1. The partial differential equation of temperature change in a beam with its initial andboundary conditions are given, determine equation T(x,t),T₁=5 Txx t>0,0 ≤x≤1T(1,t)=10tT(0,t) = 3tT(x,0) = 5.1 diffamantinl

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Answered: 1. The partial differential equation of…

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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The following partial differential equation describes the diffusion of salt in a vessel. Assuming that diffusion happens only along the x-axis, calculate the salt concentration (C(x, t)). D is the diffusion constant. ∂?/∂t = D*(∂2?/∂x2) Initial condition: ?(x,0,) = ?0 ?(?, t) = 0 Boundary conditions: {∂?(0, ?)/∂x = 0
For a linear first-order differential equation dy/dt=F(y,t) , we know that the existence and uniqueness theorem tells us that a solution to an initial value problem exists if F is continuous on some interval (because of the intermediate value theorem), and that the solution is unique if the partial derivative with respect to y of F is also continuous on the same interval. However, partial derivatives measure the change of a function in one direction while the other(s) are held constant, which in this case would mean t is held constant while y varies. But if we are talking about an "interval," how can this be? Wouldn't our t interval be restricted? I'm presuming the reasoning behind the uniqueness part has to do with the fact that coplanar curves are said to be parallel and do not intersect, thus meaning only one of a family of coplanar curves can pass through a given point (our initial point). But I'm having trouble understanding how the partial derivative with respect to y relates to…

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1. The partial differential equation of temperature change in a beam with its initial and boundary conditions are given, determine equation T(x,t), T₁=5 Txx t>0,0 ≤x≤1 T(1,t)=10t T(0,t) = 3t T(x,0) = 5. 1 diffamantinl