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I'm currently working on a thesis in Economics. I have stumbled upon a system of differential equations that needs to be solved. I am stuck, and have trouble getting the right help from my advisor who is also not very acquainted with numerical methods. For the past couple of days I have been searching the net for a way to solve my problem, but since all else failed, I am here.

This is the system:

[itex]\dot{K}=F(K,R)-G(S)R-C-\delta K[/itex]

[itex]\dot{S}=-R[/itex]

[itex]\dot{C}=\sigma C(F_{K}(K,R)-\delta-\rho)[/itex]

[itex]\dot{R}=\frac{(\alpha K^{\alpha-1} R^{\beta}-\delta)(\beta K^{\alpha} R^{\beta-1}-\delta-\rho)-\alpha\beta K^{\alpha-1} R^{\beta-1} \dot{K}}{\beta (\beta-1)K^{\alpha} R^{\beta-2}}

[/itex]

with boundary terms K(0)=K_{0}, S(0)=S_{0}, K(T)=K_{1}, C(T)=C_{1}, that determine the entire system. F and G are continuous function (F cobb douglas and G=constant/S, for instance)

If all the constraints would have been at t=0, i.e. K(0), S(0), C(0) and R(0) had been known, then I could use Euler linearization:

[itex]

y'(t)\approx \frac{y(t+h)-y(t)}{h},

y(t+h)\approx y(t)+h*y'(t)

[/itex]

And hence if I know the values of my variables at t=0, then I can find out by iteratively caluting the dotted variables at times t what the value of my variables will be at later times t+h.

But now the boundary constraints are given at time t and I need to transform my system somehow, in order to find the solution paths. Since my system is not linear I don't know how to write it in the form y'=Ay+b and then decompose A into MDM^-1, where D is diagonal. If I could do that, that would already help, but I don't know how to linearize.

Does someone know a good method to solve this system? Thanks a thousand.

Jacob

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# Numerical PDE boundary problem methodology

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