Explanation Given: The given Laplace equation is ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 = 0 , 0 < x < a , 0 < y < b and the given boundary conditions are u ( 0 , y ) = 0 , u ( a , y ) = 0 , u ( x , 0 ) = f ( x ) and u ( x , b ) = g ( x ) . Calculation: The given Laplace equation for rectangular plate is, ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 = 0 Consider the equation, u ( x , y ) = X ( x ) Y ( y ) (1) Separate the variables of the above equation, X ″ X = − λ and − Y ″ Y = − λ The equation of the variable X is, X ″ + λ X = 0 (2) The equation of the variable Y is, Y ″ − λ Y = 0 (3) When λ = α 2 > 0 , Substitute λ = α 2 in equation (2), X ″ + α 2 X = 0 The general solution of the above equation is, X ( x ) = c 1 cos α x + c 2 sin α x (4) At boundary conditions u ( 0 , y ) = 0 , u ( a , y ) = 0 , u ( x , 0 ) = f ( x ) and u ( x , b ) = g ( x ) , so the value of c 1 is 0 also written as c 1 = 0 . For non-trivial solution c 2 ≠ 0 , α = n π a Substitute c 1 = 0 and α = n π a in the equation (4), X ( x ) = c 2 sin n π a x (5) Substitute λ = α 2 in equation (3), Y ″ − α 2 Y = 0 The general solution of the above equation is, Y ( y ) = c 3 cosh α y + c 4 sinh α y (6) At boundary conditions u ( 0 , y ) = 0 , u ( a , y ) = 0 , u ( x , 0 ) = f ( x ) and u ( x , b ) = g ( x ) , Substitute α = n π a in the equation (6), Y ( y ) = c 3 cosh n π a y + c 4 sinh n π a y (7) Substitute the value of X ( x ) from equation (5) and the value of Y ( y ) from equation (7) in equation (1), u ( x , y ) = ( c 2 c 3 cosh n π a y + c 2 c 4 sinh n π a y ) sin n π a x Rewrite the term c 2 c 3 = A n and c 2 c 4 = B n in the above equation, u ( x , y ) = ∑ n = 1 ∞ (

Differential Equations with Boundary-Value Problems

9th Edition

ISBN: 9781337632515

Author: Dennis G. Zill

Publisher: Cengage Learning US

See similar textbooks

Videos

Question

Chapter 12.5, Problem 13E

To determine

To solve: The Laplace equation for a rectangular plate under given boundary conditions.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 12.5, Problem 12E

Chapter 12.5, Problem 14E

Students have asked these similar questions

Solve no chatgpt already got wrong chatgpt answer

Not use ai please

You may need to use the appropriate appendix table or technology to answer this question. You are given the following information obtained from a random sample of 4 observations. 24 48 31 57 You want to determine whether or not the mean of the population from which this sample was taken is significantly different from 49. (Assume the population is normally distributed.) (a) State the null and the alternative hypotheses. (Enter != for ≠ as needed.) H0: Ha: (b) Determine the test statistic. (Round your answer to three decimal places.) (c) Determine the p-value, and at the 5% level of significance, test to determine whether or not the mean of the population is significantly different from 49. Find the p-value. (Round your answer to four decimal places.) p-value = State your conclusion. Reject H0. There is insufficient evidence to conclude that the mean of the population is different from 49.Do not reject H0. There is sufficient evidence to conclude that the…

Chapter 12 Solutions

Differential Equations with Boundary-Value Problems

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

The Laplace equation for a rectangular plate under given boundary conditions.

Solution Summary: The author explains the Laplace equation for a rectangular plate under given boundary conditions.

Videos

To solve: The Laplace equation for a rectangular plate under given boundary conditions.

Want to see the full answer?

Chapter 12 Solutions