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The two-dimensional heat equation reduces to Laplace's equation to = 0 if the temperature u is...
The function u(x, t) satisfies the partial differential equation with the boundary conditions u(0,t) = 0...
The function u(x, t) satisfies the partial differential equation with the boundary conditions u(0,t) = 0 , u(1,t) = 0 and the initial condition u(x,0) = f(x) = 2x if 0<x<} 2(1 – x) if}<x< 1 . The initial velocity is zero. Answer the following questions. (1) Obtain two ODES (Ordinary Differential Equations) by the method of separation of variables and separating variable -k? (2) Find u(x, t) as an infinite series satisfying the boundary condition and the initial condition.
In a steady-state two-dimensional heat flow problem, the temperature, u, at any point in the doma...
In a steady-state two-dimensional heat flow problem, the temperature, u, at any point in the domain (x, y) satisfies the following differential equation. With the given temperature boundary condition, find the internal temperature at points a, b, and c using a numerical method. 0 4 4
In a steady-state two-dimensional heat flow problem, the temperature, u, at any point in the domain (x, y) satisfies the following differential equation. With the given temperature boundary condition, find the internal temperature at...
Question 3. Separation of variables. Consider Laplace's Equation in two dimensions: 77 0-קר. (a) Write Ф(z,y)-F(x)G...
Question 3. Separation of variables. Consider Laplace's Equation in two dimensions: 77 0-קר. (a) Write Ф(z,y)-F(x)G(y) and use separation of variables to get ordinary differential equa- tions for F and G (b) Consider the rectangular region ,y)ER2:0SSa,0S y S b) with three boundary conditions on obtain conditions on F and G on those boundaries where conditions on Ф are given.
Question 3. Separation of variables. Consider Laplace's Equation in two dimensions: 77 0-קר. (a) Write Ф(z,y)-F(x)G(y) and use separation of...
Question 3. Separation of variables. Consider Laplace's Equation in two dimensions: (a) Write Φ(x,y) F(x)G(y) and u...
Question 3. Separation of variables. Consider Laplace's Equation in two dimensions: (a) Write Φ(x,y) F(x)G(y) and use separation of variables to get ordinary differential equa- tions for F and G (b) Consider the rectangular region {(x,y) є R2 : 0 a, 0-y-b} with three boundary x conditions on Ф: obtain conditions on F and G on those boundaries where conditions on Ф are given. (c) (i) Solve the differential equations found in (a), subject to the conditions found in (b)...
Question 3. Separation of variables Consider Laplace's Equation in two dimensions (a) Write Ф(r,y)-F(x)G(y) and use...
Question 3. Separation of variables Consider Laplace's Equation in two dimensions (a) Write Ф(r,y)-F(x)G(y) and use separation of variables to get ordinary differential equa- tions for F and G (b) Consider the rectangular region {(x, y) E R2: 0Ka, 0 y b with three boundary conditions on Ф об obtain conditions on F and G on those boundaries where conditions on Ф are given (c) (i) Solve the differential equations found in (a), subject to the conditions found in (b)...
(a) Use separation of variables to rewrite the partial differential equation below into a pair 1....
(a) Use separation of variables to rewrite the partial differential equation below into a pair 1. of ordinary differential equations. (b) Suppose the above partial differential equation has boundary condition uz (0,t) 0, u(20, t)0. Use separations of variables to determine the corresponding bound- ary conditions that the ordinary differential equations found in (a) must satisfy. (c) (Yes or no) Could the partial differential equation, u -2uzt-5utt, be separated into two ordinary differential equations?
(a) Use separation of variables to...
Question 1 [Total 20 marks] (a) [5 marks] In a steady-state two-dimensional heat flow problem, th...
Question 1 [Total 20 marks] (a) [5 marks] In a steady-state two-dimensional heat flow problem, the temperature, u, at any point in the domain (t, ) satisfies the differential equation u y(2-y) u= U0F With the given temperature boundary condition as follows: u(x, 0) = 0, u(x, 2) = x(4-x), 0 < x < 4 Calculate the temperature at the interior points a, b, and c using a mesh size h-1.
Question 1 [Total 20 marks] (a) [5 marks] In...
6. a) For a thin conducting rod of length L = π, the temperature U(x, t) at a point 0 Sx S L at timet>0 is determined by the differential equation U, Uxx with boundary data U(x, 0) fx) and U(0...
6. a) For a thin conducting rod of length L = π, the temperature U(x, t) at a point 0 Sx S L at timet>0 is determined by the differential equation U, Uxx with boundary data U(x, 0) fx) and U(0,) UL, t)- 0 for all0. Show that for any positive integer k, the function U(x, t)- exp (-ak21) sin kx is a solution. It follows that Σ exp (-ak2 t) Bk sin kx is the general solution where Σ...
(a) Find the solution u(x, y) of Laplace's equation in the semi-infinite strip 0 0, and the addit...
(a) Find the solution u(x, y) of Laplace's equation in the semi-infinite strip 0<x<a, y>0, that satisfies the boundary conditions u(0, y)-0 u(a, y)-0, y > 0, and the additional condition that u(x, y) -0 as yoo, etnyla sin nTX where Cn X where Cn- NTX) where Cn = u(x, y) - -Ttny/a sin(where Cn u(x, y) n=1 u(x, y) - (b) Find the solution if f(x) = x(a-x) V(x)- (c) Let a9. Find the smallest value of yo for...
please help me solve all the question. please. thank you. Question 3. Separation of variables. Consider Laplace's Eq...
please help me solve all the question. please. thank
you.
