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Question 19 Using the shooting method for the following second-order differential equation governing the boundary value...
Question 25 1 pts Using the shooting method for the following second-order differential equation governing the...
Question 25 1 pts Using the shooting method for the following second-order differential equation governing the boundary value problem G.E: + EA (c) a + 2 = L(x) * € (0,L] B.C's: u (0) = 0 and EA (x) din le=L= F. An appropriate algebraic equation to use in the finite difference of the boundary condition at = Lis There is no suitable finite difference equation that can be obtained. u(L) - u (L - Ax) F.A BAL) None of...
Question 24 1 pts Using the shooting method for the following second-order differential equation governing the...
Question 24 1 pts Using the shooting method for the following second-order differential equation governing the boundary value problem G.E: + EA (x) +u = L (x) € (0,L] B.C's: u () = 0 and EA (2) de Iz-L=F, the trapezoidal method is used to converts the problem into coupled integral equations solved at the quadrature points. None of the above. finite differences are used to convert the governing equation and boundary conditions of the problem into an analog set...
Set up and solve a boundary value problem using the shooting method using Matlab
Set up and solve a boundary value problem using the shooting
method using Matlab
A heated rod with a uniform heat source may be modeled with Poisson equation. The boundary conditions are T(x = 0) = 40 and T(x = 10) = 200 dTf(x) Use the guess values shown below. zg linspace (-200,100,1000); xin-0:0.01:10 a) Solve using the shooting method with f(x) = 25 . Name your final solution "TA" b) Solve using the shooting method with f(x)-0.12x3-2.4x2 + 12x....
Section 1.3 3. a. Solve the following initial boundary value problem for the heat equation 0x<L...
Section 1.3 3. a. Solve the following initial boundary value problem for the heat equation 0x<L t0 at u(r, 0) f() u(0, t)u(L, t) 0, t>0, 9Tr when f(r)6 sin L b. Solve the following initial boundary value problem for the diffusion equation au D 0 L t0 at u(r, 0) f() (0, t) (L, t) 0, t 0, x < L/2 0. when f(r) r > L/2. 1
Section 1.3 3. a. Solve the following initial boundary value problem...
Partial Differential Equation - Wave equation : Vibrating spring Question 2 A plucked string, Figure 2 shows the initial position function f (x) for a stretched string (of length L) that is set in mot...
Partial Differential
Equation
- Wave equation : Vibrating spring
Question 2 A plucked string, Figure 2 shows the initial position function f (x) for a stretched string (of length L) that is set in motion by moving t at midpoint x =-aside the distance-bL and releasing it from rest timet- 0. f (x) bL Figure 2 (a) If the length of string is 10cm with amplitude 5cm was set initially, state the initial condition and the boundary conditions for the...
4. Consider the following initial value problem of the 1D wave equation with mixed boundary condition...
4. Consider the following initial value problem of the 1D wave equation with mixed boundary condition IC: u(z, t = 0) = g(x), ut(z, t = 0) = h(z), BC: u(0, t)0, u(l,t) 0, t>0 0 < x < 1, (a)Use the energy method to show that there is at most one solution for the initial-boundary value problem. (b)Suppose u(x,t)-X()T(t) is a seperable solution. Show that X and T satisfy for some λ E R. Find all the eigenvalues An...
solve problem #1 depending on the given information Consider the following 1D second order elliptic equation with Dirichlet boundary conditions du(x) (c(x)du ) = f(x) (a $15 b), u(a) = ga, u(b) = gb...
solve problem #1 depending on the given information
Consider the following 1D second order elliptic equation with Dirichlet boundary conditions du(x) (c(x)du ) = f(x) (a $15 b), u(a) = ga, u(b) = gb dr: where u(x) is the unknown function, ga and gb are the Dirichlet boundary values, c(x) is a given coefficient function and f(x) is a given source function. See the theorem 10.1 in the textbook for the existence and uniqueness of the solution. 1.1 Weak Formulation...
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...
(1 point) Solve the boundary value problem by using the Laplace transform az w &w 16-...
(1 point) Solve the boundary value problem by using the Laplace transform az w &w 16- dx2 x > 0, t> 0 at2 w(0,t) = 0, lim w(x, t) = 0, t> 0, X+0 w(x,0) = 2xe-*, dw F(x,0) = 0, x > 0, дt First take the Laplace transform of the partial differential equation. Let W be the Laplace transform of w. Then W satisfies the ordinary differential equation W" = subject to W(0) = and limxW(x) = Solve...
Question 3 (1 mark) Attempt 1 Consider the following second order differential equation: 2 6 d2y...
