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Question 5 2 pts The propagation of waves in a dissipative mediurn is governed by...
4. (50 pts) Consider the following partial differential equation: 1du au Ət22 Ətər2 (BC) u7,t) =...
4. (50 pts) Consider the following partial differential equation: 1du au Ət22 Ətər2 (BC) u7,t) = 0 20,t) = 0 0 <t (IC) u(3,0) = 0 0 <r <a Follow the steps below to solve it: (a) (8 pts) Separate variables as u(x,t) = X(2)T(t) to derive the following differential equations for X and T, with an unknown parameter 1: T" - T' + XT = 0, X" + 1X = 0.
1. If Ea) 2. The Fourier series expansion of the function f() which is defined over one period by , 1<zc2 is f(...
1. If Ea) 2. The Fourier series expansion of the function f() which is defined over one period by , 1<zc2 is f(z) = ao + Find the coefficients an and simplify you answer. 1 z sin ax cos ar Jzcos az dz = Hint: f(x) cos(n") dz and a.-Th 3. The propagation of waves along a particular string is governed by the following bound- ary value problem u(0,t) 0 ue(8,t)0 u(x,0) = f(x) u(x,0) g(x) Use the separation of...
Question 2 The function u(x,t) is governed by the wave equation 82u 182u 8x2 c2 8t2 Subject to th...
Fourier Wave Equation
Question 2 The function u(x,t) is governed by the wave equation 82u 182u 8x2 c2 8t2 Subject to the following conditions having c2 3 i. At x 0 and x-1, u 0 for all t 2 0. i. Whent 0, for sx s 1 Use the method of separation of variables to establish that the solution for u(x, t). From the solution established, given a condition t0, usinx(1 + cosx) for 0 s x s1. Find the...
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.
This is a partial differential equations question. Please help me solve for u(x,t): Find the eigenvalues/eigenfunction...
This is a partial differential equations question. Please help
me solve for u(x,t):
Find the eigenvalues/eigenfunction and then use the initial
conditions/boundary conditions to find Fourier coefficients for the
equation.
3. (10 pts) Use the method of separating variables to solve the problem utt = curr u(0,t) = 0 = u(l,t) ur. 0) = 3.7 - 4, u(3,0) = 0 for 0 <r<l, t>0 fort > 0 for 0 <r<1
1. (10 points, part I) Consider the following initial boundary value problem lU (la) (1b) (1c) 0L, t> 0 3 cos ( a(x, 0) (a) Classify the partial differential equation (1a) (b) What do the equation...
1. (10 points, part I) Consider the following initial boundary value problem lU (la) (1b) (1c) 0L, t> 0 3 cos ( a(x, 0) (a) Classify the partial differential equation (1a) (b) What do the equations (la)-(1c) model? (Hint: Give an interpretation for the PDE, boundary conditions and intial condition.) c) Use the method of separation of variables to separate the above problem into two sub- problems (one that depends on space and the other only on time) (d) What...
I need help with these! 3. (1 point) a) Compute the general solution of the differential equation y"5 12y" 0 b) Determine the test function Y (t) with the fewest terms to be used to obtain...
I need help with these!
3. (1 point) a) Compute the general solution of the differential equation y"5 12y" 0 b) Determine the test function Y (t) with the fewest terms to be used to obtain a particular solution of the following equation via the method if undetermined coefficients. Do not attempt to determine the coefficients.5y 12y"2 10e-tesin(V3t) Spring 2011) 4. (1 point) Compute the general solution of the following differential equations dz dy dt ii)(1y iv) (z cos(y) +...
Apply the method of separation of variables to the PDE below to derive a pair of...
Apply the method of separation of variables to the PDE below to derive a pair of ODEs, one of which involves only x and the other of which involves only y. (You do not need to solve the ODE.) 23 u дх3 + x 23 u dy3 = 0 6 u=o L10)=0 Cha: Supplemental information -Linearity satisfies the property Leau, uz)=C.L(ui) +C₂L(42) - Heat Egn. is a linear partial differential equation : L(a)= eu-kay = f(xt) Linear homogeneous = L()...
Question 9 2 pts dt -ay, where Using Euler's method on a system of the form...
Question 9 2 pts dt -ay, where Using Euler's method on a system of the form dy a is a positive constant, results in the equation Yi+1 yi (1 - ah). What is the condition for the solution to be stable? 0 |1 - ah| > 1 0 |1 - ah| > 0 o 1 - ah] < 0 o 1 - ahl <1 Question 10 2 pts Partial differential equations have a single independent variable. O True False
Thank You! I always rate for clear answers! :) 7. (7 pts) Consider the initial value...
