Explanation Given: The given partial differential equation is ∂ 2 u ∂ x 2 + x 2 = ∂ 2 u ∂ t 2 , t > 0 and the given boundary conditions are u ( 0 , t ) = 0 , u ( 1 , t ) = 0 , u ( x , 0 ) = f ( x ) and ∂ u ∂ t | t = 0 = 0 . Calculation: The given partial differential equation is, ∂ 2 u ∂ x 2 + x 2 = ∂ 2 u ∂ t 2 , t > 0 . (1) Consider the equation, u ( x , t ) = v ( x , t ) + ψ ( x , t ) . (2) Partially differentiate both sides of the equation (2) with respect to x as follows. ∂ u ∂ x = ∂ v ∂ x + ψ ′ ( x ) Partially differentiate both sides of the above equation with respect to x as follows. ∂ 2 u ∂ x 2 = ∂ 2 v ∂ x 2 + ψ ″ ( x ) Substitute ∂ 2 u ∂ x 2 = ∂ 2 v ∂ x 2 + ψ ″ ( x ) and ∂ 2 u ∂ t 2 = ∂ 2 v ∂ t 2 in equation (1), ∂ 2 v ∂ x 2 + ψ ″ ( x ) + x 2 = ∂ 2 v ∂ t 2 Second derivative of ψ is zero in the above equation by homogeneous system, ψ ″ ( x ) + x 2 = 0 Integrate the above equation with respect to x as follows. ψ ′ ( x ) = − x 3 3 + c 1 Integrate the above equation with respect to x as follows. ψ ( x , t ) = − x 4 12 + c 1 x + c 2 . (3) At boundary conditions u ( 0 , t ) = 0 , u ( 1 , t ) = 0 and ∂ u ∂ t | t = 0 = 0 , ψ ( 0 ) = 0 and ψ ( 1 ) = 0 c 1 = − 11 12 and c 2 = 1 Substitute c 1 = − 11 12 and c 2 = 1 in equation (3), ψ ( x , t ) = − x 4 12 − 11 12 x + 1 Substitute ψ ( x , t ) = − x 4 12 − 11 12 x + 1 in equation (2), u ( x , t ) = − x 4 12 − 11 12 x + 1 + v ( x , t ) . (4) When λ = α 2 , α > 0 . Consider the equation, v ( x , t ) = X ( x ) T ( t ) . (5) Separate the variables of equation (5), X ″ X = − λ and T ′ k T = − λ The equation of the variable X is, X ″ + λ X = 0 . (6) The equation of the variable T is, T ′ + λ k T = 0

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

See similar textbooks

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

Videos

Question

Chapter 13, Problem 6CR

To determine

The displacement u(x,t) under given boundary conditions.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 13, Problem 5CR

Chapter 13, Problem 7CR

Students have asked these similar questions

Theorem: show that XCH) = M(E) M" (6) E + t Mcfic S a Solution of ODE -9CA)- x = ACE) x + g (t) + X (E) - E

5. (a) State the Residue Theorem. Your answer should include all the conditions required for the theorem to hold. (4 marks) (b) Let y be the square contour with vertices at -3, -3i, 3 and 3i, described in the anti-clockwise direction. Evaluate に dz. You must check all of the conditions of any results that you use. (5 marks) (c) Evaluate L You must check all of the conditions of any results that you use. ཙ x sin(Tx) x²+2x+5 da. (11 marks)

3. (a) Lety: [a, b] C be a contour. Let L(y) denote the length of y. Give a formula for L(y). (1 mark) (b) Let UCC be open. Let f: U→C be continuous. Let y: [a,b] → U be a contour. Suppose there exists a finite real number M such that |f(z)| < M for all z in the image of y. Prove that < ||, f(z)dz| ≤ ML(y). (3 marks) (c) State and prove Liouville's theorem. You may use Cauchy's integral formula without proof. (d) Let R0. Let w € C. Let (10 marks) U = { z Є C : | z − w| < R} . Let f UC be a holomorphic function such that 0 < |ƒ(w)| < |f(z)| for all z Є U. Show, using the local maximum modulus principle, that f is constant. (6 marks)

Chapter 13 Solutions

Advanced Engineering Mathematics

Ch. 13.1 - Prob. 1E Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E Ch. 13.1 - Prob. 5E Ch. 13.1 - Prob. 6E Ch. 13.1 - Prob. 7E Ch. 13.1 - Prob. 8E Ch. 13.1 - Prob. 9E Ch. 13.1 - Prob. 10E

Knowledge Booster

$Advanced Math$

Learn more about

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

The displacement u ( x , t ) under given boundary conditions.

Solution Summary: The author calculates the displacement of the string u(x,t) under given boundary conditions.

Concept explainers

Videos

The displacement u(x,t) under given boundary conditions.

Want to see the full answer?

Chapter 13 Solutions