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3. Consider a Laplacian equation in a two-dimensional Cartesian coordinate as a2u + ax2 =0 in...
34. Consider a two-dimensional Cartesian coordinate system and a two-dimensional uv-system with the coordinates related by y = (1/2)(112-U2). In general, Laplace's equation in two dimensions can be written as with ох ду Zi (a) In the xy-plane, sketch lines of constant u and constant v. (b) Express Laplace's equation using the uw-coordinates. (c) Use the method of separation of variables to separate Laplace's equation in the v-system and obtain the general solution for ų'(u, u). 34. Consider a two-dimensional...
9) Solve the following partial differential equation au a2u ax2 n(0, t) = u(2, t)-0 t > 0 (x, 0) = 0 au at It=0 =x(2-x) 9) Solve the following partial differential equation au a2u ax2 n(0, t) = u(2, t)-0 t > 0 (x, 0) = 0 au at It=0 =x(2-x)
Problem 9-A Consider the following steady, three-dimensional velocity field in Cartesian coordinate: u,V, W where a, b, c, and d are constants. Under what conditions is this flow field incompressible? Problem 9-A Consider the following steady, three-dimensional velocity field in Cartesian coordinate: u,V, W where a, b, c, and d are constants. Under what conditions is this flow field incompressible?
(40 marks) Find the solution of the two-dimensional Laplace equation$$ u_{x x}+u_{y y}=0 \quad 0<x<1,0<y<1 $$with the boundary conditions$$ u(x, 1)=x, u(x, 0)=u(0, y)=0, u(1, y)=y $$
The two-dimensional heat equation reduces to Laplace's equation to = 0 if the temperature u is steady-state. u(x, y) is defined in 0<x<2 and 0 Sys2 and satisfy u(x,0) = u(x, 2) = u(0, y) = 0 and u(2, y) = 80 sin my. Answer the following questions. (1) Obtain two ODES (Ordinary Differential Equations) by the method of separation of variables. (2) Find u(x, y) satisfying the boundary condition. (3) Obtain the value of u(1,5).
9. Solve - cos(x) for 0 <x < 27, t > 0 ax2 at2 y(0, t) y(27, t) = 0 for t 0 y(x, 0) y(x.0)= 0 for 0 <x < 27. at Graph the fortieth partial sum for some values of the time. 11. Solve the telegraph equation au A Bu= c2- at ax2 at2 for 0 x < L, t > 0. A and B are positive constants The boundary conditions are u(0, t) u(L, t)=0 for t...
11. Consider the parabolic coordinate system (u, v) related to the Cartesian coordi- nates (r, y) by х — 2иv, y — u? — u? for (и, v) € [0, оо) х [0, оо) 1 u = 1, u 2' (a) Sketch in the ry-plane the curves given u = 2. Then sketch in 1 v = 1, v = 2. Shade in the region R the xy-plane the curves given v = 2' bounded by the curves given by...
[&r, a1]7, that V2u = V.Vu = 6.4. Verify directly from the gradient operator that V ux+u-see Definition 6.5 Definition 6.5 (Two-Dimensional Heat or Diffusion Equation). Consider the open do- main (x,y) W. Using the continuity equation (1.4) the flux rule (6.13) yields DV u+R. (6.14) where V2u V.Vu u +lyy is the linear Laplacian operator The boundary conditions come in the three types: conditions on u, conditions on flux, and mixed as we are familiar with from Chapter 4....
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...
3. This question is about non-homogeneous boundary conditions (a) Consider Laplace's equation on a rectangle, with fully inhomogeneous boundary conditions =0 0 a, 0< y <b u(x, 0) fi() u(, b) f2(a) u(0, y)g (x) ua, y) = 92(r) 0 ra Homogenise the boundary conditions to convert the problem to one of the form 2 F(x, y) 0 xa,0 y < b + (x, 0)= fi() b(x, b) f2(x) b(0, y)0 (a, y) = 0 0y b 0 y sb...