Thank You! I always rate for clear answers! :)
7. (7 pts) Consider the initial value problem y" +4y' +8y=80), (0=6, yO=0, ſo if 0 <i<6 where g(t) = 8e-21-6) if6 <i<e. (1) Take the Laplace transform of both sides of the given differential equation to create the corresponding alge- braic equation. Denote the Laplace transform of y(t) by Y(s). Do not move any terms from one side of the equation to the other (until you get to part (b)...
4. (50 pts) Consider the following partial differential equation: 1du au Ət22 Ətər2 (BC) u7,t) = 0 20,t) = 0 0 <t (IC) u(3,0) = 0 0 <r <a Follow the steps below to solve it: (a) (8 pts) Separate variables as u(x,t) = X(2)T(t) to derive the following differential equations for X and T, with an unknown parameter 1: T" - T' + XT = 0, X" + 1X = 0.
1. If Ea) 2. The Fourier series expansion of the function f() which is defined over one period by , 1<zc2 is f(z) = ao + Find the coefficients an and simplify you answer. 1 z sin ax cos ar Jzcos az dz = Hint: f(x) cos(n") dz and a.-Th 3. The propagation of waves along a particular string is governed by the following bound- ary value problem u(0,t) 0 ue(8,t)0 u(x,0) = f(x) u(x,0) g(x) Use the separation of...
Fourier Wave Equation
Question 2 The function u(x,t) is governed by the wave equation 82u 182u 8x2 c2 8t2 Subject to the following conditions having c2 3 i. At x 0 and x-1, u 0 for all t 2 0. i. Whent 0, for sx s 1 Use the method of separation of variables to establish that the solution for u(x, t). From the solution established, given a condition t0, usinx(1 + cosx) for 0 s x s1. Find the...
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.
This is a partial differential equations question. Please help
me solve for u(x,t):
Find the eigenvalues/eigenfunction and then use the initial
conditions/boundary conditions to find Fourier coefficients for the
equation.
3. (10 pts) Use the method of separating variables to solve the problem utt = curr u(0,t) = 0 = u(l,t) ur. 0) = 3.7 - 4, u(3,0) = 0 for 0 <r<l, t>0 fort > 0 for 0 <r<1
1. (10 points, part I) Consider the following initial boundary value problem lU (la) (1b) (1c) 0L, t> 0 3 cos ( a(x, 0) (a) Classify the partial differential equation (1a) (b) What do the equations (la)-(1c) model? (Hint: Give an interpretation for the PDE, boundary conditions and intial condition.) c) Use the method of separation of variables to separate the above problem into two sub- problems (one that depends on space and the other only on time) (d) What...
I need help with these!
3. (1 point) a) Compute the general solution of the differential equation y"5 12y" 0 b) Determine the test function Y (t) with the fewest terms to be used to obtain a particular solution of the following equation via the method if undetermined coefficients. Do not attempt to determine the coefficients.5y 12y"2 10e-tesin(V3t) Spring 2011) 4. (1 point) Compute the general solution of the following differential equations dz dy dt ii)(1y iv) (z cos(y) +...
Apply the method of separation of variables to the PDE below to derive a pair of ODEs, one of which involves only x and the other of which involves only y. (You do not need to solve the ODE.) 23 u дх3 + x 23 u dy3 = 0 6 u=o L10)=0 Cha: Supplemental information -Linearity satisfies the property Leau, uz)=C.L(ui) +C₂L(42) - Heat Egn. is a linear partial differential equation : L(a)= eu-kay = f(xt) Linear homogeneous = L()...
Question 9 2 pts dt -ay, where Using Euler's method on a system of the form dy a is a positive constant, results in the equation Yi+1 yi (1 - ah). What is the condition for the solution to be stable? 0 |1 - ah| > 1 0 |1 - ah| > 0 o 1 - ah] < 0 o 1 - ahl <1 Question 10 2 pts Partial differential equations have a single independent variable. O True False
Thank You! I always rate for clear answers! :)
7. (7 pts) Consider the initial value problem y" +4y' +8y=80), (0=6, yO=0, ſo if 0 <i<6 where g(t) = 8e-21-6) if6 <i<e. (1) Take the Laplace transform of both sides of the given differential equation to create the corresponding alge- braic equation. Denote the Laplace transform of y(t) by Y(s). Do not move any terms from one side of the equation to the other (until you get to part (b)...
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