BCA Semester 4 – Numerical Methods – Unit 6: Solution of Partial Differential Equations (5Hrs.)

Question 1

Solve the Poisson equation \[\quad \quad ∇^2f=2x^2y^2\] over the square domain \(0≤x≤3,0≤y≤3\) with f=0 on the boundary and \(h=1\).

Marks: 5

Year: 2021 Final TU FOHSS

Given: \[ \nabla^2 f = 2x^2 y^2 \] Domain: \[ 0 \le x \le 3,\quad 0 \le y \le 3 \] Boundary condition: \[ f = 0 \quad \text{on all boundaries} \] Step size: \[ h = 1 \] --- Finite Difference Approxim

Question 2

Write a short note on (Any Two):\ \(\quad \)a) Types of Errors\ \(\quad \)b) Convergent Methods\ \(\quad \)c) PDE

Marks: 5

Year: 2021 Final TU FOHSS

(a) Types of Errors In numerical methods, errors are the difference between the exact value and the approximate value. Types of Errors: 1. Round-off Error - Occurs due to rounding of numbers dur

Question 3

Solve the Poisson equation \[ \begin{aligned} \Delta^2 f &= 2x^2 y^2 \end{aligned} \] over the square domain \(0≤x≤3and0≤y≤3\) with \(f=0\) on the boundary and \(h=1\).

Marks: 5

Year: 2022 Final TU FOHSS

Given: \[ \nabla^2 f = 2x^2 y^2 \] Domain: \(0 \le x \le 3, \; 0 \le y \le 3\) Boundary condition: \(f = 0\) on all boundaries Step size: \(h = 1\) --- Step 1: Finite Difference Approximation The

Question 4

Solve for the steady state temperatures in a rectangular plate of 8cm x 10cm, if one 10cm side is held at 50°C, and the other 10cm side is held at 30°C and other two sides are held at 10°C. Assume gri

Marks: 5

Year: 2023 Final TU FOHSS

Problem Statement - Rectangular plate: \(8\,\text{cm} \times 10\,\text{cm}\) - Boundary conditions: | Side | Temperature | |------|------------| | Left (x=0) | 50°C | | Right (x=10) | 30°C | | Top (

Question 5

Write a short note on (Any Two):\ \(\quad\) a) Partial Differential Equations\ \(\quad\) b) Linear Interpretation\ \(\quad\) c) Boundary value problems

Marks: 5

Year: 2023 Final TU FOHSS

a) Partial Differential Equations (PDEs) - A partial differential equation (PDE) involves an unknown function of two or more independent variables and its partial derivatives. - General form: \[ F

Question 6

Solve the laplace equation \(Uxx + Uyy = 0\) for the following square mesh with the boundary values.

Marks: 5

Year: 2024 Final TU FOHSS

1. Problem Statement Solve the Laplace equation: \[ \frac{\partial^2 U}{\partial x^2} + \frac{\partial^2 U}{\partial y^2} = 0 \] on a square domain using a finite difference mesh with given boundary v