$ as base function.  The system of collocation equations has been solved numerically using Newton Raphson method. Numerical examples have been discussed to check the reliability and efficiency of the scheme. Numerically calculated results have been compared to the exact values as well as the values already given in the literature to check the compatibility of the scheme. Error analysis has been studied by calculating the absolute error, $L_2- norm$ and $L_{\infty}- norm$. Computer codes have been prepared using MATLAB.

This paper presents a numerical scheme for solving the non-linear time fractional Klein-Gordon equation. To approximate spatial derivatives, we employ the pseudo-spectral method based on Lagrange polynomials at Chebyshev points, while using the finite difference method for time discretization. Our analysis demonstrates that this scheme is unconditionally stable, with a time convergence order of $\mathcal{O}({3 \alpha})$. Additionally, we provide numerical results in one, two, and three dimensions, highlighting the high accuracy of our approach. The significance of our proposed method lies in its ability to efficiently and accurately address the non-linear time fractional Klein-Gordon equation. Furthermore, our numerical outcomes validate the effectiveness of this scheme across different dimensions.

The motive of this paper is to investigate the SEIQRD model of the COVID-19 outbreak in Indonesia with the help of a fractional modeling approach. The model is described by the nonlinear system of six fractional order differential equations (DE) incorporating the Caputo-Fabrizio Fractional derivative (CFFD) operator. The existence and uniqueness of the model are proved by applying the well-known Banach contraction theorem. The reproduction number ($R_0$) is calculated, and its sensitivity analysis is conducted concerning each parameter of the model for the prediction and persistence of the infection. Moreover, the numerical simulation for various fractional orders is performed using the Adams-Bashforth technique to analyze the transmission behavior of disease and to get the approximated solutions. At last, we represent our numerical simulation graphically to illustrate our analytical findings.

In this paper, the finite-time sliding mode controller design problem of a class of conic-type nonlinear systems with time-delays, mismatched external disturbance and uncertain coefficients is investigated. The time-delay conic nonlinearities are considered to lie in a known hypersphere with an uncertain center. Conditions have been obtained to design a linear quadratic regulator based on sliding mode control. For this purpose, by applying Lyapunov- Krasovskii stability theory and linear matrix inequality approach, sufficient conditions are derived to ensure the finite-time boundedness of the closed-loop systems over the finite-time interval. Thereafter, an appropriate control strategy is constructed to drive the state trajectories onto the specified sliding surface in a finite time. Finally, an example related to the time-delayed Chua’s circuit is given to demonstrate the effectiveness of the suggested method. Also, the efficiency of the suggested method is compared with other methods by using an another numerical example

The basic necessities of life are food, shelter and clothing. Food is more necessary because the existence of life depends on food. In order to foster global food security, integrated pest management (IPM), an environmentally-friendly program, was designed to maintain the density of pest population in the equilibrium level below the economic damage. For years, mathematics has been an ample tool to solve and analyze various real-life problems in science, engineering, industry and so on but the use of mathematics to quantify ecological phenomena is relatively new. While efforts have been made to study various methods of pest control, the extent to which pests’ enemies as well as natural treatment can reduce crop damage is new in the literature. Based on this, deterministic mathematical models are designed to investigate the prey-predator dynamics on a hypothetical crop field in the absence or presence of natural treatment. The existence and uniqueness of solutions of the models are examined using Derrick and Grossman’s theorem. The equilibria of the models are derived and the stability analysed following stability principle of differential equations and Bellman and Cooke’s theorem. The theoretical results of the models are justified by a means of numerical simulations based on a set of reasonable hypothetical parameter values. Results from the simulations reveal that the presence of pests’ enemies on a farm without application of natural treatment may not avert massive crop destruction. It is also revealed that the application of natural treatment may not be enough to keep the density of the pest population below the threshold of economic damage unless the rate of application of natural treatment exceeds the growth rate of the pest.

The purpose of this paper is to obtain numerical solutions of fractional interval optimal control problems. To do so, first, we obtain a system of fractional interval differential equations through necessary conditions for the optimality of these problems, via the interval calculus of variations in the presence of interval constraint arithmetic. Relying on the trapezoidal rule, we obtain a numerical approximation for the interval Caputo fractional derivative. This approach causes the obtained conditions to be converted to a set of algebraic equations which can be solved using an iterative method such as the interval Gaussian elimination method and interval Newton method. Finally, we solve some examples of fractional interval optimal control problems in order to evaluate the performance of the suggested method and compare the past and present achievements in this manuscript.

Our primary goal is to address the q-deformed wave equation, which serves as a mathematical framework for characterizing physical systems with symmetries that have been violated. By incorporating a q-deformation parameter, this equation expands upon the traditional wave equation, introducing non-commutativity and nonlinearity to the dynamics of the system. In our investigation, we explore three distinct approaches for solving the generalized q-deformed wave equation: the reduced q-differential transform method (RqDTM) [17], the separation method (SM), and the variational iteration method (VIM). The RqDTM is a modified version of the differential transform method specially designed to handle q-deformed equations. The SM aims to identify solutions that can be expressed as separable variables, while the VIM employs an iterative scheme to refine the solution. We conduct a comparative analysis of the accuracy and efficiency of the solutions obtained through these methods and present numerical results. This comparative analysis enables us to evaluate the strengths and weaknesses of each approach in effectively solving the q-deformed wave equation, providing valuable insights into their applicability and performance. Additionally, this paper introduces a generalization of the q-deformed wave equation, as previously proposed in [13], and investigates its solution using two different analytical RqDTM, SM, and an approximation method is known as VIM.

