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This article investigates the RBF partition of unity method for solving the two-dimensional Sobolev equation with a Burgers-type nonlinearity on irregular domains. To the best of our knowledge, there are limited results in solving the Sobolev equation with a Burgers-type nonlinearity on complex irregular domains. Two variants of the radial basis functions partition of unity method, namely the classical and direct methods with smooth and discontinuous weight functions, are employed for spatial discretization, and their results are compared in terms of accuracy, execution time, and sparsity of their computational matrices. We Utilized the polyharmonic radial kernels due to their remarkable properties, including their ability to approximate and scalability. The scalability attribute of these radial kernels is used to regulate the instability of the method. To ensure the accuracy and reliability of our proposed approach, we carried out numerical simulations and provided several examples for comparison. Our comparisons of the radial basis functions partition of unity procedures reveal that the direct method is computationally more efficient than the classical approach, while both versions have admissible accuracy. [ABSTRACT FROM AUTHOR]

In this paper, we study the thermoelastic Timoshenko system with diffusion effect, memory, and distributed delay terms when they act on the second equation. Firstly, we use the Faedo–Galerkin method to show the well-posedness of the system. Then, we study the exponential stability using the Lyapunov functional. Next, we introduce a finite element scheme of the problem, and then we establish the discrete energy decay. Finally, we give a priori error estimates and some numerical simulations. [ABSTRACT FROM AUTHOR]

We use a numerical-analytic technique to construct a sequence of successive approximations to the solution of a system of fractional differential equations, subject to Dirichlet boundary conditions. We prove the uniform convergence of the sequence of approximations to a limit function, which is the unique solution to the boundary value problem under consideration, and give necessary and sufficient conditions for the existence of solutions. The obtained theoretical results are confirmed by a model example. [ABSTRACT FROM AUTHOR]

In this paper, the fractional-order Chelyshkov functions (FCHFs) and Riemann-Liouville fractional integrals are utilized to find numerical solutions to fractional delay differential equations, by transforming the problem into a system of algebraic equations with unknown FCHFs coefficients. An error bound of FCHFs approximation is estimated and its convergence is also demonstrated. The effectiveness and accuracy of the presented method are established through several examples. The resulting solution is accurate and agrees with the exact solution, even if the exact solution is not a polynomial. Moreover, comparisons between the obtained numerical results and those recently reported in the literature are shown. [ABSTRACT FROM AUTHOR]

In this study, a multipoint boundary value problem for Volterra-Fredholm integro-differential equations is considered. The addition of a new function converts the system of Volterra-Fredholm integro-differential equations to a system of Fredholm integro-differential equations. In contrast to the original problem, the dimension of a Fredholm integro-differential equation is determined by the number of matrices in the degenerate kernel of the Volterra integral. A numerical algorithm of Dzhumabaev parameterization method for addressing a multipoint boundary value problem for Volterra-Fredholm integro-differential equations is proposed. The main advantage of the proposed method is splitting the problem into auxiliary Cauchy problems for ordinary differential equations and a system of algebraic equations with respect to the parameters. The conditions for the unique solvability of the multipoint boundary value problem for Fredholm integro-differential equations are established. Finally, various numerical examples are provided to demonstrate the efficiency and correctness of the suggested technique.

In this research, the constant type of neutral delay Volterra integro-differential equations (NDVIDEs) are currently being resolved by applying the proposed technique in numerical analysis namely, two-point two off-step point block multistep method (2OBM4). This new technique is being applied in solving NDVIDE, identified as a hybrid block multistep method, developed using Taylor series interpolating polynomials. To complete the algorithm, two alternative numerical approaches are introduced to resolve the integral and differential parts of the problems. Note that the differentiation is approximated by the divided difference formula while the integration is interpolated using composite Simpson’s rule. The proposed method has been analysed thoroughly in terms of its order, consistency, zero stability and convergence. The suitable stability region for 2OBM4 in solving NDVIDE has been constructed and the stability region is built based on the stability polynomial obtained. Consequently, numerical results are presented to demonstrate the effectiveness of the proposed 2OBM4. [ABSTRACT FROM AUTHOR]

