[Second Order Equations] How do you solve this? 7. (a)(b)More generally, if ut — kuxx = f, Vt − kVxx = g, f ≤ g, and u ≤ vat 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.

Answered: 7. (a) (b) More generally, if ut — kuxx…

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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Related questions

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.

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:

$u_{t} - k u_{x x} = f$ .
$v_{t} - k v_{x x} = g$ .
$f \leq g$ .
$u \leq v at x = 0 x = l, t = 0$ .

Also,

$v_{t} - v_{x x} \geq \sin x for 0 \leq x \leq π, 0 < t < \infty$ .
$v (0, t) \geq 0$ .
$v (π, t) \geq 0$ .
$v (x, 0) \geq \sin x$ .

To Determine:

We prove that for all $0 \leq x \leq l, t \geq 0$ , $u \leq v$ ,
We establish that for all $0 \leq x \leq π, t > 0$ , $v (x, t) \geq (1 - e^{- t}) \sin x$ .