## 1491 Reputation

8 years, 232 days

## I doubt there's a closed form for the in...

See attached file.

## I doubt there are simpler form....

Another formula:

`-(-1 + 2*k)*sqrt(1 - csc(omega*T)^2)*(csc(omega*T)^2)^(-1/2 + k)*int(t^(k - 1)/sqrt(1 - t), t = 0 .. 1/csc(omega*T)^2)/sqrt(-1 + csc(omega*T)^2) + 2*sqrt(Pi)*sqrt(1 - csc(omega*T)^2)*(csc(omega*T)^2)^(-1/2 + k)*GAMMA(k)/(sqrt(-1 + csc(omega*T)^2)*GAMMA(-1/2 + k))`

Integral is incomplete beta function ,but Maple dosen't know, solve as hypergeometric function.

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## Maple is not figure it out to give solut...

 > restart;
 > eq1 := 2^(-m/2-n/2)*exp(lambda^2*sigma^2/4)/sqrt(n!)/sqrt(m!*Pi)*int(HermiteH(m,s+lambda*sigma/2)*HermiteH(n,s+lambda*sigma/2)*exp(-s^2), s=-infinity..infinity);;
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## Only simple equation...

We can use Laplace transform and Inverse Laplace transfrom to solve  for: Simple linear-differential  fractional-equations with initial conditions.

differential_equations_with_fractional_order.mw

## Solve delay differential system of equat...

Maple can solve only numerically.Adding random missing values to parameters:

dsolve-delay_sys_example_1.mw

I use Maple version 2023.2,I don't have version 18 !

Regards

## For n=10000...

 > restart;
 > ee := unapply((-1)^n*((-4*n^2 - 16*n - 28)*JacobiP(-1 + n, -1 - 2*n, 2, -1/2) + JacobiP(-2 + n, -2*n, 3, -1/2)*(3 + n)*(-1 + n))*4^n/(48*(1 + n)*n),n):
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 > L:=seq(evalb(expand(ee(i))=eee(i+1)), i=1..10000):
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After 30 min computation on my hardware we see that for n =10000 are True.

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## Try....

kernelopts(version);

#`Maple 2023.1, X86 64 WINDOWS, Jul 07 2023, Build ID 1723669`

Try:

```allvalues(value(solll));

If want solve for: y(x) :

```solve(simplify(allvalues(value(solll))), [y(x)]);

```

## Syntax.....

Try:

```with(DynamicSystems):
lambda := 15.4;
f := N1 -> sum(exp(-lambda)*lambda^n/n!, n = 1 .. N1);
Ns := 50;
T := Vector(Ns, t -> t);
A := Vector(Ns, t -> f(t));
DiscretePlot(T, A, style = stair, legend = "stair", color = red, labels = ["time", "signal"]);
plot(f(x), x = 0 .. Ns);```

.

```with(DynamicSystems);
f := N1 -> sum(1/(n^3*sin(n)^2), n = 1 .. N1);
Ns := 400;
T := Vector(Ns, t -> t);
A := Vector(Ns, t -> f(t));
DiscretePlot(T, A, style = stair, legend = "stair", color = red, labels = ["time", "signal"]);```

## Maybe you what:...

From help pages the fractional derivative using the Davison-Essex (D-E) definition:

`diff(f(x),[x\$nu]) = 1/GAMMA(n-nu)*Int((x-t)^(n-nu-1)*diff(f(t),[t\$n]),t = 0 .. x);`

```U1 := t -> (1/2*1/M - 1/4*1/(M*K))*t + 1/2;
eq := Int((t - z)^(ceil(alpha) - alpha - 1)*diff(U1(z), [z \$ ceil(alpha)]), z = 0 .. t)/GAMMA(ceil(alpha) - alpha) + U1(t)/M - U1(t)^2/(M*K) + diff(U1(t), t) - diff(U1(t), t)/epsilon;
(value(eq) assuming (0 < alpha and alpha < 1));
int(%, t);

#t/(2*M) - t/(4*M*K) - ((2/M - 1/(M*K))*t)/(4*epsilon) + (2*M*K*t + t^2*K - 1/2*t^2)/(4*M^2*K) - (1 + (1/M - 1/(2*M*K))*t)^3/(12*M*K*(1/M - 1/(2*M*K))) - (2*K - 1)*t^(2 - alpha)/(4*(-1 + alpha)*M*K*GAMMA(1 - alpha)*(2 - alpha))
```

## With Mathematica:...

Where QPochhammer function.

## FoxH function...

Looks like analytical solution for sum is FoxH function:

`sum(R^g*product(-B*r + N*g + 1, r = 1 .. g - 1)/(B^g*g!), g = 0 .. infinity) = -FoxH([[[1 + 1/B, 1 - N/B]], []], [[[0, 1]], [[1/B, -N/B]]], R)/B`

See attached file.

brn_ac_ver2.mw

## Analytical solution with Mathematica....

See attached file:

1_case1.mw

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