Numerical Methods · Unit 10
Partial Differential Equations
Exam-focused notes for Partial Differential Equations (Numerical Methods, BIT203): what the TU syllabus asks and how it has actually been tested, with 4 solved past questions from this unit.
What this unit covers
- Poisson's equation and finite difference method
- Laplace equation and steady state problems
- Grid discretization and boundary conditions
- Iterative solution of PDE systems
- Heat conduction and temperature distribution problems
Poisson's equation and finite difference method
Solve the Poisson's equation ∂2f/∂x2+∂2f/∂y2=2x2y2\partial^2f/\partial x^2+\partial^2f/\partial y^2 = 2x^2y^2∂2f/∂x2+∂2f/∂y2=2x2y2 over the square domain 0<=x<=3 and 0<=y<=3 with f=0 on the boundary and h = 1. [5]
- PDE: $\dfrac{\partial^2 f}{\partial x^2} + \dfrac{\partial^2 f}{\partial y^2} = 2x^2 y^2$, so $g(x,y) = 2x^2y^2$ - Domain: $0 \le x \le 3$, $0 \le y \le 3$ (square) - Boundary condition: $f = 0$ on all boundaries - Mesh spacing: $h = 1$ All data present. ...
Full solved answer →Solve the Poisson's Equation $\nabla^2 f = 2x^2 y^2$ over the square domain $0 \leq x \leq 3$ and $0 \leq y \leq 3$ with $f = 0$ on the boundary and $h = 1$. [5]
- PDE: $\nabla^2 f = f{xx} + f{yy} = 2x^2y^2$ - Domain: $0 \le x \le 3$, $0 \le y \le 3$ (square) - Boundary condition: $f = 0$ on all boundaries - Step size: $h = 1$ With $h=1$ over $[0,3]$, interior nodes are at $x=1,2$ and $y=1,2$ (four interior points)....
Full solved answer →Solve the Poisson's equation $\nabla^2 f = xy$ with $f = 2$ on boundary by assuming square domain $0 \leq x \leq 3$, $0 \leq y \leq 3$ and $h = 1$. [5]
- PDE: $\nabla^2 f = xy$, i.e. $g(x,y) = xy$ - Domain: $0 \le x \le 3$, $0 \le y \le 3$ - Step size: $h = 1$ - Boundary condition: $f = 2$ on all boundaries With $h = 1$: nodes at $x = 0,1,2,3$ and $y = 0,1,2,3$. Interior unknowns: - $f1 = f(1,1)$ - $f2 = f...
Full solved answer →Laplace equation and steady state problems
Consider a steel plate of size 24cm x 24cm. If two of the opposite sides are held at 100 degree Celsius and the other two opposite sides at 0 degree Celsius, find the steady state temperatures of interior points, assuming a grid size of 8cm x 8cm. [5]
- Plate size: $24 \text{ cm} \times 24 \text{ cm}$ - Grid spacing: $h = 8 \text{ cm}$ - Boundary conditions: - Two opposite sides held at $100^\circ$C - Other two opposite sides held at $0^\circ$C - Interior points required (steady state) With $h = 8$ cm, t...
Full solved answer →Make Unit 10 stick
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