Homework Answers
The Poisson equation is a PDE that occurs in many problems in science and engineering (such as compressible flow) and a simplified form of it is given by
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3. The Poisson equation is a PDE that occurs in many problems in science and engineering...
Solve the heat equation Ut = Uxx + Uyy on a square 0 <= x <= 2, 0<= y<= 2 with the following boundary and initial conditions 2. Solve the heat equation boundary conditions uvw on a square...
Solve the heat equation Ut = Uxx
+ Uyy on a square 0 <= x <= 2, 0<= y<= 2 with the
following boundary and initial conditions
2. Solve the heat equation boundary conditions uvw on a square O S r s 2, 0 S vS 2 with the (note the mix of u and tu) and with initial condition 0 otherwise Present your answer as a double trigonometric sum.
2. Solve the heat equation boundary conditions uvw on a...
14 points Consider the following equation : PDE: u+ 0 ,0<x <1, 0<y <1 BCs: u(0, y)= 0, u (1, y ) = 0 ,0...
14 points Consider the following equation : PDE: u+ 0 ,0<x <1, 0<y <1 BCs: u(0, y)= 0, u (1, y ) = 0 ,0<y <1 ICs: u (x,0)=0, u (x,1)=2 ,0<x <1 a) Using the PDE and the boundary conditions write the form of the solution u (x ,t) b) Now apply the initial condition to solve for the unknown coefficients in the solution from part (a)
14 points Consider the following equation : PDE: u+ 0 ,0
Solve the wave equation on the domain 0 < x < , t > 0 ?...
Solve the wave equation on the domain 0 < x < , t > 0 ? uxx Utt = with the boundary condition u (0, t) = 0 and the initial conditions u (x,0) = x2 u (x,0) = x
formulate complete PDE problems (specify the equation, space domain, time interval, boundary, and initial conditions) for...
formulate complete PDE problems (specify the equation, space domain, time interval, boundary, and initial conditions) for the following model situations: a) Conduction heat transfer occurs in a thin rod of length L with insulated side walls. Temperature is initially constant T(0) = T0. We are asked to find the temperature distribution in the time period 0 < t < t1, during which the left end of the rod is kept at the temperature T0, and the right end is subject...
b) i. Form partial differential equation from z = ax - 4y+b [4 marks] a +1...
b) i. Form partial differential equation from z = ax - 4y+b [4 marks] a +1 ii. Solve the partial differential equation 18xy2 + sin(2x - y) = 0 дх2ду c) i. Solve the Lagrange equation [4 Marks] az -zp + xzq = y2 where p az and q = ду [5 Marks] x ax ii. A special form of the second order partial differential equation of the function u of the two independent variables x and t is given...
1. Consider the insulated heat equation up = cum, 0 <r<L, t > 0 u (0,t)...
1. Consider the insulated heat equation up = cum, 0 <r<L, t > 0 u (0,t) = u (L, t) = 0, t > 0 u(x,0) = f(2). What is the steady-state solution? 2. Solve the two-dimensional wave equation (with c=1/) on the unit square (i.e., [0, 1] x [0,1) with homogeneous Dirichlet boundary conditions and initial conditions: (2, y,0) = sin(x) sin(y) (,y,0) = sin(x). 3. Solve the following PDE: Uzr + Uyy = 0, 0<<1,0 <y < 2...
a) Find the solution to the following interior Dirichlet problem with radius R=1 1 PDE Urr...
a) Find the solution to the following interior Dirichlet problem with radius R=1 1 PDE Urr + Up t 0 0 <r <1 wee p2 r BC u (1,0) = 10 + 3 sin(0) 10 cos(20) 0 <0 < 27 b) Consider the above problem on the unit square (x,y) domain PDE Urr + Uyy = 0 0<x<1 0<y <1 Transform the solution u(r, 0) from "a)" to the solution u(x, y) for "b)" Use the solution u(x,y) to calculate...
This is PDE problem. Please show all steps in detail with neat handwriting. Problem . Consider...
This is PDE problem. Please show all steps in detail with neat
handwriting.
