7. (a) (b) More generally, if ut — kuxx = f, vt − kVxx = g, f ≤ g, and u ≤ v at x = 0, x= 1 and t = 0, prove that u ≤ v for 0 ≤ x ≤ 1,0 ≤ t <∞. If vt − Vxx ≥ sin x for 0 ≤ x ≤ π, 0 < t < ∞, and if v(0, t) ≥ 0, Vt v(π, t) ≥ 0 and v(x, 0) ≥ sin x, use part (a) to show that v(x, t) ≥ (1 - e-¹) sinx.

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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Question

[Second Order Equations] How do you solve this?

7. (a)
(b)
More generally, if ut — kuxx = f, Vt − kVxx = g, f ≤ g, and u ≤ v
at x = 0, x = 1 and t = 0, prove that u ≤ v for 0≤x≤ 1,0 ≤t<∞.
If vt - Uxx ≥ sin x for 0 ≤ x ≤ ñ, 0 < t < ∞, and if v(0, t) ≥ 0,
v(π, t) ≥ 0 and v(x, 0) ≥ sin x, use part (a) to show that v(x, t) ≥
(1 - e-¹) sin x.
Transcribed Image Text:7. (a) (b) More generally, if ut — kuxx = f, Vt − kVxx = g, f ≤ g, and u ≤ v at x = 0, x = 1 and t = 0, prove that u ≤ v for 0≤x≤ 1,0 ≤t<∞. If vt - Uxx ≥ sin x for 0 ≤ x ≤ ñ, 0 < t < ∞, and if v(0, t) ≥ 0, v(π, t) ≥ 0 and v(x, 0) ≥ sin x, use part (a) to show that v(x, t) ≥ (1 - e-¹) sin x.
Expert Solution
Step 1

What is Partial Differential Equation:

An equation that imposes relationships between the many partial derivatives of a multivariable function is known as a partial differential equation (PDE) in mathematics. A significant portion of research in pure mathematics is also devoted to the study of partial differential equations. Generally speaking, these studies focus on the identification of general qualitative characteristics of solutions to various partial differential equations, such as existence, uniqueness, regularity, and stability.

Given:

Given that:

  • ut-kuxx=f.
  • vt-kvxx=g.
  • fg.
  • uv at x=0 x=l, t=0

Also, 

  • vt-vxxsinx for 0xπ, 0<t<.
  • v0,t0.
  • vπ,t0.
  • vx,0sinx.

To Determine:

  • We prove that for all 0xl, t0uv,
  • We establish that for all 0xπ, t>0,vx,t1-e-tsinx.
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