PDE to Ax=b

PDE to Ax=b

Wanho Choi

(wanochoi.com)

y = f(x)

x y

0

h

x₀ x₁ x₂ x₃ x_i−1 x_i x_i+1 x_n−2 x_n−1 x_n

y = f(x)

⋯

⋯

h

x₀ x₁ x₂ x₃ x_i−1 x_i x_i+1 x_n−2 x_n−1 x_n

y = f(x)

⋯

⋯

y₀ y₁ y₂ y₃ y_i−1 y_i y_{i+1 yn−2 yn−1 yn}

h x₀ x₁ x₂ x₃ x_i−1 x_i x_i+1 x_n−2 x_n−1 x_n y = f(x)

⋯

⋯

Finite Differencing for the i-th Point

y′(x_i) = y(xi+1) − y(xi) h

y′(x_i) = y(xi) − y(xi−1) h

y′(x_i) = y(xi+1) − y(x_2h i−1)

The 2nd Derivative for the i-th Point

y′′(x_i) = y′(xi+0.5) − y′(xi−0.5)

h =

y(x_i+1) − y(xi)

h −

y(x_i) − y(x_i−1) h

h ∴ y′′ = y(xi−1) − 2y(xi) + y(xi+1)

h2

x₀ x₁ x₂ x₃ x_i−1 x_i x_i+1 x_n−2 x_n−1 x_n

y = f(x)

⋯

⋯

Taylor Expansion Series

1! + y′′(x_i)h2 2! + y′′′(x_i)h3 3! + ⋯ http://xaktly.com/TaylorSeries.html

Taylor Expansion Series

1! +

y′′(x_i)h2

2! +

y′′′(x_i)h3

3! + ⋯

Taylor Expansion Series

1! +

y′′(x_i)h2

2! +

y′′′(x_i)h3

3! + ⋯

y(x_i − h) = y(x_i) − y′(x_1!i)h + y′′(x_2!i)h2 − y′′′(x_3!i)h3 + ⋯

(a)

(b)

(a)-(b): y′(x_i) = y(xi + h) − y(xi − h)

2h + O(h3)

(a)+(b): y′′(x_i) = y(xi − h) − 2y(xi) + y(xi − h)

A Simple PDE Problem

y′′ = f(x) on [0,1] with y(0) = y(1) = 0 y(x_i) = ? (i = 0,1,2,⋯, n)

Finite Differencing

y′′ = f(x) on [0,1] with y(0) = y(1) = 0 y(x_i) = ? (i = 0,1,2,⋯, n)

y(x_i−1) − 2y(x_i) + y(x_i+1)

Finite Differencing

y′′ = f(x) on [0,1] with y(0) = y(1) = 0 y(x_i) = ?

y(x_i−1) − 2y(x_i) + y(x_i+1)

h2 ≈ y′′(xi) = fi

∴ y(x_i−1) − 2y(x_i) + y(x_i+1) ≈ h2f_i

For Every Point

y′′ = f(x) on [0,1] with y(0) = y(1) = 0 y(x_i) = ? y₋₁ − 2y₀ + y₁ = h2_f 0 y₀ − 2y₁ + y₂ = h2_f 1 y₁ − 2y₂ + y₃ = h2_f 2 y_n−2 − 2y_n−1 + y_n = h2_f n

⋱

Linear System of Matrix Equation

y(x_i) = ? (i = 0,1,2,⋯, n) 2 −1 0 0 ! 0 0 0 0 −1 2 −1 0 ! 0 0 0 0 0 −1 2 −1 ! 0 0 0 0 0 0 −1 2 ! 0 0 0 0 " " " " # " " " " 0 0 0 0 ! 2 −1 0 0 0 0 0 0 ! −1 2 −1 0 0 0 0 0 ! 0 −1 2 −1 0 0 0 0 ! 0 0 −1 2 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ y₀ y₁ y₂ y₃ ! y_n−3 y_n−2 y_n−1 y_n ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ = h2 f₀ h2 f₁ h2 f₂ h2 f₃ ! h2 f_n−3 h2 f_n−2 h2 f_n−1 h2 f_n ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

Linear System of Matrix Equation

y(x_i) = ? (i = 0,1,2,⋯, n) 2 −1 0 0 ! 0 0 0 0 −1 2 −1 0 ! 0 0 0 0 0 −1 2 −1 ! 0 0 0 0 0 0 −1 2 ! 0 0 0 0 " " " " # " " " " 0 0 0 0 ! 2 −1 0 0 0 0 0 0 ! −1 2 −1 0 0 0 0 0 ! 0 −1 2 −1 0 0 0 0 ! 0 0 −1 2 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ y₀ y₁ y₂ y₃ ! y_n−3 y_n−2 y_n−1 y_n ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ = h2 f₀ h2 f₁ h2 f₂ h2 f₃ ! h2 f_n−3 h2 f_n−2 h2 f_n−1 h2 f_n ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

Ax

= b