Numerical Study of Soliton Solutions of KdV, Boussinesq, and Kaup-Kuperschmidt Equations Based on Jacobi Polynomials

In this paper, a numerical method is developed to approximate the soliton solutions of some nonlinear wave equations in terms of the Jacobi polynomials. Wave are very important phenomena in dispersion, dissipation, diffusion, reaction, and convection. Using the wave variable converts these nonlinear equations to the nonlinear ODE equations. Then, the operational Collocation method based on Jacobi polynomials as bases is applied to approximate the solution of ODE equation resulted. In addition, the intervals of the solution will be extended using an rational exponential approximation (REA). The KdV, Boussinesq, and Kaup–Kuperschmidt equations are studied as the test examples. Finally, numerical computation of the conservation values shows the effectiveness and stability of the proposed method

A high-order nonlinear Schrödinger equation with the weak non-local nonlinearity and its optical solitons

Abstract The present paper explores a high-order nonlinear Schrodinger equation in a non-Kerr law media with the weak non-local nonlinearity describing solitons' propagation through nonlinear optical fibers. To this end, the real and imaginary parts of the model are firstly extracted using a wave variable transformation. The modified Kudryashov method and symbolic computations are then adopted to successfully retrieve optical solitons of the model. The results presented in the current study demonstrate the great performance of the modified Kudryashov method in handling high-order nonlinear Schrodinger equations

Shifted Jacobi Collocation Method Based on Operational Matrix for Solving the Systems of Fredholm and Volterra Integral Equations

This paper aims to construct a general formulation for the shifted Jacobi operational matrices of integration and product. The main aim is to generalize the Jacobi integral and product operational matrices to the solving system of Fredholm and Volterra equations. These matrices together with the collocation method are applied to reduce the solution of these problems to the solution of a system of algebraic equations. The method is applied to solve system of linear and nonlinear Fredholm and Volterra equations. Illustrative examples are included to demonstrate the validity and efficiency of the presented method. Also, several theorems, which are related to the convergence of the proposed method, will be presented

A new operational method to solve Abel's and generalized Abel's integral equations

Based on Jacobi polynomials, an operational method is proposed to solve the generalized Abel's integral equations (a class of singular integral equations). These equations appear in various fields of science such as physics, astrophysics, solid mechanics, scattering theory, spectroscopy, stereology, elasticity theory, and plasma physics. To solve the Abel's singular integral equations, a fast algorithm is used for simplifying the problem under study. The Laplace transform and Jacobi collocation methods are merged, and thus, a novel approach is presented. Some theorems are given and established to theoretically support the computational simplifications which reduce costs. Also, a new procedure for estimating the absolute error of the proposed method is introduced. In order to show the efficiency and accuracy of the proposed method some numerical results are provided. It is found that the proposed method has lesser computational size compared to other common methods, such as Adomian decomposition, Homotopy perturbation, Block-Pulse function, mid-point, trapezoidal quadrature, and product-integration. It is further found that the absolute errors are almost constant in the studied interval

A new numerical method for delay and advanced integro-differential equations

A general formulation is constructed for Jacobi operational matrices of integration, product, and delay on an arbitrary interval. The main purpose of this study is to improve Jacobi operational matrices for solving delay or advanced integro–differential equations. Some theorems are established and utilized to reduce the computational costs. All algorithms can be used for both linear and nonlinear Fredholm and Volterra integro-differential equations with delay. An error estimator is introduced to approximate the absolute error when the exact solution of a given problem is not available. The error of the proposed method is less compared to other common methods such as the Taylor collocation, Chebyshev collocation, hybrid Euler–Taylor matrix, and Boubaker collocation methods. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments

Low cost numerical solution for three-dimensional linear and nonlinear integral equations via three-dimensional Jacobi polynomials

In recent years, numerical methods have been introduced to solve two-dimensional Volterra and Fredholm integral equations. In this study, a numerical scheme is constructed to solve classes of linear and nonlinear three-dimensional integral equations (Volterra, Fredholm, and mixed Volterra–Fredholm). This operational approach is proposed to easily and directly solve these equations at low computational costs. The scheme is based on the Jacobi polynomials on the interval [0, 1] where three-variable Jacobi polynomials are introduced and their operational matrices of integration and product are derived. Compared to other existing methods for multidimensional problems, the Jacobi operational method eliminates the time-consuming computations and solely employs the one-dimensional operational matrix to construct corresponding multidimensional operational matrices. The absolute error of the proposed method is almost constant on the studied interval even at higher dimensions, confirming the stability of the proposed operational Jacobi method. Required theorems on the convergence of the method are proved in Jacobi-weighted Sobolev space. It is established that the error function vanishes as NN increases. The method is evaluated using several illustrative examples which indicate the proposed method with lesser computational size compared to the Block–Pulse functions, differential transform, and degenerate kernel methods