In this paper, we focus on solving nonlinear fractional logistic differential equation within the Caputo–Fabrizio fractional derivative operator. First, we present a new numerical scheme called new decomposition transform method (NDTM) for solving general nonlinear fractional differential equations. The proposed scheme is obtained by combining the Jafari transform method and a new decomposition method. Furthermore, we establish the uniqueness theorem of the solution using Banach's fixed point theorem. Finally, after the convergence analysis of the NDTM, we provide two numerical examples of nonlinear fractional logistic differential equations involving the Caputo–Fabrizio fractional derivative operator to validate the results. The obtained results have shown that the new method for analytical solutions is simple to implement and very effective for analyzing the complex fractional problems that arise in related fields of science and engineering. [ABSTRACT FROM AUTHOR]

The search for Skiba points in infinite-time optimal control problems with several steady states is an active research area. In this paper, we apply the backward integration method and show its power and relative simplicity as a tool for this task. We apply the method for the classical model of the economy shallow lake based on the phosphorous dynamics and explicitly compute the optimal trajectories and the Skiba points. We also suggest a more natural welfare function, which takes into account the finite value of any economic resource, and show, again in an explicit manner, that it also possesses Skiba points. [ABSTRACT FROM AUTHOR]

The Green's function method is applied to second, third, and fourth order two-point boundary value problems with deviating argument. At each iterative step, the Picard-Green's method is combined with suitable quadrature rule and interpolation procedures. The Hermite type cubic spline is applied to second and third order boundary value problems, while complete cubic splines are used at fourth order boundary value problems. The corrected trapezoidal quadrature rule is involved at third and fourth order boundary value problems with appropriate error bounds. The convergence of the method was proved, and the maximal order of convergence is obtained for each problem. The theoretical results are tested on some numerical examples. [ABSTRACT FROM AUTHOR]

This article adopts a novel technique to numerical solution for fractional time-delay diffusion equation with variable-order derivative (VFDDEs). As a matter of fact, the variable-order fractional derivative (VFD) that has been used is in the Caputo sense. The first step of this technique is constructive on the construction of the solution using the shifted Legendre–Laguerre polynomials with unknown coefficients. The second step involves using a combination of the collocation method and the operational matrices (OMs) of the shifted Legendre–Laguerre polynomials, as well as the Newton–Cotes nodal points, to find the unknown coefficients. The final step focuses on solving the resulting algebraic equations by employing Newton's iterative method. To illustrate and demonstrate the technique's efficacy and applicability, two examples have been provided. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the nonlinear space-fractional Klein–Gordon–Schrödinger system involving the fractional Laplacian operator of order α for 1 < α ≤ 2 . We propose an accurate numerical method with eneregy-preserving property for solving the well-known system. The problem is discretized in spatial direction by the Fourier spectral method, and in temporal direction by utilizing the fourth-order exponential time-differencing Runge–Kutta technique. We show that the proposed method satisfies both mass and energy conservation. The convergence of this method is proved, and the order of accuracy is obtained, which shows that the order of convergence is near two. Several numerical experiments are tested to validate the accuracy and reliability of the proposed method. The results are presented in tables and figures for the different values of α that show the proposed method is an efficient framework for solving nonlinear space-fractional Klein–Gordon–Schrödinger system. [ABSTRACT FROM AUTHOR]

The effective sufficient conditions are established for the numerical approximation of solutions of the initial problem with weight for linear systems of ordinary differential equations with singularities. [ABSTRACT FROM AUTHOR]

The present article concerns itself with the theoretical investigation on an interacting species model system with special emphasis on the species growth followed by Smith and a constant proportion of prey refuge. The prime objective of the study is to provide an adequate mathematical framework in order to carry out a comprehensive analytical investigation of the dynamical complexity between predator and prey species. The proposed model system is not only explored in the perception of diverse local bifurcations in a two-dimensional plane but also of the global bifurcations about coexistence equilibria under specific parametric conditions. For the purpose of validation of all the analytical outcomes together with the applicability of the model concerned, a quantitative sensitivity analysis based on numerical simulation is performed. The evolution of diffusion-driven pattern formation in two-dimensional plane in terms of spot, stripe, labyrinthine, stripe-hole mixture and hole replication as well is patently exhibited. These patterns are all influenced by both the Smith growth principle and prey refuge of the diffusive system. Finally, the influence of model parameters of significance on the dynamics of the proposed model system is, however, not ruled out to depict graphically from the present study. [ABSTRACT FROM AUTHOR]

In this paper a new method to compute approximate analytical solutions for differential equations of fractional order is presented. The proposed computational method, called the Piecewise Polynomial Least Squares Method (PWPLSM), is a combination of the Polynomial Least Squares Method and of piecewise-defined functions. Numerical results for differential equations of different fractional orders are discussed. The approximate solutions obtained with PWPLSM are compared with other existing analytical and numerical solutions. The tables and figures included demonstrate the accuracy of the new method.

