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Unformatted text preview: AIMSLectureNotes 2006 Peter J. Olver 10. Numerical Solution of Ordinary Differential Equations This part is concerned with the numerical solution of initial value problems for systems of ordinary differential equations. We will introduce the most basic one-step methods, beginning with the most basic Euler scheme, and working up to the extremely popular RungeKutta fourth order method that can be successfully employed in most situations. We end with a brief discussion of stiff differential equations, which present a more serious challenge to numerical analysts. 10.1. First Order Systems of Ordinary Differential Equations. Let us begin by reviewing the theory of ordinary differential equations. Many physical applications lead to higher order systems of ordinary differential equations, but there is a simple reformulation that will convert them into equivalent first order systems. Thus, we do not lose any generality by restricting our attention to the first order case throughout. Moreover, numerical solution schemes for higher order initial value problems are entirely based on their reformulation as first order systems. First Order Systems A first order system of ordinary differential equations has the general form du 1 dt = F 1 ( t,u 1 ,...,u n ) , du n dt = F n ( t,u 1 ,...,u n ) . (10 . 1) The unknowns u 1 ( t ) ,...,u n ( t ) are scalar functions of the real variable t , which usually represents time. We shall write the system more compactly in vector form d u dt = F ( t, u ) , (10 . 2) where u ( t ) = ( u 1 ( t ) ,...,u n ( t ) ) T , and F ( t, u ) = ( F 1 ( t,u 1 ,...,u n ) ,...,F n ( t,u 1 ,...,u n ) ) T is a vector-valued function of n + 1 variables. By a solution to the differential equation, we mean a vector-valued function u ( t ) that is defined and continuously differentiable on an interval a < t < b , and, moreover, satisfies the differential equation on its interval of definition. Each solution u ( t ) serves to parametrize a curve C R n , also known as a trajectory or orbit of the system. 2/25/07 152 c circlecopyrt 2006 Peter J. Olver In this part, we shall concentrate on initial value problems for such first order systems. The general initial conditions are u 1 ( t ) = a 1 , u 2 ( t ) = a 2 , u n ( t ) = a n , (10 . 3) or, in vectorial form, u ( t ) = a (10 . 4) Here t is a prescribed initial time, while the vector a = ( a 1 ,a 2 ,...,a n ) T fixes the initial position of the desired solution. In favorable situations, as described below, the initial conditions serve to uniquely specify a solution to the differential equations at least for nearby times. The general issues of existence and uniquenss of solutions will be addressed in the following section....
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