Mariusz Iwaniuk

1491 Reputation

14 Badges

8 years, 280 days

Social Networks and Content at Maplesoft.com

MaplePrimes Activity


These are answers submitted by Mariusz Iwaniuk

Int((1 - sigma*sin(2*Pi*x))^k, x = 0 .. n) = n*hypergeom([1/2 - k/2, -k/2], [1], sigma^2);

f := (k, n, sigma) -> int((1 - sigma*sin(2*Pi*x))^k, x = 0 .. n):
g := (k, n, sigma) -> n*hypergeom([1/2 - 1/2*k, -1/2*k], [1], sigma^2):

f(11/2, 6, -2/13):
evalf(%);

g(11/2, 6, -2/13):
evalf(%);

Solution only for:  n for Integers !!! 

Regards M.I.

 

Download Integral.mw

Solution without use rsolve:

N := 30;
B := array(0 .. N):
B[0] := 1:
B[1] := -1/2:
for n from 2 to N do
    B[n] := -add(binomial(n + 1, k)*B[k], k = 0 .. n - 1)/(n + 1):
end do:
[B[i] $ (i = 0 .. N)];

[1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, -691/2730, 0, 7/6, 0, -3617/510, 0, 43867/798, 0, -174611/330, 0, 854513/138, 0, -236364091/2730, 0, 8553103/6, 0, -23749461029/870, 0, 8615841276005/14322]

Analytical solution: for:n >=1 is: -n*Zeta(1 - n).

[seq(-n*Zeta(1 - n), n = 2 .. N)];

[1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, -691/2730, 0, 7/6, 0, -3617/510, 0, 43867/798, 0, -174611/330, 0, 854513/138, 0, -236364091/2730, 0, 8553103/6, 0, -23749461029/870, 0, 8615841276005/14322, 0, -7709321041217/510, 0, 2577687858367/6, 0, -26315271553053477373/1919190, 0, 2929993913841559/6, 0, -261082718496449122051/13530, 0, 1520097643918070802691/1806, 0, -27833269579301024235023/690, 0, 596451111593912163277961/282, 0, -5609403368997817686249127547/46410, 0, 495057205241079648212477525/66]

 Why doesn’t this give me any solution ?, because Maple is not a magic box that'll spit out a solution to any problem..All computer algebra systems, including Maple, are limited in their capabilities.

See attached file.

Download Int.mw

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.


 

restart

kernelopts(version)

`Maple 2024.0, X86 64 WINDOWS, Mar 01 2024, Build ID 1794891`

(1)

F := int(1/(u*sqrt(1+p1*u^2/(2*p2))), u)

-arctanh(2^(1/2)/(2+p1*u^2/p2)^(1/2))

(2)

G := `assuming`([simplify(F)], [p1 > 0, p2 < 0])

arctanh(2^(1/2)*p2/((p1*u^2+2*p2)*p2)^(1/2))

(3)

S := solve(G = x-x[0], u)

(-2*p1*p2*(tanh(x-x[0])^2-1))^(1/2)/(p1*tanh(x-x[0])), -(-2*p1*p2*(tanh(x-x[0])^2-1))^(1/2)/(p1*tanh(x-x[0]))

(4)

`assuming`([simplify({evalc(Im(S[1])), evalc(Im(S[2]))})], [p1 > 0, p2 < 0, x-x[0] > 0])

{2^(1/2)*(-p2)^(1/2)*csch(x-x[0])/p1^(1/2), -2^(1/2)*(-p2)^(1/2)*csch(x-x[0])/p1^(1/2)}

(5)

NULL


 

Download start_v1.mw

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);;

2^(-(1/2)*m-(1/2)*n)*exp((1/4)*lambda^2*sigma^2)*(int(HermiteH(m, s+(1/2)*lambda*sigma)*HermiteH(n, s+(1/2)*lambda*sigma)*exp(-s^2), s = -infinity .. infinity))/(factorial(n)^(1/2)*(factorial(m)*Pi)^(1/2))