In this study, two numerical methods [(a) artificial neural network method with three layers (input layer, hidden layer, output layer) and (b) least squares support vector regression (LS-SVR) method] are suggested for solving three classes of differential equations. For the first method, we use the Genocchi wavelets, inverse trigonometric functions and hyperbolic functions as activation functions. In the second method, we apply the Genocchi wavelets kernel and the collocation LS-SVR method for training the network. Then, for the two methods, the formulation of the methods gives rise to an optimization problem. Finally, the classical optimization and Newton's iterative method are applied to train these networks. Also, some useful theorems concerning the error analysis associated with the LS-SVR scheme are proved in our article. Finally, some test problems are included to show the efficiency and accuracy of the current methods. [ABSTRACT FROM AUTHOR]

High-order linear Volterra delay integro-differential equations are examined in the present paper. Proposed approach, which will be provided for solving high-order linear Volterra delay integro-differential equations, expresses certain key elements of determining the equations' stability bounds and exact solutions. Furthermore, stability inequalities can be generated for each order of derivative using the proposed method. [ABSTRACT FROM AUTHOR]

Viscous Burgers' equations with a small viscosity are considered and convergence of vanishing viscosity limit problem is investigated. We examine interior layers of a solution to viscous Burgers' equations, u ϵ , as a viscosity parameter ε tends to zero. The inviscid model, i.e. when ϵ = 0 , possesses the structure of scalar hyperbolic conservation laws, hence our studies deliver an important idea that arises in the field of shock discontinuities of nonlinear hyperbolic waves. The heart of the paper is to establish asymptotic expansions and utilize inner solutions of sharp transition, which are called a corrector function. With aid of corrector functions and energy estimates, we improve the convergence rate of u ϵ to u 0 as O (ϵ 1 / 2 ) in L 2 (R) ( O (ϵ) in L l o c 1 (R)) in the regions including shocks under an entropy condition. [ABSTRACT FROM AUTHOR]

The main aim of this article is to present an a priori estimate of an exponential layer and thereby a new mesh (G-mesh) for a class of singularly perturbed problems with discontinuous data. We have also proved that the standard upwind scheme on the proposed mesh is parameter uniformly convergent and is of almost first order. To demonstrate the efficiency of the proposed mesh we carried out a few numerical experiments comparing the performance of the proposed G-mesh with other standard meshes available in the literature. [ABSTRACT FROM AUTHOR]

The goal of this paper is to present a high-order numerical scheme for solving fourth-order time-fractional partial differential equations (TFPDEs). The fractional derivative in the considered model is in Caputo's sense. In the proposed approach, the time variable is approximated by the Legendre polynomials, and space discretisation is based on the modified basis constructed from a combination of the Legendre polynomials. We study the stability and convergence of the proposed method. Some numerical examples are investigated to validate the efficiency, accuracy, and theoretical results of the given method. [ABSTRACT FROM AUTHOR]

The Green's function method is a powerful technique to investigate the existence and uniqueness of the solution for two-point boundary value problems. The main task of this paper is to establish the maximal order of convergence of the Green's function method applied in the approximation of the solution for third- and fourth-order two-point boundary value problems with deviating argument. The developed iterative method uses a complete cubic spline interpolation procedure being able to provide both discrete and continuous approximation of the solution. Some numerical examples confirm the obtained theoretical results and illustrate the accuracy of the algorithm. [ABSTRACT FROM AUTHOR]

This paper discusses two efficient collocation methods for solving the Hammerstein integral equations on the semi-infinite domain, where the underlying solutions decay to zero at infinity. These methods are based upon modified Legendre rational and exponential functions, and reduce the Hammerstein integral equation to a nonlinear algebraic system. The error between the approximate and exact solutions in the usual L 2 -norm is estimated. Finally, some numerical experiments are presented to examine and demonstrate the effectiveness and accuracy of the proposed methods in comparison to other approaches. [ABSTRACT FROM AUTHOR]

