Non-Linear BVPs

Consider the following non-linear BVP in the x domain [0,1] given by

$$\frac{d^2v}{dx^2} = 3v+x^2+10v^3,\:\:\: v(0)=v(1)=0.$$ (37)

Once again, let's divide the domain into n equal intervals of length h. Using the finite difference approximation given in Eq. 32, we get

$$v_{i+1} - 2v_i + v_{i-1} - h^2 (3 v_i + {(i-1)}^2h^2 + 10 v_i^3) = 0, \:\:\: i=2,3,\cdots n.$$ (38)

The boundary conditions give the remaining two equations, i.e., v₁ = 0 and v_n+1 = 0. The FD equations for the non-linear problem above differ from those obtained for the linear BVP (compare Eqs. 36 with 39). The FD approximation of the linear BVP results in a system of linear equations whereas that of the non-linear BVP results in a system of non-linear equations. In Eq. 39, the non-linearity arises from the cubic term in v present in the BVP. The the solution of the n+1 non-linear equations can be obtained using Newton's method where the unknowns are $v_1, v_2, \cdots , v_{n+1}$. Recall that Newton's method is iterative, and it requires the solution of a system of linear equations at every iteration step. Once again we see that the solution of a system of linear equations is indeed a central problem of scientific computing.

Exercise Problems

1a. Show that the FD equations (Eq. 39) can be rewritten as n-1 equations in n-1 unknowns $v_2,v_3, \cdots v_n$ using the fact that the boundary conditions imply v₁ = v_n+1 = 0. Name the n-1 unknowns as $y_1, y_2, \cdots , y_{n-1}$ where y₁ = v₂, $y_2 = v_3, \cdots , y_{n-1} = v_n$.
1b. Express the equations using the vector notation in the form ${\bf {f}}({\bf {y}})={\bf {0}}$, where ${\bf {f}}={(f_1,f_2,\cdots ,f_{n-1})}$ and ${\bf {y}}={(y_1,y_2,\cdots y_{n-1})}^T$. For instance, the the first equation (f₁ = 0) should read y₂ - 2y₁ - h²(3y₁ + h² + 10y₁³) = 0.
1c. Derive the Jacobian matrix ${\bf {J}}=\{J_{ij}\}=\{ \partial f_i/\partial y_j \}$ for the above system of equations (refer to the notes on Newton's method).

2. The non-linear BVP given below represents a reaction-diffusion process with non-linear chemical reaction kinetics.

$$\frac{d^2 C}{dx^2} - \frac{\alpha C}{1+\beta C} = 0,$$ C(x=0) = 1,

$$\frac{dC}{dx}(x=1) - \gamma C (x=1) = 0,$$ (39)

where $\alpha$, $\beta$ and $\gamma$ are positive constants and $0\leq x\leq 1$. Develop the FD equations (based on Eqs. 32 and 33) for the above BVP. How can we solve the system of equations?