(1)

eq2 := 2^(-(1/2)*n+(1/2)*m)*exp((1/4)*lambda^2*sigma^2)*sqrt(factorial(m))*(lambda*sigma)^(-m+n)*LaguerreL(m, -m+n, -(1/2)*lambda^2*sigma^2)/sqrt(factorial(n))

eq1 = eq2

2^(-(1/2)*m-(1/2)*n)*exp((1/4)*lambda^2*sigma^2)*(int(HermiteH(m, s+(1/2)*lambda*sigma)*HermiteH(n, s+(1/2)*lambda*sigma)*exp(-s^2), s = -infinity .. infinity))/(factorial(n)^(1/2)*(factorial(m)*Pi)^(1/2)) = 2^(-(1/2)*n+(1/2)*m)*exp((1/4)*lambda^2*sigma^2)*factorial(m)^(1/2)*(lambda*sigma)^(-m+n)*LaguerreL(m, -m+n, -(1/2)*lambda^2*sigma^2)/factorial(n)^(1/2)

(2)

NULL

NULL

m := 2; n := 5; lambda := 1/6; sigma := 2/3

evalf[20](eq1)

0.62998679107780885597e-3

(3)

evalf[20](eq2)

0.62998679107780885597e-3

(4)

``

Download Q2.mw

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

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

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):

eee := rsolve({a(1) = 1, a(2) = 2, a(n) = ((10*n-16)*a(n-1)-(9*n-27)*a(n-2))/(n-1)}, a, 'makeproc')

L:=seq(evalb(expand(ee(i))=eee(i+1)), i=1..10000):

ListTools:-Occurrences(true, [L])

10000

(1)

Download hg_ex.mw

After 30 min computation on my hardware we see that for n =10000 are True.

restart

ElzakiTransform := proc (f, t) simplify(inttrans:-laplace(f*v, t, 1/v)) end proc; f := exp(n*t); g := ElzakiTransform(f, t)

v/(1/v-n)

(1)

InverseElzakiTransform := proc (g, v) inttrans:-invlaplace(eval(g/v, v = 1/v), v, t) end proc

`assuming`([InverseElzakiTransform(g, v)], [n > 0])

exp(n*t)

(2)

NULL

restart

ElzakiTransform := proc (f, t) simplify(inttrans:-laplace(f*v, t, 1/v)) end proc; f := exp(-t^2); g := ElzakiTransform(f, t)

-(1/2)*v*Pi^(1/2)*exp((1/4)/v^2)*(-1+erf((1/2)/v))

(3)

InverseElzakiTransform := proc (g, v) inttrans:-invlaplace(eval(g/v, v = 1/v), v, t) end proc

InverseElzakiTransform(g, v)

exp(-t^2)

(4)

restart

ElzakiTransform := proc (f, t) simplify(inttrans:-laplace(f*v, t, 1/v)) end proc; f := sin(t); g := ElzakiTransform(f, t)

v^3/(v^2+1)

(5)

InverseElzakiTransform := proc (g, v) inttrans:-invlaplace(eval(g/v, v = 1/v), v, t) end proc

InverseElzakiTransform(g, v)

sin(t)

(6)

restart

ElzakiTransform := proc (f, t) simplify(inttrans:-laplace(f*v, t, 1/v)) end proc; f := 1/(1+t); g := ElzakiTransform(f, t)

v*exp(1/v)*Ei(1, 1/v)

(7)

InverseElzakiTransform := proc (g, v) inttrans:-invlaplace(eval(g/v, v = 1/v), v, t) end proc

InverseElzakiTransform(g, v)

1/(1+t)

(8)

NULL

Download Elzaki_Transfrom.mw

 

kernelopts(version);

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

Try:

allvalues(value(solll));

# long answer

If want solve for: y(x) :

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

#very long answer

 

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"]);

 

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))

 

 

 

Where QPochhammer function.

1 2 3 4 5 6 7 Last Page 1 of 20