how to solve a second order differential equation using Euler's method?

2018 4 17
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更新时间：2018 4 18
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How to solve a second order differential equation (boundary value problem) using Euler's Method without using inbuilt matlab functions such as ode45?

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James Tursa
James Tursa 2018-4-17
What instructions were you given in class? What have you done so far? What specific problems are you having with your code?
Remston Martis
Remston Martis 2018-4-18
编辑：Remston Martis 2018-4-18
I have a second order differential equations with 2 boundary conditions.
I have to use Euler's method(the shooting method) to solve the equation.
I am able to code for a first order differential equation but not for a second order differential equation.
function Eout = Eulers(F, yint,h,yfinal,x0)
% F is the desired differential equation
% yint and yfinal are the boundary value conditions
% h is the step size
% x0 is the initial guess
x = x0;
Eout = x;
for y = yint : h : tfinal-h
s = F(y,x);
x = x + h*s;
Eout = [Eout; x];
end
The code works for a simple first order differential equation.
My differential equation is d^2y/dx^2 = 2y +8x(9-x)
I have managed to solve it mathematically on paper but am struggling with the code. I can provide you with the mathematical solution if you require it.

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Torsten
Torsten 2018-4-18

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If you define

F=@(y,x)[y(2);2*y(1)+8*x*(9-x)]

you can work with the code from above.

Best wishes

Torsten.

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Remston Martis
Remston Martis 2018-4-18
When I do that I receive the error "Index exceeds array bounds"
Torsten
Torsten 2018-4-18
You've set x0 as a (2x1) vector, haven't you ?
Jan
Jan 2018-4-18
编辑：Jan 2018-4-18
@Remston Matris: Please post the complete error message, such that the readers do not have to guess, in which line the error occurs.
Remston Martis
Remston Martis 2018-4-18
Sorry for being excessively terse.
error message "Error in @(y,x)[y(2);2*y(1)+8*x*(9-x)]
Error in Eulers(line 14) s=F(y,x)
Remston Martis
Remston Martis 2018-4-18
编辑：Remston Martis 2018-4-18
I have not set x0 as a vector. It is an integer.
x0 is my initial "shooting" guess. I do not understand why it should be a vector.
Torsten
Torsten 2018-4-18
编辑：Torsten 2018-4-18
You have a second order differential equation.
This has to be written as a system of two first-order ODEs:
y1'=y2
y2'=y1+8*x*(9-x)
So you will need two initial values.
That's the reason why x0 (and x) are (2x1).
But I think, according to your notation y is the independent variable and x is the dependent variable to be solved for. Thus, F has to read
F=@(y,x)[x(2);2*x(1)+8*y*(9-y)]
Remston Martis
Remston Martis 2018-4-18
No, y is my dependent variable. I'll try your code and revert if it works. I think I understand where you're getting at. Thank you .
Torsten
Torsten 2018-4-18
y can't be your dependent variable because you update x as
x = x + h*s;
where s is obtained by
s = F(y,x);
Jan
Jan 2018-4-18
@Remston Martis: Using y, x and t causes confusion here. You have e.g. "yfinal" and "tfinal". Use either "x and y" or "x and t". Then:
function dx = F(t, x)
dx = [x(2); x(1) + 8 * t * (9 - t)];
end
Now "yint and yfinal are the boundary value conditions" confuses me, but I assume you want something like:
x = x0;
tvec = tint:h:tfinal-h;
Eout = zeros(2, size(tvec));
Eout(:, 1) = x;
for idx = 1:length(tvec)
t = tvec(idx);
dx = F(t,x);
x = x + h * dx;
Eout(:, idx) = x;
end

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更多回答（1 个）

Jan
Jan 2018-4-18

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It is worth to be nitpicking:

 % x0 is the initial guess

No, x0 is the initial value of the trajectory when you consider the integration. To solve a boundary value problem, you need an additional layer around the integration: e.g. a single shooting or multiple shooting method. Then this x0 is the initial guess of the shooting method.

To solve your problem, convert the 2nd order equation to a system of two equations of order 1. Then y has 2 components: The initial position and velocity. Converting higher order equations to order 1 is the first step for almost all integrators.

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Remston Martis
Remston Martis 2018-4-18
Hi, Thank you for the information. I have actually solved the entire question on paper but mathematically so I guess I'm on the same page with what you have stated.
However, I do not know how to apply the 2nd order equation conversion into my code. I am unable to translate it into the code.
Would greatly appreciate if you could just help with those couple lines of code that I could add into the above.
Thank you!

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