Question 3 (1 mark) Attempt 1 Consider the following second order differential equation: 2 6 d2y +4 dr2 If we let u y and v y' then as an AUTONOMOUS first order system, the second order differential equation is correctly expressed as: dy d.r -y 1+112, with y(1) 9 and y'(1)5 d.r du 6 А dt1, to)1, dtv, u(to)=9, -1+11r2+2u-4v2, v(to)=5. du 6 B dry )9, ar=1+112+y-4(y), v(1)5 du d.r dt du C 1, (1) 0, =v, u(1)=9, -1+11r2+u-4v2, v(1)=5...
Question 25 1 pts Using the shooting method for the following second-order differential equation governing the boundary value problem G.E: + EA (c) a + 2 = L(x) * € (0,L] B.C's: u (0) = 0 and EA (x) din le=L= F. An appropriate algebraic equation to use in the finite difference of the boundary condition at = Lis There is no suitable finite difference equation that can be obtained. u(L) - u (L - Ax) F.A BAL) None of...
Question 24 1 pts Using the shooting method for the following second-order differential equation governing the boundary value problem G.E: + EA (x) +u = L (x) € (0,L] B.C's: u () = 0 and EA (2) de Iz-L=F, the trapezoidal method is used to converts the problem into coupled integral equations solved at the quadrature points. None of the above. finite differences are used to convert the governing equation and boundary conditions of the problem into an analog set...
Set up and solve a boundary value problem using the shooting
method using Matlab
A heated rod with a uniform heat source may be modeled with Poisson equation. The boundary conditions are T(x = 0) = 40 and T(x = 10) = 200 dTf(x) Use the guess values shown below. zg linspace (-200,100,1000); xin-0:0.01:10 a) Solve using the shooting method with f(x) = 25 . Name your final solution "TA" b) Solve using the shooting method with f(x)-0.12x3-2.4x2 + 12x....
Section 1.3 3. a. Solve the following initial boundary value problem for the heat equation 0x<L t0 at u(r, 0) f() u(0, t)u(L, t) 0, t>0, 9Tr when f(r)6 sin L b. Solve the following initial boundary value problem for the diffusion equation au D 0 L t0 at u(r, 0) f() (0, t) (L, t) 0, t 0, x < L/2 0. when f(r) r > L/2. 1
Section 1.3 3. a. Solve the following initial boundary value problem...
Partial Differential
Equation
- Wave equation : Vibrating spring
Question 2 A plucked string, Figure 2 shows the initial position function f (x) for a stretched string (of length L) that is set in motion by moving t at midpoint x =-aside the distance-bL and releasing it from rest timet- 0. f (x) bL Figure 2 (a) If the length of string is 10cm with amplitude 5cm was set initially, state the initial condition and the boundary conditions for the...
4. Consider the following initial value problem of the 1D wave equation with mixed boundary condition IC: u(z, t = 0) = g(x), ut(z, t = 0) = h(z), BC: u(0, t)0, u(l,t) 0, t>0 0 < x < 1, (a)Use the energy method to show that there is at most one solution for the initial-boundary value problem. (b)Suppose u(x,t)-X()T(t) is a seperable solution. Show that X and T satisfy for some λ E R. Find all the eigenvalues An...
solve problem #1 depending on the given information
Consider the following 1D second order elliptic equation with Dirichlet boundary conditions du(x) (c(x)du ) = f(x) (a $15 b), u(a) = ga, u(b) = gb dr: where u(x) is the unknown function, ga and gb are the Dirichlet boundary values, c(x) is a given coefficient function and f(x) is a given source function. See the theorem 10.1 in the textbook for the existence and uniqueness of the solution. 1.1 Weak Formulation...
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...
(1 point) Solve the boundary value problem by using the Laplace transform az w &w 16- dx2 x > 0, t> 0 at2 w(0,t) = 0, lim w(x, t) = 0, t> 0, X+0 w(x,0) = 2xe-*, dw F(x,0) = 0, x > 0, дt First take the Laplace transform of the partial differential equation. Let W be the Laplace transform of w. Then W satisfies the ordinary differential equation W" = subject to W(0) = and limxW(x) = Solve...
Question 3 (1 mark) Attempt 1 Consider the following second order differential equation: 2 6 d2y +4 dr2 If we let u y and v y' then as an AUTONOMOUS first order system, the second order differential equation is correctly expressed as: dy d.r -y 1+112, with y(1) 9 and y'(1)5 d.r du 6 А dt1, to)1, dtv, u(to)=9, -1+11r2+2u-4v2, v(to)=5. du 6 B dry )9, ar=1+112+y-4(y), v(1)5 du d.r dt du C 1, (1) 0, =v, u(1)=9, -1+11r2+u-4v2, v(1)=5...
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