I am trying to solve a system of coupled ODEs:$$  \begin{align}    \frac{dn_A}{dt} & = e\left[j(t) - f\, _H\sinh\left(\frac{g\,n_A}{T}\right)\right] \\    \frac{d_H}{dt} & = a\left[bP\,(1 - _H )^2 - c~(_H)^2 - f\,_H\,\sinh\left(\frac{g\,n_A}{T}\right)\right] \\    \frac{dP}{dt} & =h(i - P) + d\,T\,(P-P_a)\left[bP~(1 - _H )^2 - c~(_H)^2\right].  \end{align}$$Here $e, f, g, T, a, b, c,h,i,d$ are constants, $j$ is a time dependent variable, and $P_a$ = 101325. The initial values for the three parameters $n_A, _H, P$ are $0, 0.5$, and $8106$ respectively.I was able to solve the equations using the explicit Euler method, but I am having trouble applying the implicit Euler method to do the same. For the first equation I can use the explicit Euler method to estimate the initial value for $n_A$ at time point $t_{n+1}$, but I am not able to figure out what should I take as a value for the $_H$ term that also gets involved in the Newton-Raphson iteration scheme for calculation at $t_{n+1}$.
How can I implement the implicit Euler method for a small nonlinear system of ODEs?
matlab;ode;implicit methods
Define$$ \begin{align}  A^{n+1} & = e\left[j^{n+1} - f\, _H^{n+1}\sinh\left(\frac{g\,n_A^{n+1}}{T}\right)\right] \\  B^{n+1} & = a\left[bP^{n+1}\,(1 - _H^{n+1} )^2 - c~(_H^{n+1})^2 - f\,_H^{n+1}\,\sinh\left(\frac{g\,n_A^{n+1}}{T}\right)\right] \\  C^{n+1} & = h(i - P^{n+1}) + d\,T\,(P^{n+1}-P_a)\left[bP^{n+1}~(1 - _H^{n+1} )^2 - c~(_H^{n+1})^2\right]  \end{align}$$Then backward Euler gives you$$ \begin{align}   n_A^{n+1} & = n_A^n + \Delta t\,A^{n+1} \\   \theta_H^{n+1} & = \theta_H^n + \Delta t\,B^{n+1} \\   P^{n+1} & = P^n + \Delta t\,C^{n+1} \,. \end{align}$$Reorganize into the form$$ \begin{align} F^{n+1} &:= n_A^{n+1} - \Delta t\,A^{n+1} -  n_A^n = 0 \\ G^{n+1} &:= \theta_H^{n+1} - \Delta t\,B^{n+1} - \theta_H^n = 0 \\ H^{n+1} &:= P^{n+1} -  \Delta t\,C^{n+1} - P^n = 0  \end{align}$$Let us assume that you have an equation for $j^{n+1}$ that can be solved explicitly.  Then the above system of equations has the form$$  \mathbf{r}(\mathbf{u}^{n+1}) = 0$$where $\mathbf{u}^{n+1} = (n_A^{n+1}, \theta_H^{n+1}, P^{n+1})$.  Newton's method can then be expressed as$$  \mathbf{u}^{n+1}_{k+1} = \mathbf{u}^{n+1}_k - \mathbf{T}^{-1}(\mathbf{u}^{n+1}_k)\,\mathbf{r}(\mathbf{u}^{n+1}_k)$$where$$  \mathbf{T}(\mathbf{u}^{n+1}_k) = \frac{\partial \mathbf{r}(\mathbf{u}^{n+1}_k)}{\partial \mathbf{u}^{n+1}_k} \,.$$The components of the matrix $\mathbf{T}$ are found using the usual relations$$  T_{ij} = \frac{\partial r_i}{\partial u_j} \,.$$You already have initial values for $\mathbf{u}$.  I'm not sure why you can't use those for your Newton iterations.  If Newton iterations don't converge you may have to use an alternative algorithm (such as bisection/line search etc.)