In this paper, we are going to approximate the dynamical system of the SIR mathematical model, numerically. The expression for basic reproduction number R0 is obtained which plays main role in the local and global stability. On the other hand, if R0 > 1 then the equilibrium point is unstable. Our numerical simulations show that our proposed NSFD method provides reliable and consistent, positive and converging results at all step sizes as compared to the traditional RK-4 method and Euler method which diverge at large step sizes and produce negative and unstable results with large oscillations. The local stability of the equilibrium point is proved by Jacobean method,however, Schur-Cohn conditions are used to discuss the local stability for the discrete NSFD scheme. To prove the global stability of equilibria, the Enatsu criterion and Lyapunov function are used. This paper presents theoretical findings and numerical simulations that can be used as a powerful instrument for forecasting the spread of added infectious illnesses.

This paper considers the collocation method for the numerical solution of the Volterra integro- differential equation using polynomial basis functions. The modeled equation was converted into a linear algebraic system of equations and matrix inversion was employed to solve the algebraic equation. We substitute the result of algebraic into the approximate solution to obtain the numerical result. Some numerical problems are solved to show the method's efficiency and consistency.

In this article, a numerical scheme is constructed to approximate the generalized fractional Volterra integro-differential equations with the regularized Prabhakar derivative. The solution of the problem is represented in the form of inverse Laplace transform in the complex plane. Then we select the parabolic contour as an optimal contour and use trapezoidal rule to approximate the inverse Laplace transform. Next, the performance of the numerical scheme is implemented for an example. Further, we obtain the absolute errors for various parameters by using our numerical scheme on parabolic contour and show that the proposed algorithm for the solution of inverse Laplace transform is a very well algorithm with high order accuracy.

In this work we constructed a numerical scheme to approximate the Volterra integro-differential equations of convolution type using Laplace transform. The solution of the problem is recovered using inverse Laplace transform as contour integral in the complex plane. The integral is then approximated along a suitable contour using the trapezoidal rule with equal step size. The solution accuracy depends on optimal contour of integrations to compute accurately the inverse Laplace transform. For better accuracy two types of contour parabolic and hyperbolic are used which are available in the literature. The performance of the numerical scheme is tested for different examples. The actual error well agree with the corresponding error estimates of the proposed numerical scheme for both parabolic as well as hyperbolic contours.

In this article, Ritz approximation have been employed to obtain the numerical solutions of a class of the fractional optimal control problems based on the Caputo fractional derivative. Using polynomial basis functions, we obtain a system of nonlinear algebraic equations. This nonlinear system of equation is solved and the coefficients of basis polynomial are derived. The convergence of the numerical solution is investigated. Some numerical examples are presented which illustrate the theoretical results and the performance of the method.

In this paper, a practical matrix method is presented for solving a particular type of telegraph equations. This procedure is based on Bernouli Polynomials. This matrix method with collocation suited nodes, decreases the supposed equations into system of algebric equations with unknown Bernouli coefficients. The obtained system is solved and approximate solutions are achieved. The well-conditioning of problems is also considered. The indicated method creates the well-onditioned problems. Some numerical problems are comprised to confirm the efficacy and fitting of the suggested method. The presented technique is easy to implement and produces accurate results. The precision of the method is demonstrated by measuring the errors between exact solutions and approximate solutions for each problem.