Problem #1: (a) Consider the Laplace equation શક Oy² = 0, x² +1² <1, (1) with the boundary condition v(x, y) = 11x²y² + 813 on the circle x² + y² = 1. (2) Use polar coordinates to solve (1) (i.e. consider the function u(r, 0) = v(r cos(8), r sin(0)). You may assume that u(r, 0) is bounded at the origin, i.e., u(0, 0) <. Replacing the variable by t, enter the function u(r, t). (b) Express the solution found in (a) in rectangular coordinates, i.e. compute the function v(x, y) that is a solution of (1) and satisfies the boundary condition (2). Enter v(x, y).
Problem #1: (a) Consider the Laplace equation શક Oy² = 0, x² +1² <1, (1) with the boundary condition v(x, y) = 11x²y² + 813 on the circle x² + y² = 1. (2) Use polar coordinates to solve (1) (i.e. consider the function u(r, 0) = v(r cos(8), r sin(0)). You may assume that u(r, 0) is bounded at the origin, i.e., u(0, 0) <. Replacing the variable by t, enter the function u(r, t). (b) Express the solution found in (a) in rectangular coordinates, i.e. compute the function v(x, y) that is a solution of (1) and satisfies the boundary condition (2). Enter v(x, y).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Solve a and b part only
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![Problem #1: (a) Consider the Laplace equation
Ox²
Oy²
= 0, x² +1² <1, (1)
with the boundary condition
v(x, y) = 11x²y² + 813 on the circle x² + y² = 1.
ox
(2)
Use polar coordinates to solve (1) (i.e. consider the function u(r, 0) = v(r cos(8), r sin(0)). You may assume
that u(r, 0) is bounded at the origin, i.e., u(0, 0) <. Replacing the variable by t, enter the function u(r, t).
(b) Express the solution found in (a) in rectangular coordinates, i.e. compute the function v(x, y) that is a solution
of (1) and satisfies the boundary condition (2). Enter v(x, y).
(c) Consider the Laplace equation
+ = 0, ₁²2² +1² > 1,
(3)
with the boundary condition (2). Use polar coordinates to solve (3) (i.e. consider the function
u(r, 8) = v(r cos(8), r sin(0)). You may assume that u(r, 0) is bounded near infinity, i.e. that lim u(r, 0)| < 0.
Replacing the variable by t, enter the function u(r, t).
(d) Express the solution found in (c) in rectangular coordinates, i.e. compute the function v(x, y) that is a solution
of (3) and satisfies the boundary condition (2). Enter v(x, y).
Hint: Use standard trigonometric formulas to write the initial condition in a more convenient form when
solving (a) and (c).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4b416b9e-1151-42d9-91e6-c99f7088d177%2Fb023bb10-0e30-4878-b349-56c835149cd0%2Fafks52p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem #1: (a) Consider the Laplace equation
Ox²
Oy²
= 0, x² +1² <1, (1)
with the boundary condition
v(x, y) = 11x²y² + 813 on the circle x² + y² = 1.
ox
(2)
Use polar coordinates to solve (1) (i.e. consider the function u(r, 0) = v(r cos(8), r sin(0)). You may assume
that u(r, 0) is bounded at the origin, i.e., u(0, 0) <. Replacing the variable by t, enter the function u(r, t).
(b) Express the solution found in (a) in rectangular coordinates, i.e. compute the function v(x, y) that is a solution
of (1) and satisfies the boundary condition (2). Enter v(x, y).
(c) Consider the Laplace equation
+ = 0, ₁²2² +1² > 1,
(3)
with the boundary condition (2). Use polar coordinates to solve (3) (i.e. consider the function
u(r, 8) = v(r cos(8), r sin(0)). You may assume that u(r, 0) is bounded near infinity, i.e. that lim u(r, 0)| < 0.
Replacing the variable by t, enter the function u(r, t).
(d) Express the solution found in (c) in rectangular coordinates, i.e. compute the function v(x, y) that is a solution
of (3) and satisfies the boundary condition (2). Enter v(x, y).
Hint: Use standard trigonometric formulas to write the initial condition in a more convenient form when
solving (a) and (c).
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