CODEWITHSUNDEEP
pythondifferential-equationspde
vlovero
vlovero

Reputation: 300

Implementing Numerical Solution to Fitzhugh-Nagumo Model With Spatial Diffusion Factor

I'm trying to come up with a python implementation for the Fitzhugh-Nagumo model. 

V_t = V_xx + V(V - a)(1 - V) - W + I
W_t = eps(beta*V - W)

Using the really basic code for eps = 0.05, a = 0.2, beta = 5, I = .1 I can numerically solve the system(with out the V_xx), but I can't seem to figure out how to implement the spacial diffusion.

def func_v(v, w):
    return v * (1 - v) * (v - .2) - w + .1


def func_w(v, w):
    return .05 * (5 * v - w)


def get_yn(t0, v, w, h, t):
    while t0 < t:
        w += h * func_w(v, w)
        v += h * func_v(v, w)
        t0 += h
    return v, w

I know the centered difference formula for second order derivatives is

V_xx(x_i, t) = (V(x_i+1, t) - 2*V(x_i, t) + V(x_i-1, t)) / dx^2

but how would I implement the different values for x_i(let's say from x=0 to 10) in order to get the wave to propagate along the x-axis?

enter image description here

The results should give a wave that propagates something like this.

Upvotes: 1

Views: 1648

Answers (1)

Lutz Lehmann
Lutz Lehmann

Reputation: 26040

An ODE solver (any computer program really) only can treat problems that have a finite dimensional state. The state in this PDE is a pair of functions v,w of x. These can not, in the necessary generality, be represented in a computer. Thus you need to work with finite approximations. A first one that is deemed sufficient in many contexts is the simple function table. Then the x derivatives are computed using finite difference formulas.

x = np.linspace(0,L,N+1);
dx = x[1]-x[0];
v0,w0 = initial_functions(x);

def func_v(v, w):
    d2v = -2*v;
    d2v[0] += v[-1]+v[1];
    d2v[1:-1] += v[:-2] + v[2:]
    d2v[-1] += v[-2]+v[0];
    return d2v/dx**2 + v * (1 - v) * (v - .2) - w + .1

etc.

For a proof-of-concept the Euler method may be sufficient, but the values obtained will be questionable. Use a higher order method to get usable results without employing ridiculously small time steps.

Upvotes: 1

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