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ax? ²2 alu 041 Given for 0 SxST, t> 0 acz Boundary conditions are: u(0,t) = u(te, t) = 0 for t20 Initial conditions are
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عر ax? ²2 alu 041 Given for 0 SxST, t> 0 acz Boundary conditions are: u(0,t)...
Consider the partial differential equation together with the boundary conditions u(0, t) 0 and u(1,t)0 for t20 and the initial condition u(z,0) = z(1-2) for 0 < x < 1. (a) If n is a positive in...
Consider the partial differential equation together with the boundary conditions u(0, t) 0 and u(1,t)0 for t20 and the initial condition u(z,0) = z(1-2) for 0 < x < 1. (a) If n is a positive integer, show that the function , sin(x), satisfies the given partial differential equation and boundary conditions. (b) The general solution of the partial differential equation that satisfies the boundary conditions is Write down (but do not evaluate) an integral that can be used to...
=T 20 marks) Consider the following PDE with boundary and initial conditions: U = Upx +...
=T 20 marks) Consider the following PDE with boundary and initial conditions: U = Upx + ur, for 0<x< 1 and to with u(0,t) = 1, u(1,t) = 0, u(1,0) = (a) Find the steady state solution, us(1), for the PDE. (b) Let Uſz,t) = u(?, t) – us(T). Derive a PDE plus boundary and initial conditions for U(2,t). Show your working. (c) Use separation of variables to solve the resulting problem for U. You may leave the inner products...
9. Solve the wave equation subject to the boundary and initial conditions u(0,t) = 0, u(x,0)...
9. Solve the wave equation subject to the boundary and initial conditions u(0,t) = 0, u(x,0) = 0, U(TT, t) = 0, t> 0 $ (3,0) = sin(x), 0<x<a
solve the PDE +u= at2 on 3 € (0,L), t > 0, with boundary conditions au...
solve the PDE
+u= at2 on 3 € (0,L), t > 0, with boundary conditions au 2x2 u(0,t) = 0, u(L, t) = 0 au and initial condition u(x,0) = f(x), at (x,0) = g(x) following the steps below. (a) Separate the variables and write differential equations for the functions (x) and h(t); pick the separation constant so that we recover a problem already studied. (b) Find the eigenfunctions and eigenvalues. (c) Write the general solution for this problem. (d)...
QUESTION 2 Consmder the problem ди 2k, 0<r< 1, t>O оt and the boundary conditions u(0,t)=...
QUESTION 2 Consmder the problem ди 2k, 0<r< 1, t>O оt and the boundary conditions u(0,t)= 1, u (1,t) = 3, t > 0 (a) Find the equiltbrium solutiou ug (r) (b) Find the solution u (z.t) of the PDE and the boundary condition which also satisfies the mitial condition (,0)-1+++sin (3wx), 0<o< 1 [25]
Let u be the solution to the initial boundary value problem for the Heat Equation au(t,) -48Fu(t,), te (0,oo), z (0,5); with boundary conditions u(t,0) 0, u(t,5) 0, and with initial condition 5 15 15...
Let u be the solution to the initial boundary value problem for the Heat Equation au(t,) -48Fu(t,), te (0,oo), z (0,5); with boundary conditions u(t,0) 0, u(t,5) 0, and with initial condition 5 15 15 The solution u of the problem above, with the conventions given in class, has the form with the normalization conditions vn(0)-1, u Find the functions vnwn and the constants cn n(t) wnr)
Let u be the solution to the initial boundary value problem for the...
3) (25 marks) Consider the following problem: u2(0,t) 3, u(2,t)u(2,t), t>0 u(,0) 0, 0
3) (25 marks) Consider the following problem: u2(0,t) 3, u(2,t)u(2,t), t>0 u(,0) 0, 0<2 (a) Find the steady state solution u,(x) of this problem. b) Write a new PDE, boundary conditions and initial conditions for U(x, t) - u(x, t)- Cox) (c) Use separation of variables to find a solution to the PDE, boundary conditions and initial conditions. You must justify each step of your solution carefully to get full marks. (Hint: if you are unable to write the eigenvalues...
Let u be the solution to the initial boundary value problem for the Heat Equation, au(t,z 382u(t,z), tE (0,oo), E (0,3); with initial condition u(0,x)-f(x)- and with boundary conditions Find the solu...
Let u be the solution to the initial boundary value problem for the Heat Equation, au(t,z 382u(t,z), tE (0,oo), E (0,3); with initial condition u(0,x)-f(x)- and with boundary conditions Find the solution u using the expansion u(t,x) n (t) wn(x), with the normalization conditions vn (0)1, Wn (2n -1) a. (3/10) Find the functionswn with index n 1. b. (3/10) Find the functions vn, with index n 1 C. (4/10) Find the coefficients cn , with index n 1.