In this paper, by using the techniques of measures of non-compactness and the Petryshyn fixed point theorem, we investigate the existence of solutions of a Caputo fractional functional integro-differential equation and obtain some new results. These existing results involve particular results gained from earlier studies under weaker conditions.

This study employs the cubic B-spline collocation strategy to address the solution challenges posed by the nonlinear generalized Burgers-Fisher’s equation (gBFE), with some improvisation. This approach incorporates refinements within the spline interpolants, resulting in enhanced convergence rates along the spatial dimension. Temporal integration is achieved through the Crank-Nicolson methodology. The stability of the technique is assessed using the rigorous von Neumann method. Convergence analysis based on Green’s function reveals a fourth-order convergence along the space domain and a second-order convergence along the temporal domain. The results are validated by taking a number of examples. MATLAB 2017 is used for computational work.

In this paper, we developed a $SEI_aI_sQRS$ epidemic model for COVID-19 by using compartmental analysis. In this article, the dynamics of COVID-19 are divided into six compartments: susceptible, exposed, asymptomatically infected, symptomatically infected, quarantined, and recovered. The positivity and boundedness of the solutions have been proven. We calculated the basic reproduction number for our model and found both disease-free and endemic equilibria. It is shown that the disease-free equilibrium is globally asymptotically stable. We explained under what conditions, the endemic equilibrium point is locally asymptotically stable. Additionally, the center manifold theorem is applied to examine whether our model undergoes a backward bifurcation at $R_0 = 1$ or not. To finish, we have confirmed our theoretical results by numerical simulation

In this article, with the help of the new Kudryashov method, we examine general solutions to the (2+1)-dimensional Sawada-Kotera equation (SKE) and Kaup-Kupershmidt (KK) equation. Using Maple, a symbolic computing application, it was shown that all obtained solutions are given by hyperbolic, exponential and logaritmic function solutions which obtained solutions are useful for fluid dynamics, optics and so on. Finally, we have presented some graphs for general solutions of these equations with special parameter values. The reliability and scope of programming provide eclectic applicability to high-dimensional nonlinear evolution equations for the development of this method. The results found gave us important information regarding the applicability of the new Kudryashov method.

The cubic spline in tension method is taken into consideration to solve the singularly perturbed delay differential equations of convection diffusion type with integral boundary condition. Simpson’s 1/3 rule is used to the non-local boundary condition and three model problems are examined for numerical treatment and are addressed using a variety of values for the perturbation parameter  and the mesh size to verify the scheme’s applicability. The computational results and rate of convergence are given in tables, and it is seen that the proposed method is more precise and improves the methods used in the literature.

The multi-step differential transform method (DTM) adopted from the standard DTM is employed in this case study to solve a model of the transesterification reaction. The DTM is considered in a sequence of time intervals. The accuracy of the proposed method is confirmed by comparing its results with those of the fourth-order RungeKutta (RK4) method. In addition, the experimental results are investigated with the Multi-step DTM to demonstrate the efficiency and effectiveness of these chemical reactions obtained in the laboratory. The present findings confirmed the effectiveness of using the multi-step DTM in validating the chemical models obtained in laboratories.

One of the most important natural phenomena that has been studied extensively in engineering, oceanography,meteorology, and other fields is called fluid turbulence (FT). FT stands for irregular flow of fluid. Scientists detected models to describe this phenomenon, among these models is the (3+1)-dimensional Vakhnenko-Parkes (VP) equation. In this research, the high-frequency waves’ dynamical behavior through the relaxation medium is explored by considering two semi-analytic methods, the $(G^'/G)$ and the tanh-coth (TC) expansion methods. Nineteen different solutions have been detected and some of these solutions have been illustrated graphically. Figures show a range of degenerate, periodic, and complex propagating soliton wave solutions.

Genocchi polynomials have exciting properties in the approximation of functions. Their derivative and integral calculations are simpler than other polynomials and, in practice, they give better results with low degrees. For these reasons, in this article, after introducing the important properties of these polynomials, we use them to approximate the solution of different population balance models. In each case, we first discuss the solution method and then do the error analysis. Since we do not have an exact solution, we compare our numerical results with those of other methods. The comparison of the obtained results shows the efficiency of our method. The validity of the presented results is indicated using MATLAB-Simulink.

This paper studies dynamical systems in product Lukasiewicz semirings and we generalize the results of Markechovaand Riecan concerning the logical entropy. Also, the notion of logical entropy of a product Lukasiewicz semiring is introduced and it is shown that entropy measure is invariant under isomorphism.

This paper examines a high-dimensional non-linear partial differential equation called the generalized Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation exists in three dimensions. The Lie symmetry analysis of the equation is carried out step-by-step. As a result, we found symmetries from which various group-invariant solutions arise, leading to numerous solutions of interest that satisfy the KP-BBM equation. Secured solutions of interest include hyperbolic functions and elliptic functions, with the latter being the more general of the two solutions. Additionally, a significant number of algebraic solutions with arbitrary functions are also obtained. Furthermore, the dynamics of the solutions are further explored diagrammatically using computer software. In the concluding section, various conservation laws of the underlying model are derived via the multiplier method and the Noether theorem.

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