Mariusz Iwaniuk

1461 Reputation

14 Badges

8 years, 138 days

Social Networks and Content at Maplesoft.com

MaplePrimes Activity


These are answers submitted by Mariusz Iwaniuk

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.

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

See attached file:

1_case1.mw

INV := invztrans((z - 1)^2/(a*z^2 + b*z + c), z, n);
simplify(allvalues(INV));

#(-2^(-n)*((a - c)*sqrt(-4*a*c + b^2) + (b + 4*c)*a + c*b)*((-b + sqrt(-4*a*c + b^2))/a)^n + ((-a + c)*sqrt(-4*a*c + b^2) + (b + 4*c)*a + c*b)*(-(b + sqrt(-4*a*c + b^2))/(2*a))^n + 2*charfcn[0](n)*a*sqrt(-4*a*c + b^2))/(2*sqrt(-4*a*c + b^2)*a*c)

 

 Because dsolve doesn't know whether to compute  T(x) or T(L ) .

dsolve({ics2, ode}, T(x));

 

If we use MMA to solve series:

inttrans:-fourier(sech(x), x, -k);

#Pi*sech(k*Pi/2)

 

L := 1;
g := 9.81;
f := 4*sqrt(L/g)*EllipticF(Pi/2, sin(theta0/2));
plot([Re(f), Im(f)], theta0 = 1 .. 20);
1 2 3 4 5 6 7 Last Page 1 of 19