- #1
jjr
- 51
- 1
Hello
I am having some trouble understanding what the difference is between the reference solution and the "true" solution of a set of certain differential equation. The book I'm working with is Numerical Analysis by Walter Gautschi. Equation 5.16 from this book is a set of first order differential equations with a set of initial conditions: [tex] \frac{d\textbf{y}}{dx} = \textbf{f} (x, \textbf{y} ), a \leq x \leq b; \textbf{y} (a) = \textbf{y}_0. [/tex]
3 pages later he introduces this reference solution that is confusing me a bit. I quote:
"[...] we consider the solution [itex] \textbf{u} (t) [/itex] of the differential equation (5.16) passing through the point [itex] (x, \textbf{y} ) [/itex], that is, the local intial value problem [tex] \frac{d\textbf{u}}{dt} = \textbf{f}(t, \textbf{u} ), x \leq t \leq x + h; \textbf{u} (x) = \textbf{y}." [/tex]
This is for discrete-variable methods of approximation, so h here is the step length in x between the points one is trying to approximate.
So as I stated in the beginning, I'm having some trouble understanding the difference between these solutions. The function [itex] \textbf{f} [/itex] is the same in both cases, but in the former case it is valid for the whole interval [itex] [a,b] [/itex], whereas in the latter it is only valid locally between [itex] x [/itex] and [itex] x+h [/itex]. Does this mean that one can find a solution of the differential equation 5.16 that is only valid on a small interval, but is not equal to a true solution on the whole interval? Is [itex] \textbf{f} = \textbf{u} [/itex] for the whole interval [itex] [x,x+h] [/itex] or just at the very start of the interval? In the book they are talking about a local description of one-step method, in which one is trying to find an approximation to [itex] \textbf{u} [/itex], which suggests that [itex] \textbf{u} [/itex] is indeed an acceptable solution (but only on a small interval?). So does one use different [itex] \textbf{u} [/itex]'s for every new interval [itex] [x,x+h] [/itex] in one's attempt to reach an approximation to the global solution?
I hope I made the source of my confusion clear, and I will be happy to elaborate if needed.
J
I am having some trouble understanding what the difference is between the reference solution and the "true" solution of a set of certain differential equation. The book I'm working with is Numerical Analysis by Walter Gautschi. Equation 5.16 from this book is a set of first order differential equations with a set of initial conditions: [tex] \frac{d\textbf{y}}{dx} = \textbf{f} (x, \textbf{y} ), a \leq x \leq b; \textbf{y} (a) = \textbf{y}_0. [/tex]
3 pages later he introduces this reference solution that is confusing me a bit. I quote:
"[...] we consider the solution [itex] \textbf{u} (t) [/itex] of the differential equation (5.16) passing through the point [itex] (x, \textbf{y} ) [/itex], that is, the local intial value problem [tex] \frac{d\textbf{u}}{dt} = \textbf{f}(t, \textbf{u} ), x \leq t \leq x + h; \textbf{u} (x) = \textbf{y}." [/tex]
This is for discrete-variable methods of approximation, so h here is the step length in x between the points one is trying to approximate.
So as I stated in the beginning, I'm having some trouble understanding the difference between these solutions. The function [itex] \textbf{f} [/itex] is the same in both cases, but in the former case it is valid for the whole interval [itex] [a,b] [/itex], whereas in the latter it is only valid locally between [itex] x [/itex] and [itex] x+h [/itex]. Does this mean that one can find a solution of the differential equation 5.16 that is only valid on a small interval, but is not equal to a true solution on the whole interval? Is [itex] \textbf{f} = \textbf{u} [/itex] for the whole interval [itex] [x,x+h] [/itex] or just at the very start of the interval? In the book they are talking about a local description of one-step method, in which one is trying to find an approximation to [itex] \textbf{u} [/itex], which suggests that [itex] \textbf{u} [/itex] is indeed an acceptable solution (but only on a small interval?). So does one use different [itex] \textbf{u} [/itex]'s for every new interval [itex] [x,x+h] [/itex] in one's attempt to reach an approximation to the global solution?
I hope I made the source of my confusion clear, and I will be happy to elaborate if needed.
J