Question

How to prove the truncation error of the Dufort Frankel numerical scheme for partial differential equations?

Answer #1

Define ordinary differential equations and partial differential
equations. How do they differ? What are the most important aspects
of each?

Use mathematical concepts to explain the differences between
ordinary differential equations and partial differential
equations.\
When does a particular solution
required? What does that mean in a real system?

(a) Separate the following partial differential equation into
two ordinary differential equations: Utt + 4Utx − 2U = 0. (b) Given
the boundary values U(0,t) = 0 and Ux (L,t) = 0, L > 0, for all
t, write an eigenvalue problem in terms of X(x) that the equation
in (a) must satisfy.

(a) Separate the following partial differential equation into
two ordinary differential equations: e 5t t 6 Uxx + 7t 2 Uxt − 6t 2
Ut = 0. (b) Given the boundary values Ux (0,t) = 0 and U(2π,t) = 0,
for all t, write an eigenvalue problem in terms of X(x) that the
equation in (a) must satisfy. That is, state (ONLY) the resulting
eigenvalue problem that you would need to solve next. You do not
need to actually solve...

Partial differential equations
Solve using the method of characteristics
ut +1/2 ux + 3/2 vx = 0 , u(x,0) =cos(2x)
vt + 3/2 ux + 1/2 vx = 0 , v(x,0) = sin(2x)

Explain how Taylor series are used to estimate truncation
error.

Walter A . Strauss- Partial Differential Equations (2nd
Edition)
Chapter 7.1, Problem 10E
Let u(x，y) be the harmonic function in the unit disk with
boundary
values u(x， y) = x^2 on {x^2 + y^2 = 1}. Find the Rayleigh-Ritz
approximation of the
form, w0 + c1w1 = x^2 + c1(1 - x^2 - y^2).

Show how to solve a non-separable partial differential equation
using fourier transforms, and briefly explain each step.

On question 37 in 2.5 of A First Course in Differential
Equations with Modeling Applicatiosn 11th edition,
how do we decide that the substitution to use is u=v^(1-n)?
In general, how do we decide what to substitute when solving
differential equations or using Bernoulli's equation?

What are Singular solutions in differential equations..
How to find them.Can we find them using the general
solution.
Could you give me a explanation

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