Question 3. Separation of variables. Consider Laplace's Equation in two dimensions: (a) Write Φ(x,y) F(x)G(y) and u...
Question 3. Separation of variables Consider Laplace's Equation in two dimensions (a) Write Ф(r,y)-F(x)G(y) and use separation of variables to get ordinary differential equa- tions for F and G (b) Consider the rectangular region {(x, y) E R2: 0Ka, 0 y b with three boundary conditions on Ф об obtain conditions on F and G on those boundaries where conditions on Ф are given (c) (i) Solve the differential equations found in (a), subject to the conditions found in (b)...
please help me solve all the question. please. thank you. Question 3. Separation of variables. Consider Laplace's Equation in two dimensions:-+-- (a) Write φ(z y) F(x)G(y) and use separation of variables to get ordinary differential equa- tions for F and CG (b) Consider the rectangular region ((r,y) E R:0 conditions on Φ < a, 0 y b with three boundary 0(x, 0) = 0, D(x, b) = 0, (0,y) = 0 Obtain conditions on F and G on those boundaries...
Question 3. Separation of variables. Consider Laplace's Equation in two dimensions: 77 0-קר. (a) Write Ф(z,y)-F(x)G(y) and use separation of variables to get ordinary differential equa- tions for F and G (b) Consider the rectangular region ,y)ER2:0SSa,0S y S b) with three boundary conditions on obtain conditions on F and G on those boundaries where conditions on Ф are given. Question 3. Separation of variables. Consider Laplace's Equation in two dimensions: 77 0-קר. (a) Write Ф(z,y)-F(x)G(y) and use separation of...
10. [18 Marks] Using separation of variables, solve Laplace's equation for {(x,y): 0 < x < 2,0 < y < 2), subject to the boundary conditions 0 (0, y) = d(x, 2) 6 + cos(nz) = In your solution, you must consider all three cases for the separation constant λ. 10. [18 Marks] Using separation of variables, solve Laplace's equation for {(x,y): 0
Complete i, ii, and iii. Use the method of separation of variables to solve Laplace's Equation (V V-O)forleither V(xy) in 2-D Cartesian coordinates with Vix,0) conditions on the y-axis) V(s, ø) in 2-D cylindrical coordinates, or V(r, 6) in 2-D spherical coordinates. Vx,a) 0 (homogeneous boundary y( Use the method of separation of variables to solve Laplace's Equation (V V-O)forleither V(xy) in 2-D Cartesian coordinates with Vix,0) conditions on the y-axis) V(s, ø) in 2-D cylindrical coordinates, or V(r, 6)...
3. Consider the Laplace's equation on a rectangular domain subject to the following boundary conditions that represents the steady-state heating of a plate. A temperature probe shows that (1/2, 1/4) = 0. Solve this problem using the method of separation of variables. (7) byllyy = 0 0 <I<41 and O y <21 U-(0,y)=0, 1-(41, y) = cos(2), 4(1,0) = cos(2), 4(1,2)=0. (total 25 marks
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).
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...
3. Using separation of variables to solve the heat equation, u -kuxx on the interval 0x<1 with boundary conditions u(0 and ur(1, t)-0, yields the general solution, u(x, t) =A0 + Σ Ane-k,t cos(nm) (with A, = ㎡π2) 0<x<l/2 0〈x〈1,2 u(x,0)=f(x)-.., , . . .) when u(x,0) = f(x)- Determine the coefficients An (n - 0, 1,2,
1. (a) Derive the solution u(x, y) of Laplace's equation in the rectangle 0 < x <a, 0 <y <b, that satisfies the boundary conditions u(0,y) = 0, u(a, y) = 0, u(x,0) = 0, u(x,b) = g(x), 0 0 0 < a. (b) Find the solution if a = 4, b = 2, and g(x) = 0 <r <a/2, a-r, a/2 < x <a.