I have been trying to solve this differential equation using Runge Kutta method on Python. I haven't been able to get the right result with the code I wrote. What could be wrong here? The equation is:
$$ \frac{du_l}{dx} = u, \frac{du}{dx} = (V-w^2)u_l $$The code that I wrote is: (Here, Rd is the second derivative, x0 and x1 are the boundary x coordinate values, V is a dictionary, V[l]=some matrix (V varies with l), ul0 is the initial value for the solution, n is a variable which decides the initial value of the first derivative(1,-1). h is the step size, l is some range over which I need to calculate values). [The indentation here is not right, but it works fine on Python].
def Rd(ul,V): return (V-w**2)*uldef sol(Rd,x0,x1,V,ul0,n,h,l): ulR_final={} n1=int((x1-x0)/h) xVs1=xV(x0,x1,V,h,l) xVs2=xV(x0,x1,V,h/2,l) for j in range(1,l,1): xV1=xVs1[j] xV2=xVs2[j] #x1data=pd.DataFrame(xV1[:,0]) #x2data=pd.DataFrame(xV2[:,0]) #data=pd.concat([x1data,x2data],axis=1) #print(data.head(50)) #print(len(xV1),len(xV2)) x_values=xV1[:,0] ul_values=[ul0] u0=math.sqrt(abs(w**2-xV1[0,1]))*ul0*1j*n u_values=[u0] ulR_values=np.zeros((n1,4), dtype=complex) for i in range(n1-1): x=xV1[i,0] V=xV1[i,1] x1=xV1[i+1,0] V_1=xV1[i+1,1] x2=xV2[(2*i)+1,0] V_2=xV2[(2*i)+1,1] #print(x,'x+h/2',x2,x+(h/2),'x+h',x1,x+h) ul=ul_values[-1] u=u_values[-1] k1ul=h*u k2ul=h*(u+(k1ul/2)) k3ul=h*(u+(k2ul/2)) k4ul=h*(u+(k3ul/2)) k1u=h*Rd(ul,V) k2u=h*Rd(ul+(k1u/2),V_2) k3u=h*Rd(ul+(k2u/2),V_2) k4u=h*Rd(ul+(k3u),V_1) ul_new=ul+((k1ul+(2*k2ul)+(2*k3ul)+k4ul)/6) u_new=u+((k1u+(2*k2u)+(2*k3u)+k4u)/6) ul_values.append(ul_new) u_values.append(u_new) ulR_values[i,0]=x ulR_values[i,1]=np.round(ul_new,decimals=5) ulR_values[i,2]=np.round(u_new,decimals=5) ulR_values[i,3]=j ulR_final[j]=ulR_values[:-1] return ulR_final