A mathematical system of COVID-19 disease model:existence, uniqueness, numerical and sensitivity analysis

Abstract A compartmental mathematical model of spreading COVID-19 disease in Wuhan, China is applied to investigate the pandemic behaviour in Iran. This model is a system of seven ordinary differential equations including individual behavioural reactions, governmental actions, holiday extensions, travel restrictions, hospitalizations, and quarantine. We fit the Chinese model to the Covid-19 outbreak in Iran and estimate the values of parameters by trial-error approach. We use the Adams-Bashforth predictor-corrector method based on Lagrange polynomials to solve the system of ordinary differential equations. To prove the existence and uniqueness of solutions of the model we use Banach fixed point theorem and Picard iterative method. Also, we evaluate the equilibrium points and the stability of the system. With estimating the basic reproduction number R₀, we assess the trend of new infected cases in Iran. In addition, the sensitivity analysis of the model is assessed by allocating different parameters to the system. Numerical simulations are depicted by adopting initial conditions and various values of some parameters of the system

Designing a Matrix Collocation Method for Fractional Delay Integro-Differential Equations with Weakly Singular Kernels Based on Vieta–Fibonacci Polynomials

In the present work, the numerical solution of fractional delay integro-differential equations (FDIDEs) with weakly singular kernels is addressed by designing a Vieta–Fibonacci collocation method. These equations play immense roles in scientific fields, such as astrophysics, economy, control, biology, and electro-dynamics. The emerged fractional derivative is in the Caputo sense. By resultant operational matrices related to the Vieta–Fibonacci polynomials (VFPs) for the first time accompanied by the collocation method, the problem taken into consideration is converted into a system of algebraic equations, the solving of which leads to an approximate solution to the main problem. The existence and uniqueness of the solution of this category of fractional delay singular integro-differential equations (FDSIDEs) are investigated and proved using Krasnoselskii’s fixed-point theorem. A new formula for extracting the VFPs and their derivatives is given, and the orthogonality of the derivatives of VFPs is easily proved via it. An error bound of the residual function is estimated in a Vieta–Fibonacci-weighted Sobolev space, which shows that by properly choosing the number of terms of the series solution, the approximation error tends to zero. Ultimately, the designed algorithm is examined on four FDIDEs, whose results display the simple implementation and accuracy of the proposed scheme, compared to ones obtained from previous methods. Furthermore, the orthogonality of the VFPs leads to having sparse operational matrices, which makes the execution of the presented method easy

A generalized Chebyshev operational method for Volterra integro-partial differential equations with weakly singular kernels

Volterra integro-partial differential equations with weakly singular kernels (VIPDEWSK) are utilized to model diverse physical phenomena. A matrix collocation method is proposed for determining the approximate solution of this functional equation category. The method employs shifted Chebyshev polynomials of the fifth kind (SCPFK) to construct two-dimensional pseudo-operational matrices of integration, avoiding the need for explicit integration and thereby speeding up computations. Error bounds are examined in a Chebyshev-weighted space, providing insights into approximation accuracy. The approach is applied to several experimental examples, and the results are compared with those obtained using the Bernoulli wavelets and Legendre wavelets methods

Designing a Matrix Collocation Method for Fractional Delay Integro-Differential Equations with Weakly Singular Kernels Based on Vieta–Fibonacci Polynomials

In the present work, the numerical solution of fractional delay integro-differential equations (FDIDEs) with weakly singular kernels is addressed by designing a Vieta–Fibonacci collocation method. These equations play immense roles in scientific fields, such as astrophysics, economy, control, biology, and electro-dynamics. The emerged fractional derivative is in the Caputo sense. By resultant operational matrices related to the Vieta–Fibonacci polynomials (VFPs) for the first time accompanied by the collocation method, the problem taken into consideration is converted into a system of algebraic equations, the solving of which leads to an approximate solution to the main problem. The existence and uniqueness of the solution of this category of fractional delay singular integro-differential equations (FDSIDEs) are investigated and proved using Krasnoselskii’s fixed-point theorem. A new formula for extracting the VFPs and their derivatives is given, and the orthogonality of the derivatives of VFPs is easily proved via it. An error bound of the residual function is estimated in a Vieta–Fibonacci-weighted Sobolev space, which shows that by properly choosing the number of terms of the series solution, the approximation error tends to zero. Ultimately, the designed algorithm is examined on four FDIDEs, whose results display the simple implementation and accuracy of the proposed scheme, compared to ones obtained from previous methods. Furthermore, the orthogonality of the VFPs leads to having sparse operational matrices, which makes the execution of the presented method easy