A finite difference scheme along with a fourth-order approximation is examined in this article for finding the solution of time-fractional diffusion equation with fourth-order derivative in space subject to homogeneous and non-homogeneous boundary conditions. Caputo fractional derivative is used to describe the time derivative. The time-fractional diffusion equation of order 0 < γ < 1 is transformed into Volterra integral equation which is then approximated by linear interpolation. A novel numerical scheme based on fractional trapezoid formula for time discretization followed by Stephenson's scheme to discretize the fourth order space derivative is developed for the linear diffusion equation. Afterward, convergence and stability are investigated thoroughly showing that the proposed scheme is unconditionally stable and hold convergence accuracy of order O (τ 2 + h 4) . The numerical examples are presented in accordance with the theoretical results showing the efficiency and accuracy of the presented numerical technique. [ABSTRACT FROM AUTHOR]

In this study, by combining the generalized pseudospectral method, which is a new numerical method, with the quasi-linearization method (QLM), an efficient hybrid method to solve nonlinear differential equations in applied sciences and engineering is presented. Given that this method requires generalized Lagrange functions and their derivative operational matrices, we first introduce them and then implement the generalized pseudospectral method at each iteration of the quasi-linearization method and obtain a system of linear algebraic equations. In the presented method, derivative operational matrices are used, and also, the nonlinear differential equation is converted into a sequence of linear differential equations using QLM, so there is no need to calculate the analytical derivative and solve nonlinear systems during the implementation of the method, which reduces computational costs and increasing the efficiency of the method. The efficiency and accuracy of the method have been demonstrated by applying it to several important applied equations; the Blasius equation, the Falkner–Skan problem over an isothermal moving wedge, and the third-grade fluid in a porous half-space. Then the obtained results are compared with other researchers. [ABSTRACT FROM AUTHOR]

We analyze and describe algorithm of Levenberg-Marquardt training (TrianLM) and how to modify it (MLM) for solving nonlinear equation systems under the condition of local error bounds and Jacobian continuity.We propose a new μk computation, which is Start by looking at the nonlinear processing equations. E(ω) = 0 does not have an empty solution. W*, as a sequence formed by MLM, quadratically converges to the solution set, where the system of nonlinear equation E(ω): Rn → Rm has a local error bound: is continuously differentiable and E(x) is Lipeschitz continuous. [ABSTRACT FROM AUTHOR]

In this paper we have introduced a new method, by using fully fuzzy neural network (FFNN) to find the numerical solution for solving fuzzy singular perturbation problems (FSPPS) for ordinary differential equations. It uses the Levenberg-Marquardt training machine algorithm (TrianLM) for calculating the Fuzzy neural network parameters and the sigmoid function of the hidden units is hyperbolic tangent activation function, that is:. The fuzzy trial solution of FSPPs is written as a sum of two parts. The first part satisfies the fuzzy conditions, it contains no fuzzy adjustable parameters. The second part involves a feed-forward fuzzy neural network containing fuzzy adjustable parameters. In comparison with existing numerical methods, shows upp that the use of fuzzy neural networks provides solutions with good generalization and high accuracy. The proposed method is illustrated by several examples. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a modified method for solving singular perturbation problem for ordinary differential equations by using artificial neural networks. This modified method is called modified artificial neural network method (MANN). This new method is based on replacing each in the input vector training set) by a polynomial of first degree which will be as. Using MANN makes that training points should be selected over the open interval (a, b) without training the neural network in the range of first and end points. We illustrate this modified method by solving some numerical examples and provide a comparison between our results and those calculated by other numerical methods, this method demonstrates that neural networks produce solutions with good generalization and accuracy. Several examples are provided to demonstrate the suggested strategy. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a novel numerical technique for solving a class of nonlinear multi-term fractional differential equations. The proposed method mainly relies on using the Dickson polynomial to derive the Dickson operational matrix of fractional derivative, which is then used to solve the given problem. The fundamental significance of this strategy is that it converts the fractional problem into a system of nonlinear equations that can be numerically solved. In addition, the error bound of the suggested method is deduced. Numerical experiments are presented to verify and confirm the accuracy of the suggested technique for dealing with such problems. [ABSTRACT FROM AUTHOR]

In this paper, we show that the solution of an infinite delay equations in a Banach space given by non-linear semigroups in Fréchet spaces can be approximated by finite delay equations. [ABSTRACT FROM AUTHOR]

This article revisits the approximation problem of systems of nonlinear delay differential equations (DDEs) by a set of ordinary differential equations (ODEs). We work in Hilbert spaces endowed with a natural inner product including a point mass, and introduce polynomials orthogonal with respect to such an inner product that live in the domain of the linear operator associated with the underlying DDE. These polynomials are then used to design a general Galerkin scheme for which we derive rigorous convergence results and show that it can be numerically implemented via simple analytic formulas. The scheme so obtained is applied to three nonlinear DDEs, two autonomous and one forced: (i) a simple DDE with distributed delays whose solutions recall Brownian motion; (ii) a DDE with a discrete delay that exhibits bimodal and chaotic dynamics; and (iii) a periodically forced DDE with two discrete delays arising in climate dynamics. In all three cases, the Galerkin scheme introduced in this article provides a good approximation by low-dimensional ODE.

