# Finite difference method without fiction points

I need to use finite difference method without introducing fiction points to solve the following problem:

$$−\mu u′′(x)+\eta u′(x)+\sigma u(x)=f(x)$$, $$a subject to the boundary conditions $$u′(a) = \alpha$$ and $$u′(b) = \beta$$.

I discretize the derivatives with second order central differences formulas at interior points ($$a < x < b$$), but use one-sided forward and backward differences to approximate the boundary conditions (at $$x = a$$ and at $$x = b$$).

$$u_1 - u_0 = \alpha h$$;

$$u_N - u_{N-1} = \beta h$$

At $$i=0$$ I have to introduce a fiction point of $$x_{-1}$$:

$$(-\mu + \eta h/2)u_{1} + (2 \mu + \sigma h^2)u_0 + (-\mu - \eta h/2)u_{-1}=f_0 h^2$$

What am I doing wrong? Thank you

Those one-sided differences aren't what you want because the central difference approximations to the derivatives have error $$O(h^2)$$ and the one-sided differences have error $$O(h)$$. So you can expand in Taylor series about $$x=a$$: \begin{align}x_0&=x_0\\ x_1&=x_0+hx_0^{\prime}+\frac12h^2x_0^{\prime\prime}+O(h^3)\\ x_2&=x_0+2hx_0^{\prime}+2h^2x_0^{\prime\prime}+O(h^3)\end{align} You want $$c_0x_0+c_1x_1+c_2x_2=x_0^{\prime}+O(h^2)$$. Comparing coefficients of like derivatives of $$x_0$$, we get \begin{align}c_0+c_1+c_2&=0\\ c_1+2c_2&=\frac1h\\ \frac12c_1+2c_2&=0\end{align} With solution $$c_0=-\frac3{2h}$$, $$c_1=\frac2h$$, and $$c_2=-\frac1{2h}$$, so now your boundary condition reads $$\frac{-3u_0+4u_1-u_2}{2h}=\alpha$$ The other boundary condition reads something like $$\frac{u_{N-2}-4u_{N-1}+3u_N}{2h}=\beta$$ You don't need the equations with $$h^2f_0$$ and $$h^2f_N$$ because you are using the boundary conditions at $$x=a$$ and $$x=b$$.
• I just plugged in the Taylor series for $x_0$, $x_1$, and $x_2$ to get \begin{align}c_0x_0+c_1x_1+c_2x_2&=(c_0+c_1+c_2)x_0+(c_1+2c_2)hx_0^{\prime}+\left(\frac12c_1+2c_2\right)h^2x_0^{\prime\prime}\\ &=(0)x_0+(1)x_0^{\prime}+(0)x_0^{\prime\prime}\end{align} So really I was comparing coefficients of $x_0$, $x_0^{\prime}$, and $x_0^{\prime\prime}$ to get the $3$ equations for $c_0$, $c_1$, and $c_2$. Feb 26 '19 at 3:29