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A bar with a length of 1 m has a fixed temperature of 100°C at one...
(1) Use finite the difference method to solve for the temperature profile given by the equation b...
(1) Use finite the difference method to solve for the temperature profile given by the equation below. The thin rod is one (1) meter long. The temperature on the left end is 100 and 0 on the right end. Set up the problem for three internal nodes (unknowns). Set up the augmented matrix for Gauss elimination solution (do not solve). Roughly sketch the five T values (including BC's).
(1) Use finite the difference method to solve for the temperature profile...
1. [12 marks] In the following parts of this question, write a MATLAB code to solve a linear syst...
1. [12 marks] In the following parts of this question, write a MATLAB code to solve a linear system A b (A is a square nonsingular matrix) using Jacobi and Gauss-Seidel algorithms. Do not use the built-in Matlab functions for solving linear systems (a) Write a Matlab function called Jacobi that consumes a square n x n matrix A, and an n x 1 vector b, and uses the Jacobi technique to solve the system Ax-b, starting with the zero...
50 70 100 180 100 70 50 20 20 20 10mm 20 40 60 90 140...
50 70 100 180 100 70 50 20 20 20 10mm 20 40 60 90 140 90 60 40 Figure 1: A rectangular plate with temperature defined on the boundaries 4 Figure 1 shows a rectangular plate made of an homogeneous isotropic material The temperature distribution in this plate satisfies the indicated boundary condi- tions (given in degrees centigrade) and has reached a steady-state condition so that the temperature is described by Laplace's equation a2T a2T (i) Draw a sketch...
LT An aluminum rod of diameter D 2.5 cm and of length from a wall maintained at T, 300"C. The.coovective heat coefficient h 17 W/m2-K with an ambie.t air temperature T Thermal conductivity k =...
LT An aluminum rod of diameter D 2.5 cm and of length from a wall maintained at T, 300"C. The.coovective heat coefficient h 17 W/m2-K with an ambie.t air temperature T Thermal conductivity k = 204 WmK. Assume there is e luid by convection from the end of the rod, which is at temperature Ta. Using 2. -38c. nergy transfer to the a finite difference method, with Ax = 5 cm, the fo be derived for the solution of the...
Problem 1: A metallic fin of length is attached to a hot surface (at To) and is exposed to cold a...
Problem 1: A metallic fin of length is attached to a hot surface (at To) and is exposed to cold air at To. The temperature at the end of the fin is T1. The governing equation for the temperature along the length of the fin is given below where h is the heat transfer coefficient, k is the thermal conductivity of metal, P is the perimeter of the fin, and A is the cross-sectional area of the fin. (a) Do...
Question 1 (10 marks) For a linear system Ax b with 1 0-1 A-1 2-1 2-13 and b4 18 compute by hand ...
Question 1 (10 marks) For a linear system Ax b with 1 0-1 A-1 2-1 2-13 and b4 18 compute by hand the first four iterations with the Jacobi method, usg0 Hint: for the ease of calculation, keep to rational fractions rather than decimals. (10 marks) Question 2 For the same linear svstem as in Question 1. compute by hand the first three iterations with the Gauss Seidel method, us0 Hint: for the ease of calculation, keep to rational fractions...
Partial Differential Equations
QUESTION 22.1 Assume you have a 2D concrete slab that has dimensions of 1 x 1. The boundaries x=0 and y=0 both have a temperature of 00C, while the boundaries x=1 and y=1 both have a temperature of 150x and 120y respectively. (i) Plot/draw the computational domain taking step sizes in both the x-direction and the y-direction to be 0.1 units. Show all the boundary conditions. (10)(ii) How many computational nodes are in this problem? (5)2.2 Assuming the problem is a steady state heat...
Partial Differential Equations Question: A homogeneous cylindrical rod of length L = 1 is insulated along...
