Using the notation from the text, the finite difference equation for solving the boundary value problem $y ^ { \prime \prime } + P ( x ) y ^ { \prime } + Q ( x ) y = f ( x ) , y ( a ) = \alpha , y ( b ) = \beta$ is
A) $\left( 1 - h P _ { i } / 2 \right) y _ { i + 1 } + \left( - 2 + h ^ { 2 } Q _ { i } \right) y _ { i } + \left( 1 + h P _ { i } / 2 \right) y _ { i - 1 } = h ^ { 2 } f _ { i }$
B) $\left( 1 - h P _ { i } / 2 \right) y _ { i + 1 } + \left( - 2 + h ^ { 2 } Q _ { i } \right) y _ { i } + \left( 1 - h P _ { i } / 2 \right) y _ { i - 1 } = h ^ { 2 } f _ { i }$
C) $\left( 1 + h P _ { i } / 2 \right) y _ { i + 1 } + \left( - 2 + h ^ { 2 } Q _ { i } \right) y _ { i } + \left( 1 + h P _ { i } / 2 \right) y _ { i - 1 } = h ^ { 2 } f _ { i }$
D) $\left( 1 + h P _ { i } / 2 \right) y _ { i + 1 } + \left( 2 - h ^ { 2 } Q _ { i } \right) y _ { i } + \left( 1 - h P _ { i } / 2 \right) y _ { i - 1 } = h ^ { 2 } f _ { i }$
E) $\left( 1 + h P _ { i } / 2 \right) y _ { i + 1 } + \left( - 2 + h ^ { 2 } Q _ { i } \right) y _ { i } + \left( 1 - h P _ { i } / 2 \right) y _ { i - 1 } = h ^ { 2 } f _ { i }$

Question

Quizplus · Accepted Answer

The correct finite difference equation for the given boundary value problem incorporates both the first derivative term with a coefficient that changes sign across the $y_{i+1}$ and $y_{i-1}$ terms, and the second derivative term with a negative coefficient in front of $y_i$. This is accurately represented in option E, where the coefficients of $y_{i+1}$ and $y_{i-1}$ correctly account for the first derivative term $P(x)y'$ with differing signs, and the coefficient of $y_i$ correctly represents the second derivative term and the $Q(x)y$ term.

Using the Notation from the Text, the Finite Difference Equation y′′+P(x)y′+Q(x)y=f(x),y(a)=α,y(b)=βy ^ { \prime \prime } + P ( x ) y ^ { \prime } + Q ( x ) y = f ( x ) , y ( a ) = \alpha , y ( b ) = \betay′′+P(x)y′+Q(x)y=f(x),y(a)=α,y(b)=β

Using the Notation from the Text, the Finite Difference Equation $y ^ { \prime \prime } + P ( x ) y ^ { \prime } + Q ( x ) y = f ( x ) , y ( a ) = \alpha , y ( b ) = \beta$