Boundary Value Problem with non boundary values

For solving a Boundary Value Problem with non boundary values (for instance a second order differential defined at [0,1] with values at x=0.2 and x=1) is there a way only using bvp4c except from solving at [0.2,1] and afterwards using bvpxtend for predicting solution's behaviour ?

General suggestions are also welcomed !

[Torsten]

Your problem description is unclear.

[Charalampos Papargyriou]

Ode equation is x^2 *y''(x) = exp(x) defined in [0,1] with Dirichlet boundary values y(0.2) = -1 and y(1) = +2

[Torsten]

Solve your equation on [0,1] by taking the boundary value at x=1 and assuming a boundary value at x=0. Evaluate this solution at x=0.2. Usually, this value won't be the same as the prescribed condition at x=0.2. Use fzero to adjust the boundary condition at x=0 such that both values become the same.

By the way:

Your problem has an analytical solution

y(x)=(x-1)*Ei(x)+6.44286*x-exp(x)-1.72458

where Ei(x) is the exponential integral.

So this example is much suited to test the method I suggested above.

[Torsten]

This is a possible implementation (untested !)

The nonlinear equation solver "fzero" tries to find y'(1) such that the solution of the ODE passes through the point (0.2,-1).

function main

y0dot0 = -1.0;

y0dot = fzero(@fun,y0dot0);

tstart = 1.0;

tend = 1e-8;

tspan = [tstart tend];

y0 = [2 y0dot];

[T,Y] = ode15s(@ode,tspan,y0);

plot(T,Y(:,1))

end

function s = fun(y0dot)

tstart = 1.0;

tend = 0.2;

tspan = [tstart tend];

y0 = [2 y0dot];

[T,Y] = ode15s(@ode,tspan,y0);

s = Y(end,1) + 1;

end

function dy = ode(t,y)

dy = [y(2);exp(x)/x^2];

end

[Alex Sha]

Hi, look at your ODE function: y'' = exp(x)/x^2, if x=0, will cause the error of 0/0 .

[Lewis Fer]

Alex Sha, If you can reply on my post ( code )in

https://www.mathworks.com/matlabcentral/answers/853120-how-to-solve-a-system-of-second-order-nonlinear-differential-equations-with-boundary-conditions#comment_1591370

Answer 1

[Jan]

Yes, BVP4C can solve multi-point conditions also. See:

doc bvp4c

• https://www.mathworks.com/help/matlab/ref/bvp4c.html#mw_1119c222-d5cc-4a25-bbd5-1bc4318a650e
• https://www.mathworks.com/help/matlab/math/solve-bvp-with-multiple-boundary-conditions.html

[Torsten]

But no boundary condition is given at x=0 ...

[Jan]

@Torsten: A good point. I prefer single and multiple-shooting approachs, for which it does not matter, where the conditions appear.

[Charalampos Papargyriou]

@Jan such as ?

[Jan]

@Charalampos Papargyriou: I do not understand the question.

[Charalampos Papargyriou]

@Jan I am talking about the other "single and multiple-shooting approaches" you prefer .

Do you have any examples of such methods ?

[Jan]

• https://www.mathworks.com/matlabcentral/fileexchange/62564-single-shooting-method-example
• https://sam.pfrommer.us/tutorial-single-shooting-trajectory-optimization-with-matlab
• https://www.youtube.com/watch?v=u-b4fJGOxP0

The codes I use for the shooting methods for my scientific work are not in a state, which can be published currently.