Numerical Method · Unit 6 · 5 hrs
Solution of Partial Differential Equations
Exam-focused notes for Solution of Partial Differential Equations (Numerical Method, CSC212): what the TU syllabus asks and how it has actually been tested, with 6 solved past questions from this unit.
What this unit covers
- Review of partial differential equations
- Classification of partial differential equation
- Deriving difference equations
- Laplacian equation and Poisson's equation
- engineering examples
Laplacian equation and Poisson's equation
Solve the Poisson equation with boundary conditions:
$$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = -64xy, \quad 0 \leq x \leq 1, ; 0 \leq y \leq 1$$
$$u(0,y) = 0, ; u(x,0) = 0, ; u(1,y) = 150, ; u(x,1) = 150, ; h = \frac{1}{3}$$
[5]
PDE: $$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = -64xy, \quad 0 \le x \le 1,\ 0 \le y \le 1$$ So $f(x,y) = -64xy$. Boundary Conditions: - $u(0,y) = 0$ (left) - $u(x,0) = 0$ (bottom) - $u(1,y) = 150$ (right) - $u(x,1) = 150$ (to...
Full solved answer →Solve the Poisson's equation $\nabla^2 f = xy$ and $f = 2$ on boundary by assuming square domain $0 \leq x \leq 3$ and $0 \leq y \leq 3$ and $h = 1$. [5]
- PDE: $\nabla^2 f = xy$ - Boundary condition: $f = 2$ on all boundaries - Domain: $0 \le x \le 3$, $0 \le y \le 3$ - Step size: $h = 1$ Grid points at $x = 0,1,2,3$ and $y = 0,1,2,3$. Interior nodes at $(1,1),(2,1),(1,2),(2,2)$. $$f{i-1,j} + f{i+1,j} + f{i...
Full solved answer →Solve the Poisson's equation $\nabla^2 f = 3x^2y$ over the square domain $0 \leq x \leq 3$, $0 \leq y \leq 3$ with $f = 0$ on the boundary and $h = 1$. [5]
- PDE: $\nabla^2 f = 3x^2y$, so source $g(x,y) = 3x^2y$ - Domain: $0 \le x \le 3$, $0 \le y \le 3$ (square) - Boundary condition: $f = 0$ on all boundaries - Mesh size: $h = 1$ With $h=1$, interior nodes at $x=1,2$ and $y=1,2$. Four unknowns. Label (using s...
Full solved answer →Consider a metallic plate of size 90cm by 90cm. The two adjacent sides of the plate are maintained at a temperature of $100^\circ C$ and the remaining two adjacent sides are held at $200^\circ C$. Calculate the steady-state temperature at interior points assuming a grid size of 30 cm by 30 cm. [5]
- Plate: 90 cm × 90 cm - Grid spacing: $h = 30$ cm → interior nodes form a 2×2 arrangement (4 unknowns) - Two adjacent sides at $100^\circ C$ - Remaining two adjacent sides at $200^\circ C$ Governing equation: $$\frac{\partial^2 T}{\partial x^2} + \frac{\pa...
Full solved answer →A plate of dimension 18cm x 18cm is subjected to temperatures as follows: left side at $100^\circ C$, right side at $200^\circ C$. Upper part at $50^\circ C$, and lower at $150^\circ C$. If square grid length of 6cm x 6cm is assumed, what will be the temperature at the interior nodes? [5]
- Plate dimension: $18 \text{ cm} \times 18 \text{ cm}$ - Grid length: $6 \text{ cm} \times 6 \text{ cm}$ - Left boundary: $100^\circ C$ - Right boundary: $200^\circ C$ - Upper (top) boundary: $50^\circ C$ - Lower (bottom) boundary: $150^\circ C$ Grid divis...
Full solved answer →Solve the Poisson's equation over the square domain $0 \leq x \leq 1.5$, $0 \leq y \leq 1.5$ with $f = 0$ on the boundary and $h = 0.5$. [5]
- Domain: $0 \le x \le 1.5$, $0 \le y \le 1.5$ - Boundary condition: $f = 0$ on all boundaries - Step size: $h = 0.5$ - Poisson equation right-hand side: NOT LEGIBLE in the question. The function $f$ (source term) appears garbled. This solution takes $\nabl...
Full solved answer →Make Unit 6 stick
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