this lesson is PDE partial differential eqution 2. Given the nonlinear sine-Gordon eqn.Uz = Sin u, u(t, r), (u = u), k: constant (1)Also, given the (Bäcklund transform) equations(S).-(). - kSin (""). (2)Sina) Find vz and Vt 1using eqn. (2) – (3).b) Subtract vg and ve side by side, you have found sbove, (use v = Vz) and show thstu satisfies eqn. (1): uz = Sin u.c) Add vz and v, side by side, you have found above, (use v =Vz) and show that valso satisfies the same eqn.: Vz = Sin v.This means you can construct a new solution v to the sine-Gordon eqn. Vg = Sin vstarting from a simple, known solution u to the same eqn. ug = Sin u, using eqn. (2) - (3).For this purpose start with the trivial solution u(t, r) = 0 of eqn (1).d) Write eqn. (2) and (3) in case u = 0.e) Integrate these two equations consistently. You can use Arctanh(z) = In() for%3Da clear result.

Given: u(tx)=Sin u, u(t, x), (u(xt) = u(tx)), k=constant

2. Given the nonlinear sine-Gordon eqn. ua - Sin u, u(t, z), (4 = tig), k: constant (1) Also, given the (Bücklund transform) equations (),- Si -(), (2) -kSin Sin a) Find t, and ty tasing eqn. (2) – (3).

2. Given the nonlinear sine-Gordon eqn. ua - Sin u, u(t, z), (4 = tig), k: constant (1) Also, given the (Bücklund transform) equations (),- Si -(), (2) -kSin Sin a) Find t, and ty tasing eqn. (2) – (3).

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

this lesson is PDE partial differential eqution

2. Given the nonlinear sine-Gordon eqn.
Uz = Sin u, u(t, r), (u = u), k: constant (1)
Also, given the (Bäcklund transform) equations
(S).-
(). - kSin (""). (2)
Sin
a) Find vz and Vt 1using eqn. (2) – (3).
b) Subtract vg and ve side by side, you have found sbove, (use v = Vz) and show thst
u satisfies eqn. (1): uz = Sin u.
c) Add vz and v, side by side, you have found above, (use v =
Vz) and show that v
also satisfies the same eqn.: Vz = Sin v.
This means you can construct a new solution v to the sine-Gordon eqn. Vg = Sin v
starting from a simple, known solution u to the same eqn. ug = Sin u, using eqn. (2) - (3).
For this purpose start with the trivial solution u(t, r) = 0 of eqn (1).
d) Write eqn. (2) and (3) in case u = 0.
e) Integrate these two equations consistently. You can use Arctanh(z) = In() for
%3D
a clear result.

With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.

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