# Setting up and solving an equation

I would like to solve the equations:

$$F(\xi;x,t)=\int_0^\xi u_{0}(\xi)\:d\xi+\frac{(x-\xi)^{2}}{2t}$$ to use in evaluating: $$u(x,t)=\dfrac{\int_{-\infty}^\infty \dfrac{x-\xi}{t} e^{-1/2ReF}d\xi} {\int_{-\infty}^\infty e^{-1/2ReF}d\xi}$$

EDIT: here is what I have tried so far; first I just defined Subscript[u, 0][x_], which is a piecewise function (I apologize for explaining this poorly initially):

Subscript[u, 0][x_] := Subscript[u, 0][x] = Piecewise[{{1,
x \[LessSlantEqual] 0 && x \[GreaterSlantEqual] -1}, {0,
x \[LessSlantEqual] 1 && x > 0}}]


Then in order to evaluate the two integrals, I tried to first change Subscript[u, 0][x_] to be a function of \[Xi] instead of [x_]:

uu[\[Xi]_] := uu[\[Xi]] = Subscript[u, 0][x] /. x -> \[Xi]


Here I tried to integrate from 0 to a different variable, aa, (which I later changed to \[Xi]):

f[\[Xi]_] := f[\[Xi]] = Assuming[\[Xi] \[Element] Integers &&
aa \[Element] Integers && \[Xi] > 0 && aa > 0,
Integrate[uu[\[Xi]], {\[Xi], 0, \[Xi]}]]

ff[\[Xi]_] := f[\[Xi]] /. aa -> \[Xi]


Finally I tried to find the two integrals:

F[\[Xi]_, x_, t_] := F[\[Xi], x, t] = ((x - \[Xi])^2)/(2 t) + f[\[Xi]]

u[\[Xi]_, x_, t_] := u[\[Xi], x, t] =Integrate[((x - \[Xi])/
t) Exp[-0.5 ReF], {\[Xi], -\[Infinity], \[Infinity]}]/
Integrate[Exp[-0.5 ReF], {\[Xi], -\[Infinity], \[Infinity]}]

Plot[Evaluate[Table[u[\[Xi], x, T], {x, -1, 1}]], PlotRange -> Full,
AxesLabel -> {x, u}]


When I try to evaluate u[\[Xi]_, x_, t_], however, I get the following errors:

I know that there are errors in my initial defintions of my integrals, but that is where I am stuck because I am a relatively new user to Mathematica. Basically, I want to evaluate the integral of Subscript[u, 0][\[Xi]], use the result to evaluate the integral F[\[Xi]_, x_, t_], and then use that result to evaluate u[\[Xi]_, x_, t_].

I edited this in the hopes of adding more information and being a little clearer about what I am trying to do. Re is a constant (equal to 10), and I would like to keep time t as a variable, but I have been using it as a constant ("T=1") to try to simplify things. Thanks very much for any help!

• But, solving for what? That doesn't look like an integral equation. Is $u_0$ related to $F$ somehow? – march Mar 13 '16 at 5:17
• I would like to solve for u(x,t). u0 is a constant here. I had thought that F was an integral equation, but I could be completely misunderstanding this problem...is that the case? – Kaszt Mar 13 '16 at 5:30
• An integral equation is one where you are solving for a function, and that function appears both inside and outside the integral (it is similar in nature to a differential equation, but there are fewer methods for solving integral equations). It looks to me like you just need to compute the integral, but what is your integration variable? Something doesn't make sense, because your function $F$ is a function of all the variables that appear in the integral, but there must be some variable that is the integration variable, which should not show up. Check that your expressions are correct. – march Mar 13 '16 at 5:33
• I understand; this is not an integral equation then, because u0 does not appear outside the integral. The integration variable is included (sorry for all of the mistakes). I think I know how to do this now; thanks very much for your help! – Kaszt Mar 13 '16 at 5:39
• If u0 is a constant, why is it shown as a function in the first line of code? Also, if it is a constant, why not do the first integral analytically? There must be a typo here. – bbgodfrey Mar 13 '16 at 6:10