Let...
3. Consider the follo 2 lu The boundary conditions are: u(0,y, t) - u(x, 0,t) - 0, ou (a, y, t) =...
*Note: Please answer all parts, and explain all workings. Thank
you!
3. Consider the follo 2 lu The boundary conditions are: u(0,y, t) - u(x, 0,t) - 0, ou (a, y, t) = (x, b, t) = 0 ay The initial conditions are: at t-0,11-4 (x,y)--Yo(x,y) . ot a) Assume u(x,y,t) - X(x)Y(y)T(t), derive the eigenvalue problems: a) Apply the boundary conditions and derive all the possible eigenvalues for λι, λ2 and corresponding eigen-functions, Xm,Yn b) for any combination of...
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.
Consider the partial differential equation together with the boundary conditions u(0, t) 0 and u(1,t)0 for t20 and the initial condition u(z,0) = z(1-2) for 0 < x < 1. (a) If n is a positive integer, show that the function , sin(x), satisfies the given partial differential equation and boundary conditions. (b) The general solution of the partial differential equation that satisfies the boundary conditions is Write down (but do not evaluate) an integral that can be used to...
=T 20 marks) Consider the following PDE with boundary and initial conditions: U = Upx + ur, for 0<x< 1 and to with u(0,t) = 1, u(1,t) = 0, u(1,0) = (a) Find the steady state solution, us(1), for the PDE. (b) Let Uſz,t) = u(?, t) – us(T). Derive a PDE plus boundary and initial conditions for U(2,t). Show your working. (c) Use separation of variables to solve the resulting problem for U. You may leave the inner products...
9. Solve the wave equation subject to the boundary and initial conditions u(0,t) = 0, u(x,0) = 0, U(TT, t) = 0, t> 0 $ (3,0) = sin(x), 0<x<a
solve the PDE
+u= at2 on 3 € (0,L), t > 0, with boundary conditions au 2x2 u(0,t) = 0, u(L, t) = 0 au and initial condition u(x,0) = f(x), at (x,0) = g(x) following the steps below. (a) Separate the variables and write differential equations for the functions (x) and h(t); pick the separation constant so that we recover a problem already studied. (b) Find the eigenfunctions and eigenvalues. (c) Write the general solution for this problem. (d)...
QUESTION 2 Consmder the problem ди 2k, 0<r< 1, t>O оt and the boundary conditions u(0,t)= 1, u (1,t) = 3, t > 0 (a) Find the equiltbrium solutiou ug (r) (b) Find the solution u (z.t) of the PDE and the boundary condition which also satisfies the mitial condition (,0)-1+++sin (3wx), 0<o< 1 [25]
Let u be the solution to the initial boundary value problem for the Heat Equation au(t,) -48Fu(t,), te (0,oo), z (0,5); with boundary conditions u(t,0) 0, u(t,5) 0, and with initial condition 5 15 15 The solution u of the problem above, with the conventions given in class, has the form with the normalization conditions vn(0)-1, u Find the functions vnwn and the constants cn n(t) wnr)
Let u be the solution to the initial boundary value problem for the...
3) (25 marks) Consider the following problem: u2(0,t) 3, u(2,t)u(2,t), t>0 u(,0) 0, 0<2 (a) Find the steady state solution u,(x) of this problem. b) Write a new PDE, boundary conditions and initial conditions for U(x, t) - u(x, t)- Cox) (c) Use separation of variables to find a solution to the PDE, boundary conditions and initial conditions. You must justify each step of your solution carefully to get full marks. (Hint: if you are unable to write the eigenvalues...
Let u be the solution to the initial boundary value problem for the Heat Equation, au(t,z 382u(t,z), tE (0,oo), E (0,3); with initial condition u(0,x)-f(x)- and with boundary conditions Find the solution u using the expansion u(t,x) n (t) wn(x), with the normalization conditions vn (0)1, Wn (2n -1) a. (3/10) Find the functionswn with index n 1. b. (3/10) Find the functions vn, with index n 1 C. (4/10) Find the coefficients cn , with index n 1.
Let...
*Note: Please answer all parts, and explain all workings. Thank
you!
3. Consider the follo 2 lu The boundary conditions are: u(0,y, t) - u(x, 0,t) - 0, ou (a, y, t) = (x, b, t) = 0 ay The initial conditions are: at t-0,11-4 (x,y)--Yo(x,y) . ot a) Assume u(x,y,t) - X(x)Y(y)T(t), derive the eigenvalue problems: a) Apply the boundary conditions and derive all the possible eigenvalues for λι, λ2 and corresponding eigen-functions, Xm,Yn b) for any combination of...
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.
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