Partial Differential Equations Question: A homogeneous cylindrical rod of length L = 1 is insulated along the cylindrical side. At the end caps, heat exchange obeys Newton’s law of cooling, i.e. the flux is proportional to the difference of the temperature of the rod with that of the surrounding medium, written explicitly as ux(0,t) = u(0,t ) -T1 and ux(1,t) = T2- u(1,t) where T1 = 0 and T2 = 1. Find the steady state distribution of the temperature.
just 1,2,4 Problem 1 Consider the linear system of equations Ax = b, where x €...
just 1,2,4
Problem 1 Consider the linear system of equations Ax = b, where x € R4X1, and A= 120 b = and h= 0.1. [2+d -1 0 0 1 1 -1 2+d -1 0 h2 0 -1 2 + 1 Lo 0 -1 2+d] 1. Is the above matrix diagonally dominant? Why 2. Use hand calculations to solve the linear system Ax = b with d=1 with the following methods: (a) Gaussian elimination. (b) LU decomposition. Use MATLAB (L,...
Solve b and c only a) i. A numcrical scheme is known to perform best if...
Solve b and c only
a) i. A numcrical scheme is known to perform best if the height of the uniform subinterval is small enough in the interval (0.2]. The equation that gives the height h of the uniform subinterval is given by f(h) = 3 +h-1=0 Use the fixed point method of finding roots of equations to find the height h for which the scheme performs best. Conduct only four (4) iterations. 6 Marks ii. find algebraically all the...
(1) Use finite the difference method to solve for the temperature profile given by the equation below. The thin rod is one (1) meter long. The temperature on the left end is 100 and 0 on the right end. Set up the problem for three internal nodes (unknowns). Set up the augmented matrix for Gauss elimination solution (do not solve). Roughly sketch the five T values (including BC's).
(1) Use finite the difference method to solve for the temperature profile...
1. [12 marks] In the following parts of this question, write a MATLAB code to solve a linear system A b (A is a square nonsingular matrix) using Jacobi and Gauss-Seidel algorithms. Do not use the built-in Matlab functions for solving linear systems (a) Write a Matlab function called Jacobi that consumes a square n x n matrix A, and an n x 1 vector b, and uses the Jacobi technique to solve the system Ax-b, starting with the zero...
50 70 100 180 100 70 50 20 20 20 10mm 20 40 60 90 140 90 60 40 Figure 1: A rectangular plate with temperature defined on the boundaries 4 Figure 1 shows a rectangular plate made of an homogeneous isotropic material The temperature distribution in this plate satisfies the indicated boundary condi- tions (given in degrees centigrade) and has reached a steady-state condition so that the temperature is described by Laplace's equation a2T a2T (i) Draw a sketch...
LT An aluminum rod of diameter D 2.5 cm and of length from a wall maintained at T, 300"C. The.coovective heat coefficient h 17 W/m2-K with an ambie.t air temperature T Thermal conductivity k = 204 WmK. Assume there is e luid by convection from the end of the rod, which is at temperature Ta. Using 2. -38c. nergy transfer to the a finite difference method, with Ax = 5 cm, the fo be derived for the solution of the...
Problem 1: A metallic fin of length is attached to a hot surface (at To) and is exposed to cold air at To. The temperature at the end of the fin is T1. The governing equation for the temperature along the length of the fin is given below where h is the heat transfer coefficient, k is the thermal conductivity of metal, P is the perimeter of the fin, and A is the cross-sectional area of the fin. (a) Do...
Question 1 (10 marks) For a linear system Ax b with 1 0-1 A-1 2-1 2-13 and b4 18 compute by hand the first four iterations with the Jacobi method, usg0 Hint: for the ease of calculation, keep to rational fractions rather than decimals. (10 marks) Question 2 For the same linear svstem as in Question 1. compute by hand the first three iterations with the Gauss Seidel method, us0 Hint: for the ease of calculation, keep to rational fractions...
just 1,2,4
Problem 1 Consider the linear system of equations Ax = b, where x € R4X1, and A= 120 b = and h= 0.1. [2+d -1 0 0 1 1 -1 2+d -1 0 h2 0 -1 2 + 1 Lo 0 -1 2+d] 1. Is the above matrix diagonally dominant? Why 2. Use hand calculations to solve the linear system Ax = b with d=1 with the following methods: (a) Gaussian elimination. (b) LU decomposition. Use MATLAB (L,...
Solve b and c only
a) i. A numcrical scheme is known to perform best if the height of the uniform subinterval is small enough in the interval (0.2]. The equation that gives the height h of the uniform subinterval is given by f(h) = 3 +h-1=0 Use the fixed point method of finding roots of equations to find the height h for which the scheme performs best. Conduct only four (4) iterations. 6 Marks ii. find algebraically all the...
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