Problem . Consider the function a) Find the full Fourier Series of F(x) a(0, y, t) = u(a, y, t) 0 u(z, 0, t ) = u(z, b, l) = 0 u(z,y,0) = f(z,y), u(x, y,0)-g(x,y), 0<y< b,t0 a) b) Solve the initial-boundary value problem for 2D wave equation. What is the physical interpretation of these boundary conditions
The twisting of a beam with rectangular cross-section is described by the inhomogeneous partial differential equation...
The twisting of a beam with rectangular cross-section is described by the inhomogeneous partial differential equation (PDE) below: 024 049 = -2 əx2 + ayż Eqn 2.1 where x and y are the coordinates of the cross-section and p(x,y) is the warp or distortion of the cross-section. The cross-section is bounded by –p sx sp and —q sy sq. The boundary conditions are given by: 0(p,y) = 0, 4(-p,y) = 0, 4(x,q) = 0 and 4(x,-q) = 0. Using the...
=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...
Solve the heat equation Ut = Uxx
+ Uyy on a square 0 <= x <= 2, 0<= y<= 2 with the
following boundary and initial conditions
2. Solve the heat equation boundary conditions uvw on a square O S r s 2, 0 S vS 2 with the (note the mix of u and tu) and with initial condition 0 otherwise Present your answer as a double trigonometric sum.
2. Solve the heat equation boundary conditions uvw on a...
14 points Consider the following equation : PDE: u+ 0 ,0<x <1, 0<y <1 BCs: u(0, y)= 0, u (1, y ) = 0 ,0<y <1 ICs: u (x,0)=0, u (x,1)=2 ,0<x <1 a) Using the PDE and the boundary conditions write the form of the solution u (x ,t) b) Now apply the initial condition to solve for the unknown coefficients in the solution from part (a)
14 points Consider the following equation : PDE: u+ 0 ,0
Solve the wave equation on the domain 0 < x < , t > 0 ? uxx Utt = with the boundary condition u (0, t) = 0 and the initial conditions u (x,0) = x2 u (x,0) = x
b) i. Form partial differential equation from z = ax - 4y+b [4 marks] a +1 ii. Solve the partial differential equation 18xy2 + sin(2x - y) = 0 дх2ду c) i. Solve the Lagrange equation [4 Marks] az -zp + xzq = y2 where p az and q = ду [5 Marks] x ax ii. A special form of the second order partial differential equation of the function u of the two independent variables x and t is given...
1. Consider the insulated heat equation up = cum, 0 <r<L, t > 0 u (0,t) = u (L, t) = 0, t > 0 u(x,0) = f(2). What is the steady-state solution? 2. Solve the two-dimensional wave equation (with c=1/) on the unit square (i.e., [0, 1] x [0,1) with homogeneous Dirichlet boundary conditions and initial conditions: (2, y,0) = sin(x) sin(y) (,y,0) = sin(x). 3. Solve the following PDE: Uzr + Uyy = 0, 0<<1,0 <y < 2...
a) Find the solution to the following interior Dirichlet problem with radius R=1 1 PDE Urr + Up t 0 0 <r <1 wee p2 r BC u (1,0) = 10 + 3 sin(0) 10 cos(20) 0 <0 < 27 b) Consider the above problem on the unit square (x,y) domain PDE Urr + Uyy = 0 0<x<1 0<y <1 Transform the solution u(r, 0) from "a)" to the solution u(x, y) for "b)" Use the solution u(x,y) to calculate...
This is PDE problem. Please show all steps in detail with neat
handwriting.
Problem . Consider the function a) Find the full Fourier Series of F(x) a(0, y, t) = u(a, y, t) 0 u(z, 0, t ) = u(z, b, l) = 0 u(z,y,0) = f(z,y), u(x, y,0)-g(x,y), 0<y< b,t0 a) b) Solve the initial-boundary value problem for 2D wave equation. What is the physical interpretation of these boundary conditions
The twisting of a beam with rectangular cross-section is described by the inhomogeneous partial differential equation (PDE) below: 024 049 = -2 əx2 + ayż Eqn 2.1 where x and y are the coordinates of the cross-section and p(x,y) is the warp or distortion of the cross-section. The cross-section is bounded by –p sx sp and —q sy sq. The boundary conditions are given by: 0(p,y) = 0, 4(-p,y) = 0, 4(x,q) = 0 and 4(x,-q) = 0. Using the...
=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...
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