Orthogonal cubic splines for the numerical solution of nonlinear parabolic partial differential equations.

In this paper, a new orthogonal basis for the space of cubic splines has been introduced. A linear combination of cubic orthogonal splines is considered to approximate the functions in which the coefficients are calculated with numerically stable formulae. Applications to the numerical solutions of some parabolic partial differential equations are given, in which the approximations are obtained using the first and second integral of orthogonal splines which leads to an efficient solution procedure. The convergence analysis in the approximate scheme is investigated. A comparison of the obtained numerical solutions with some other papers indicates that the presented method is reliable and yields result with good accuracy. The main parts of our study are as follows:•We propose a robust approach based on the orthogonal cubic splines procedure in conjunction with the operational matrix.•The convergence in the approximate scheme is analyzed.•Numerical examples show that the proposed method is very accurate.

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

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 R 0 , 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.

A Novel Effective Approach for Solving Fractional Nonlinear PDEs.

The present work introduces an effective modification of homotopy perturbation method for the solution of nonlinear time-fractional biological population model and a system of three nonlinear time-fractional partial differential equations. In this approach, the solution is considered a series expansion that converges to the nonlinear problem. The new approximate analytical procedure depends only on two iteratives. The analytical approximations to the solution are reliable and confirm the ability of the new homotopy perturbation method as an easy device for computing the solution of nonlinear equations.

An efficient method for systems of variable coefficient coupled Burgers' equation with time-fractional derivative.

A new homotopy perturbation method (NHPM) is applied to system of variable coefficient coupled Burgers' equation with time-fractional derivative. The fractional derivatives are described in the Caputo fractional derivative sense. The concept of new algorithm is introduced briefly, and NHPM is examined for two systems of nonlinear Burgers' equation. In this approach, the solution is considered as a power series expansion that converges rapidly to the nonlinear problem. The new approximate analytical procedure depends on two iteratives. The modified algorithm provides approximate solutions in the form of convergent series with easily computable components. Results indicate that the introduced method is promising for solving other types of systems of nonlinear fractional-order partial differential equations.