Numerical Methods: Fundamentals and Applications 1st. Edition by Rajesh Kumar Gupta | 9781108716000 | 1108716008 | Holooly

Numerical Methods

Numerical Methods: Fundamentals and Applications

288 SOLVED PROBLEMS

Question: 16.21

Obtain an explicit finite difference scheme for the solution of following variable coefficient problem ∂u/∂t = ∂²u/∂x² + x∂u/∂x 0≤x≤1, t≥0 with following initial and boundary conditions u(x, 0) = x(2 − x) u(0, t) = 0, u(1, t) = 1 Replace temporal derivative term with forward difference and spatial ...

Verified Answer:

The spacing is Δx = 0.2 for 0 ≤ x ≤ 1; so, our nod...

Question: 16.20

Solve the wave equation, ∂²u/∂t² = 4∂²u/∂x²; 0 ≤ x ≤ 1, t ≥ 0 with initial conditions u(x, 0) = sin(πx) and ∂u/∂t|t=0 = 0 boundary conditions u(0, t) = u(1, t) = 0 t ≥ 0 Take the step size for time t is 1/6, and step size for x is 1/3. Use implicit scheme to compute the solution up to time t = 1/3. ...

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The spacing h = 1/3 and k = 1/6 for variables x an...

Question: 16.18

Solve the wave equation ∂²u/∂t² = 4∂²u/∂x² with initial conditions u(x,0) = {0.1(x) 0 ≤ x ≤ 1/2 0.1(1-x) 1/2 ≤ x ≤ 1 and ∂u/∂t|t=0 = 0 boundary conditions u(0, t) = u(1, t) = 0 t ≥ 0. Take the step size for t is 0.1, and step size for x is 0.25. Use explicit scheme to compute the solution up to ...

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It is given that step size for t is 0.1 (k = 0.1) ...

Question: 16.19

A tightly stretched flexible string has its ends fixed at x = 0 and x = 1. The string is plucked at middle point by an initial displacement 0.05 and then released from this position. Find the transverse displacement of a point at a distance x from one end and at any time t of the vibrating string. ...

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The mathematical model for this problem is exactly...

Question: 16.17

A Poisson equation uxx + uyy = 4(x + y) is defined over a domain 0 ≤ x, y ≤ 0.75 with the following boundary conditions, u = 0 on the sides x = 0, 0.75 and y = 0 (Dirichlet conditions) ∂u/∂y = u on y = 0.75 (Neumann condition) Solve the given Poisson equation by dividing the domain into squares of ...

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Nodes in the directions of variables x and y are [...

Question: 16.16

Solve the following Poisson problem ∇²u = e^x + y in ℜ, where ℜ is the square 0 ≤ x, y ≤ 0.75. Given that u = x² + y² on the boundary of the square ℜ. Take h = 0.25. ...

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The node points are as follows

\begin{alig...

Question: 16.15

Solve the Poisson equation uxx+ uyy = x² + y² for a thin rectangular plate, whose edges x = 0, x = 2 are kept at 0°C (in ice) and edges y = 0, y = 2 are kept at temperature 100°C (in boiling water). Find the values of u(x, y) at the nodal points of the rectangular region with mess length 0.5. Use ...

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The edges of the square region are x = 0, x = 2, y...

Question: 16.14

Solve the Poisson equation ∇²u = uxx + uyy = x + y for the square mesh, whose edges, x = 0, x = 0.8, y = 0, y = 0.6 are kept at the temperature shown in the following figure. Find the values of u(x, y) at the nodal points of the rectangular region with mess length 0.2. Use Gauss–Seidel iterative ...

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The edges of the rectangular region are x = 0, x =...

Question: 16.13

The Laplace equation ∇²u = uxx + uyy = 0 is defined over the following triangular region, {x = 0; y = 0; x + y = 5}. The following boundary conditions are prescribed i) u = 0 over the edges x = 0, y = 0 ii) u = 25 − x² − y² at the edge x + y = 5 Find the values of u(x, y) at the nodal points of ...

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Since we have to compute the values of u(x, y) at ...

Question: 16.12

The Laplace equation ∇²u = uxx + uyy = 0 is defined over the square region {0 ≤ x ≤ 0.6; 0 ≤ y ≤ 0.6}. The boundary conditions are defined by i) u = 0 over the edges x = 0, y = 0, y = 0.6 (Dirichlet condition) ii) ∂u/∂x = 1 at the edge x = 0.6. (Neumann condition) Find the values of u(x, y) at ...

Verified Answer:

The nodal points of the square region {0 ≤ x ≤ 0.6...

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