Comparison Between Numerical Methods for Generalized Zakharov system

We present two numerical methods to get approximate solutions for generalized Zakharov system GZS. The first one is Legendre collocation method, which assumes an expansion in a series of Legendre polynomials , for the function and its derivatives occurring in the GZS, the expansion coefficients are then determined by reducing the problem to a system of algebraic equations. The second is differential transform method DTM , it is a transformation technique based on the Taylor series expansion. In this method, certain transformation rules are applied to transform the problem into a set of algebraic equations and the solution of these algebraic equations gives the desired solution of the problem.The obtained numerical solutions compared with corresponding analytical solutions.The results show that the proposed method has high accuracy for solving the GZS.

Quantum Algorithms for Solving Ordinary Differential Equations via Classical Integration Methods

Identifying computational tasks suitable for (future) quantum computers is an active field of research. Here we explore utilizing quantum computers for the purpose of solving differential equations. We consider two approaches: (i) basis encoding and fixed-point arithmetic on a digital quantum computer, and (ii) representing and solving high-order Runge-Kutta methods as optimization problems on quantum annealers. As realizations applied to two-dimensional linear ordinary differential equations, we devise and simulate corresponding digital quantum circuits, and implement and run a 6th order Gauss-Legendre collocation method on a D-Wave 2000Q system, showing good agreement with the reference solution. We find that the quantum annealing approach exhibits the largest potential for high-order implicit integration methods. As promising future scenario, the digital arithmetic method could be employed as an "oracle" within quantum search algorithms for inverse problems.

Shifted Legendre Collocation Method for the Solution of Unsteady Viscous-Ohmic Dissipative Hybrid Ferrofluid Flow over a Cylinder

A numerical treatment for the unsteady viscous-Ohmic dissipative flow of hybrid ferrofluid over a contracting cylinder is provided in this study. The hybrid ferrofluid was prepared by mixing a 50% water (H2O) + 50% ethylene glycol (EG) base fluid with a hybrid combination of magnetite (Fe3O4) and cobalt ferrite (CoFe2O4) ferroparticles. Suitable parameters were considered for the conversion of partial differential equations (PDEs) into ordinary differential equations (ODEs). The numerical solutions were established by expanding the unknowns and employing the truncated series of shifted Legendre polynomials. We begin by collocating the transformed ODEs by setting the collocation points. These collocated equations yield a system of algebraic equations containing shifted Legendre coefficients, which can be obtained by solving this system of equations. The effect of the various influencing parameters on the velocity and temperature flow profiles were plotted graphically and discussed in detail. The effects of the parameters on the skin friction coefficient and heat transfer rates were further presented. From the discussion, we come to the understanding that Eckert number considerably decreases both the skin friction coefficient and the heat transfer rate.

Numerical Studies for Solving a Free Convection Boundary–Layer Flow Over a Vertical Plate

In this paper, The aim of this study is to present a reliable combination of the shifted Legendre collocation method to approximate of the problem of free convection boundary-layer flow over a vertical plate as produced by a body force about a flat plate in the direction of the generating body force. The proposed method is based on replacement of the unknown function by truncated series of well known shifted Legendre expansion of functions. An approximate formula of the integer derivative is introduced. Special attention is given to study the convergence analysis and derive an upper bound of the error of the presented approximate formula. The introduced method converts the proposed equation by means of collocation points to a system of algebraic equations with shift Legendre coefficients. Thus, by solving this system of equations, the shifted Legendre coefficients are obtained. Boundary conditions in an unbounded domain, i.e. boundary condition at infinity, pose a problem in general for the numerical solution methods. The obtained results are in good agreement with those provided previously by the iterative numerical method. As a result, without taking or estimating missing boundary conditions, the shifted Legendre collocation method provides a simple, non-iterative and effective way for determining the solutions of nonlinear free convection boundary layer problems possessing the boundary conditions at infinity.