# Solve Finite Difference Equation

See attached file for full problem description with equation.

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Find analytically the solution of this difference equation with the given initial values:

Without computing the solution recursively, predict whether such a computation would be stable.

(Note: A numerical process is unstable if small errors made at one stage of the process are magnified in subsequent stages and seriously degrade the accuracy of the overall calculation.)

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https://brainmass.com/math/graphs-and-functions/solve-finite-difference-equation-52144

#### Solution Preview

Please see the attached file.

Difference equation:

$$

x_{n+1} = -0.2 x_n + 0.99 x_n. eqno(1)

$$

Starting conditions:

$$

x_0 = 1, hskip 1cm x_1=0.9. eqno(2)

$$

This is a linear homogeneous equation with constant coefficients. Therefore its solutions are a linear combination of basic solutions of form

$$

B_n = cq^n,eqno(3)

$$

where $c$ and $q$ are constants.

Substituting the basic form (3) into equation ...

#### Solution Summary

The expert solves finite differences in equations. Analytical solutions of the difference equations with the given initial values are provided.

Differential Equations: Solution to Heat Equation

Consider the heat equation

delta(u)/delta(t) = (delta^2)(u)/delta(x^2)

Show that if u(c, t) = (t^alpha)psi(E) where E = x/sqrt(t) and alpha is a constant, then psi(E) satisfies the ordinary differential equation

alpha(psi) = 1/2 E(psi) = psi, where ' = d/dE

is independent of t only if alpha = - 1/2. Further, show that if alpha - 1/2 then

C - 1/2 E(psi) = psi

where C is an arbitrary constant. From this last ordinary differential equation, and assuming C = 0, deduce that

u(x, t) = A/sqrt(t) e^-x^3/At

is a solution of the heat equation (here A is an arbitrary constant).

Show that as t tends to zero from above,

lim(1/sqrt(t) * e^(-x^2/4t)) = 0 for x =/ 0

and that for all t > 0

The integral of 1/sqrt(t) * e^-x^3/4t dx = B

where B is a finite constant

Given that the integral e^-x^2 dx = sqrt(pi), find B. What physical and/or probabilistic interpretation might one give to u(x,t)?

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