This article revisits the approximation problem of systems of nonlinear delay differential equations (DDEs) by a set of ordinary differential equations (ODEs). We work in Hilbert spaces endowed with a natural inner product including a point mass, and introduce polynomials orthogonal with respect to such an inner product that live in the domain of the linear operator associated with the underlying DDE. These polynomials are then used to design a general Galerkin scheme for which we derive rigorous convergence results and show that it can be numerically implemented via simple analytic formulas. The scheme so obtained is applied to three nonlinear DDEs, two autonomous and one forced: (i) a simple DDE with distributed delays whose solutions recall Brownian motion; (ii) a DDE with a discrete delay that exhibits bimodal and chaotic dynamics; and (iii) a periodically forced DDE with two discrete delays arising in climate dynamics. In all three cases, the Galerkin scheme introduced in this article provides a good approximation by low-dimensional ODE.

The present paper deals with the numerical solution for a general form of a system of nonlinear Volterra delay integro-differential equations (VDIDEs). The main purpose of this work is to provide a current numerical method based on the use of continuous collocation Taylor polynomials for the numerical solution of nonlinear VDIDEs systems. It is shown that this method is convergent. Numerical results will be presented to prove the validity and effectiveness of this convergent algorithm. We apply models to the COVID-19 epidemic in China, Spain, and Italy and one for the Predator–Prey model in mathematical ecology. [ABSTRACT FROM AUTHOR]

The Riesz fractional advection-diffusion is a result of the mechanics of chaotic dynamics. It's of preponderant importance to solve this equation numerically. Moreover, the utilization of Chebyshev polynomials as a base in several mathematical equations shows the exponential rate of convergence. To this approach, we transform the interval of state space into the interval [-1, 1] x [-1, 1]. Then, we use the operational matrix to discretize fractional operators. Applying the resulting discretization, we obtain a linear system of equations, which leads to the numerical solution. Examples show the effectiveness of the method. [ABSTRACT FROM AUTHOR]

New iteration and Homotopy analysis methods are used to obtain the approximate solution to the nonlinear two dimensional Bogoyavlensky-Konopelchenko equation. A procedure called Multi-objective Genetic algorithm is used to find the right nonlinear coefficients for the two dimensional partial differential equation. The numerical simulations of the techniques in this paper and the outcomes which are compared with the exact solution are sufficient to show the effectivness of this technique. [ABSTRACT FROM AUTHOR]

Very recently, Yang, Abdel-Aty and Cattani (2019) introduced a new and intersting fractional derivative operator with non-singular kernel involving Rabotnov fractional-exponential function. In this paper, we present a general framework of the homotopy perturbation transform method (HPTM) for analytic treatment of time fractional partial differential equations in the sense of Yang-Abdel-Aty-Cattani. As applications, time fractional wave equations involving Yang-Abdel-Aty-Cattani fractional derivatives are solved. The solutions are obtained in the form of series involving Prabhakar functions.

In 2015, Shijun Liao introduced a new method of directly defining the inverse mapping (MDDiM) to approximate analytically a nonlinear differential equation. This method, based on the Homotopy Analysis Method (HAM) was proposed to reduce the time it takes in solving a nonlinear equation. Very recently, Dewasurendra, Baxter and Vajravelu (Applied Mathematics and Computation 339 (2018) 758–767) extended the method to a system of two nonlinear differential equations. In this paper, we extend it further to obtain the solution to a system of three nonlinear differential equations describing the HIV infection of CD4+ T-cells. In addition, we analyzed the advantages of MDDiM over HAM, in obtaining the numerical results. From these results, we noticed that the infected CD4+ T-cell density increases with the number of virions N; but decreases with the blanket death rate μI.

This work presents a numerical comparison between two efficient methods namely the fractional natural variational iteration method (FNVIM) and the fractional natural homotopy perturbation method (FNHPM) to solve a certain type of nonlinear Caputo time-fractional partial differential equations in particular, nonlinear Caputo time-fractional wave-like equations with variable coefficients. These two methods provided an accurate and efficient tool for solving this type of equations. To show the efficiency and capability of the proposed methods we have solved some numerical examples. The results show that there is an excellent agreement between the series solutions obtained by these two methods. However, the FNVIM has an advantage over FNHPM because it takes less time to solve this type of nonlinear problems without using He’s polynomials. In addition, the FNVIM enables us to overcome the diffi-culties arising in identifying the general Lagrange multiplier and it may be considered as an added advantage of this technique compared to the FNHPM.

