Advanced Search

Mariusz Iwaniuk

1526 Reputation

14 Badges

9 years, 7 days

Social Networks and Content at Maplesoft.com

Contact Mariusz Iwaniuk

MaplePrimes Activity


These are answers submitted by Mariusz Iwaniuk

MmaTranslator or File-Open!...

Mariusz Iwaniuk 1526
October 12 2024
1 0

You may use line by line for example and for simple cases:

with(MmaTranslator);
FromMma("Integrate[x, {x, 0, 1}]")#In Mathematica use InputForm

value(%)

Or: File-Open-> *.nb Mathematica file.

 

Of course, it may not work for very complex expressions.

Another way:...

Mariusz Iwaniuk 1526
September 15 2024
2 1

In Maple 2024:

p := series(x/(-b*x^2 - a*x + 1), x = infinity, oterm = false);
coeff(p, 1/x);

#-1/b

Workaround:...

Mariusz Iwaniuk 1526
August 06 2024
3 1

 

with(MultiSeries):
e := -tanh(sqrt(2)*(a*x + b)):
limit(e, x = 0);
convert(%, tanh);

#-tanh(b*sqrt(2))

Or:

restart:
e := -tanh(sqrt(2)*(a*x + b)); (limit(e, x = 0) assuming (b <> 0));

#(-exp(2*b*sqrt(2)) + 1)/(exp(2*b*sqrt(2)) + 1)

(limit(e, x = 0) assuming (a in real, b in real));

#(-exp(2*b*sqrt(2)) + 1)/(exp(2*b*sqrt(2)) + 1)

 

 

 

 

reply...

Mariusz Iwaniuk 1526
August 02 2024
1 1


 

> 

restart
> 

ODE := diff(u(x), x, x)-(v-2)*u(x)-(v+v1/mu)*(-u(x)^3+u(x)^5) = 0

diff(diff(u(x), x), x)-(v-2)*u(x)-(v+v1/mu)*(-u(x)^3+u(x)^5) = 0

 (1)
> 

with(PDEtools)
> 

A1 := `assuming`([simplify(dchange({x = c*t+xi/sqrt(k), u(x) = W(xi)^(1/2)}, ODE, [W(xi), xi], params = {c, k, t}))], [k > 0])

(1/4)*(-4*W(xi)^4*mu*v-4*W(xi)^4*v1+4*W(xi)^3*mu*v+4*W(xi)^3*v1-4*W(xi)^2*mu*v+2*W(xi)*k*(diff(diff(W(xi), xi), xi))*mu+8*W(xi)^2*mu-(diff(W(xi), xi))^2*k*mu)/(W(xi)^(3/2)*mu) = 0

 (2)
> 

collect(expand(A1*sqrt(W(xi))), W(xi))

(-v-v1/mu)*W(xi)^3+(v+v1/mu)*W(xi)^2+(-v+2)*W(xi)+(1/2)*k*(diff(diff(W(xi), xi), xi))-(1/4)*(diff(W(xi), xi))^2*k/W(xi) = 0

 (3)
> 

A2 := `assuming`([simplify(dchange({x = c*t+xi/sqrt(k), u(x) = -W(xi)^(1/2)}, ODE, [W(xi), xi], params = {c, k, t}))], [k > 0])

(-(1/2)*W(xi)*k*(diff(diff(W(xi), xi), xi))*mu+(1/4)*(diff(W(xi), xi))^2*k*mu+W(xi)^2*((mu*v+v1)*W(xi)^2+(-mu*v-v1)*W(xi)+mu*(v-2)))/(W(xi)^(3/2)*mu) = 0

 (4)
> 

collect(expand(A2*sqrt(W(xi))), W(xi))

(v+v1/mu)*W(xi)^3+(-v-v1/mu)*W(xi)^2+(v-2)*W(xi)-(1/2)*k*(diff(diff(W(xi), xi), xi))+(1/4)*(diff(W(xi), xi))^2*k/W(xi) = 0

 (5)
> 

NULL


 

Download ODE.mw

Integral solution....

Mariusz Iwaniuk 1526
July 04 2024
0 0 
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

Without use rsolve:...

Mariusz Iwaniuk 1526
June 25 2024
0 0

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.

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

Mariusz Iwaniuk 1526
May 06 2024
1 0

See attached file.

Download Int.mw

Maybe this little helps...

Mariusz Iwaniuk 1526
May 03 2024
1 0

Download SOL.mw

I doubt there are simpler form....

Mariusz Iwaniuk 1526
April 11 2024
1 0

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.

I have only:...

Mariusz Iwaniuk 1526
March 15 2024
1 0


 

> 

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

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

Mariusz Iwaniuk 1526
February 28 2024
1 5
> 

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

Only simple equation...

Mariusz Iwaniuk 1526
February 22 2024
1 0

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...

Mariusz Iwaniuk 1526
January 30 2024
0 2

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...

Mariusz Iwaniuk 1526
October 21 2023
1 0
> 

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.

Using Laplace transform...

Mariusz Iwaniuk 1526
September 04 2023
4 3
> 

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

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