Question 3. Separation of variables. Consider Laplace's Equation in two dimensions:-+-- (a) Write φ(z y) F(x)G(y) and use separation of variables to get ordinary differential equa- tions for F and CG (b) Consider the rectangular region ((r,y) E R:0 conditions on Φ < a, 0 y b with three boundary 0(x, 0) = 0, D(x, b) = 0, (0,y) = 0 Obtain conditions on F and G on those boundaries...
The function u(x, t) satisfies the partial differential equation with the boundary conditions u(0,t) = 0 , u(1,t) = 0 and the initial condition u(x,0) = f(x) = 2x if 0<x<} 2(1 – x) if}<x< 1 . The initial velocity is zero. Answer the following questions. (1) Obtain two ODES (Ordinary Differential Equations) by the method of separation of variables and separating variable -k? (2) Find u(x, t) as an infinite series satisfying the boundary condition and the initial condition.
In a steady-state two-dimensional heat flow problem, the temperature, u, at any point in the domain (x, y) satisfies the following differential equation. With the given temperature boundary condition, find the internal temperature at points a, b, and c using a numerical method. 0 4 4
In a steady-state two-dimensional heat flow problem, the temperature, u, at any point in the domain (x, y) satisfies the following differential equation. With the given temperature boundary condition, find the internal temperature at...
Question 3. Separation of variables. Consider Laplace's Equation in two dimensions: 77 0-קר. (a) Write Ф(z,y)-F(x)G(y) and use separation of variables to get ordinary differential equa- tions for F and G (b) Consider the rectangular region ,y)ER2:0SSa,0S y S b) with three boundary conditions on obtain conditions on F and G on those boundaries where conditions on Ф are given.
Question 3. Separation of variables. Consider Laplace's Equation in two dimensions: 77 0-קר. (a) Write Ф(z,y)-F(x)G(y) and use separation of...
Question 3. Separation of variables. Consider Laplace's Equation in two dimensions: (a) Write Φ(x,y) F(x)G(y) and use separation of variables to get ordinary differential equa- tions for F and G (b) Consider the rectangular region {(x,y) є R2 : 0 a, 0-y-b} with three boundary x conditions on Ф: obtain conditions on F and G on those boundaries where conditions on Ф are given. (c) (i) Solve the differential equations found in (a), subject to the conditions found in (b)...
Question 3. Separation of variables Consider Laplace's Equation in two dimensions (a) Write Ф(r,y)-F(x)G(y) and use separation of variables to get ordinary differential equa- tions for F and G (b) Consider the rectangular region {(x, y) E R2: 0Ka, 0 y b with three boundary conditions on Ф об obtain conditions on F and G on those boundaries where conditions on Ф are given (c) (i) Solve the differential equations found in (a), subject to the conditions found in (b)...
(a) Use separation of variables to rewrite the partial differential equation below into a pair 1. of ordinary differential equations. (b) Suppose the above partial differential equation has boundary condition uz (0,t) 0, u(20, t)0. Use separations of variables to determine the corresponding bound- ary conditions that the ordinary differential equations found in (a) must satisfy. (c) (Yes or no) Could the partial differential equation, u -2uzt-5utt, be separated into two ordinary differential equations?
(a) Use separation of variables to...
Question 1 [Total 20 marks] (a) [5 marks] In a steady-state two-dimensional heat flow problem, the temperature, u, at any point in the domain (t, ) satisfies the differential equation u y(2-y) u= U0F With the given temperature boundary condition as follows: u(x, 0) = 0, u(x, 2) = x(4-x), 0 < x < 4 Calculate the temperature at the interior points a, b, and c using a mesh size h-1.
Question 1 [Total 20 marks] (a) [5 marks] In...
6. a) For a thin conducting rod of length L = π, the temperature U(x, t) at a point 0 Sx S L at timet>0 is determined by the differential equation U, Uxx with boundary data U(x, 0) fx) and U(0,) UL, t)- 0 for all0. Show that for any positive integer k, the function U(x, t)- exp (-ak21) sin kx is a solution. It follows that Σ exp (-ak2 t) Bk sin kx is the general solution where Σ...
(a) Find the solution u(x, y) of Laplace's equation in the semi-infinite strip 0<x<a, y>0, that satisfies the boundary conditions u(0, y)-0 u(a, y)-0, y > 0, and the additional condition that u(x, y) -0 as yoo, etnyla sin nTX where Cn X where Cn- NTX) where Cn = u(x, y) - -Ttny/a sin(where Cn u(x, y) n=1 u(x, y) - (b) Find the solution if f(x) = x(a-x) V(x)- (c) Let a9. Find the smallest value of yo for...
please help me solve all the question. please. thank
you.
Question 3. Separation of variables. Consider Laplace's Equation in two dimensions:-+-- (a) Write φ(z y) F(x)G(y) and use separation of variables to get ordinary differential equa- tions for F and CG (b) Consider the rectangular region ((r,y) E R:0 conditions on Φ < a, 0 y b with three boundary 0(x, 0) = 0, D(x, b) = 0, (0,y) = 0 Obtain conditions on F and G on those boundaries...
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