In this study, we used the update a neural network for solve singularly perturbed partial differential equations (SPPDEs) that have been singularly disturbed. We dealt with the modernization of artiftial networks by updating the most important types of high-performance training. For nonlinear equations, the modified Levenberg-Marquardt method (MLMM) is used, in which not only an LM step but also an approximation LM step is generated at each iteration. When applying the trust region technique, a new type of projected reduction for the merit function is created to assure global convergence of the new method. The modified LM method's cubic convergence is demonstrated under the local error bound condition, which is weaker than nonsingularity. The numerical results demonstrate that the new method is quite efficient and might save numerous Jacobian calculations. [ABSTRACT FROM AUTHOR]

The article is devoted to the approximate solutions of the Fredholm integral equations of the second kind with the weak singular kernel that can have additional singularity in the numerator. We describe two problems that lead to such equations. They are the problem of minimization of small deviations and the entropy minimization problem. Both of them appear when considering a dynamical system involving a mixed fractional Brownian motion. In order to apply well-known numerical methods for weakly singular kernels, we build the continuous approximation of the solution of an integral equation with the kernel containing additional singularity by the solutions of the integral equations whose kernels are weakly singular, but the numerator is continuous. We prove that the approximated solutions tend to the solution of the original equation. We demonstrate numerically how our methods work being applied to our specific integral equations. [ABSTRACT FROM AUTHOR]

A new implicit-explicit local differential transform method (IELDTM) is derived here for time integration of the nonlinear (2 + 1)-dimensional advection–diffusion-reaction (ADR) equations. The IELDTM is adaptively constructed as a stability preserved and high order time integrator for spatially discretized ADR equations. For spatial discretization of the model equation, the Chebyshev spectral collocation method (ChCM) is utilized. A robust stability analysis and global error analysis of the IELDTM are presented with respect to the direction parameter θ . With the help of the global error analysis, adaptivity equations are derived to minimize the computational costs of the algorithms. The produced method is shown to eliminate the accuracy disadvantage of the classical θ -method and the stability disadvantages of the existing differential transform-based methods. Two examples of the Burgers equation in one and two space dimensions and the Chapman oxygen-ozone ADR model are solved via the ChCM-IELDTM hybridization. The present time integrator is proven to provide more efficient numerical characteristics than the various multi-step and multi-stage time integration methods. The IELDTM is extensively compared with the widely used MATLAB solvers, ode45 and ode15s. The adaptive IELDTM has been proven to integrate the stiff Chapman ADR equations with optimum costs over relatively long-time intervals. [ABSTRACT FROM AUTHOR]

We consider a time-delayed HIV/AIDS epidemic model with education dissemination and study the asymptotic dynamics of solutions as well as the asymptotic behavior of the endemic equilibrium with respect to the amount of information disseminated about the disease. Under appropriate assumptions on the infection rates, we show that if the basic reproduction number is less than or equal to one, then the disease will be eradicated in the long run and any solution to the Cauchy problem converges to the unique disease-free equilibrium of the model. On the other hand, when the basic reproduction number is greater than one, we prove that the disease will be permanent but its impact on the population can be significantly minimized as the amount of education dissemination increases. In particular, under appropriate hypothesis on the model parameters, we establish that the size of the component of the infected population of the endemic equilibrium decreases linearly as a function of the amount of information disseminated. We also fit our model to a set of data on HIV/AIDS in order to estimate the infection, effective response, and information rates of the disease. We then use these estimates to present numerical simulations to illustrate our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper the authors aspire to obtain the approximate analytical solution of Modified Burgers Equation with newly defined conformable derivative by employing homotopy analysis method (HAM).

In this paper, we propose a new approximate method, namely fractional natural decomposition method (FNDM) in order to solve a certain class of nonlinear time-fractional wave-like equations with variable coefficients. The fractional natural decomposition method is a combined form of the natural transform method and the Adomian decomposition method. The nonlinear term can easily be handled with the help of Adomian polynomials which is considered to be a clear advantage of this technique over the decomposition method. Some examples are given to illustrate the applicability and the easiness of this approach.