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Chapter 7: Methods for nonlinear systems of ODES
Published in Andrei D. Polyanin, Valentin F. Zaitsev, Handbook of Ordinary Differential Equations, 2017
Andrei D. Polyanin, Valentin F. Zaitsev
A solution (also an integral or an integral curve) of a system of differential equations (7.1.1.1) is a set of functions y1 = y1(x), ..., yn = yn(x) such that, when substituted into all equations (7.1.1.1), they turn them into identities. The general solution of a system of differential equations is the set of all its solutions. In the general case, the general solution of system (7.1.1.1) depends on n arbitrary constants.
Introduction to Systems of ODEs
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2022
A system of differential equations is a set of equations involving more than one unknown function and their derivatives. The order of a system of differential equations is the highest derivative that occurs in the system. In this section, we consider only first order systems of differential equations. As we saw in §7.1, certain problems lead naturally to systems of nonlinear differential equations in normal form: {dx1/dt=g1(t,x1,x2,…,xn),dx2/dt=g2(t,x1,x2,…,xn),⋮⋮dxn/dt=gn(t,x1,x2,…,xn),
Introduction
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2018
A system of differential equations is a set of equations involving more than one unknown function and their derivatives. The order of a system of differential equations is the highest derivative that occurs in the system. In this section, we consider only first order systems of differential equations. As we saw in §6.1, certain problems lead naturally to systems of nonlinear differential equations in normal form: dx1/dt=g1t,x1,x2,…,xndx2/dt=g2t,x1,x2,…,xn⋮⋮dxn/dt=gnt,x1,x2,…,xn
Binary chemically reactive flow of time-dependent Oldroyd-B nanofluid with variable properties
Published in Waves in Random and Complex Media, 2022
Muhammad Yasir, Masood Khan, Awais Ahmed
To deal with the solution of nonlinear system of differential equations, several approaches are applied. Since then, the equation arising from the estimated flow provided by the sheet extension has been exceedingly nonlinear, and the exact solution to these equations has been impossible to find. The OHAM argues that the scheme has the capacity to calculate the solutions of nonlinear-coupled problems that arise in a variety of mathematical physics phenomena for the approximate solution. There is no discretization, linearization, or perturbation in this technique, which reduces the computing cost. It is more consistent and trustworthy for nonlinear issues in mathematical physics. Thus, the outcomes of present physical phenomenon are constructed through this technique and provided in graphical view. Moreover, each result is explained physically.
Development of dynamic thermal input models for simulation of photovoltaic generators
Published in International Journal of Ambient Energy, 2020
S. N. Nnamchi, O.D. Sanya, K. Zaina, V. Gabriel
Also, Jones and Underwood (2001) presented the PV cell temperature in dynamic form by carrying out an overall thermal balance around the PV generator, which consequently unify the glass, PV cell and base_tedlar temperature. Pertinently, Tofighi (2013) presented a component thermal balance around the PV generator, but casted the nonlinear system of differential equations into state–space notation, which supposedly proffers a linear solution to the system of differential equation without appropriate assumptions to reduce it to the linear system of differential equations. Besides, the radiation model was represented with the convection equation. Thus there is an outstanding need to first solve the nonlinear system of differential equations and subsequently to make appropriate assumptions that will lead to the linear system of differential equations. Furthermore, the solutions will be validated with experimental data to select an appropriate solution.
A POD-based reduced-order model for uncertainty analyses in shallow water flows
Published in International Journal of Computational Fluid Dynamics, 2018
Jean-Marie Zokagoa, Azzeddine Soulaïmani
Consider a solution of a nonlinear system of differential equations: Now, let us select a set of discrete values (snapshots) . The snapshot-based POD method (Sirovich 1987) consists of finding a basis of functions , which are solutions of the minimisation problem: where is the inner product in H, is the norm and the Kronecker symbol. A solution to the problem (2) is given by the first eigenvectors that correspond to the largest eigenvalues of a correlation matrix R: The field can then be approximated as in which are the time-dependent modal coordinates.