Using fracdiff, Laplace Transform and Numeric Inverse Laplace Transfrom.

This method is not perfect.

Solution for z=0

Mariusz Iwaniuk

FracPDE_3.mw

Maybe if you want find solution with series.

eq := diff(y(x), x)-(Q-x*p0*exp(alpha-beta*y(x))/(1+exp(alpha-beta*y(x))))^2 = 0

Order := 5

sol := dsolve([eq, y(0) = 0], y(x), type = 'series'); convert(sol, polynom)

y(x) = Q^2*x-Q*p0*exp(alpha)*x^2/(1+exp(alpha))+(1/3)*p0*exp(alpha)*(2*Q^3*beta+p0*exp(alpha))*x^3/(1+exp(alpha))^2+(1/4)*Q^2*p0*exp(alpha)*beta*(Q^3*exp(alpha)*beta-Q^3*beta-4*p0*exp(alpha))*x^4/(1+exp(alpha))^3

EDITED:

Using Substitution y(x) = -(ln(v(x))-alfa)/beta I have:

diff(v(x), x)+v(x)*((-p0*x+Q)*v(x)+Q)^2*beta/(1+v(x))^2 = 0

but Maple can't find general solution.

Only  for the some values give a solution.

For p0=1,Q=0,beta=1

-(1/3)*x^3-ln(exp(alfa-y(x)))+(1/2)*exp(-2*alfa+2*y(x))+2*exp(-alfa+y(x))+_C1 = 0

for more information file included.

ode1.mw

In Maple 2016.1:

value(Int(sqrt(5-4*cos(x)-4*sin(x)), x = 0 .. (1/2)*Pi))

5*2^(1/4)*EllipticK((1/4)*sqrt(5*sqrt(2)+8))-5*2^(1/4)*EllipticF(2*sqrt((2+sqrt(2))/(5*sqrt(2)+8)), (1/4)*sqrt(5*sqrt(2)+8))-4*2^(3/4)*EllipticK((1/4)*sqrt(5*sqrt(2)+8))+4*2^(3/4)*EllipticF(2*sqrt((2+sqrt(2))/(5*sqrt(2)+8)), (1/4)*sqrt(5*sqrt(2)+8))+8*2^(3/4)*EllipticE((1/4)*sqrt(5*sqrt(2)+8))-8*2^(3/4)*EllipticE(2*sqrt((2+sqrt(2))/(5*sqrt(2)+8)), (1/4)*sqrt(5*sqrt(2)+8))+10*EllipticK((1/4)*sqrt(8-5*sqrt(2)))/sqrt(-2*sqrt(2))+8*sqrt(2)*EllipticK((1/4)*sqrt(8-5*sqrt(2)))/sqrt(-2*sqrt(2))-8*sqrt(2)*((5/8)*sqrt(2)+1)*EllipticE((1/4)*sqrt(8-5*sqrt(2)))/(sqrt(-2*sqrt(2))*((5/16)*sqrt(2)+1/2))

evalf(value(Int(sqrt(5-4*cos(x)-4*sin(x)), x = 0 .. (1/2)*Pi)))

0.38153745+0.61805490*I

The problem is Maple can't find Inverse of Laplace Transfom, then

I'm use numerical inverse  Lapalce transfrom Talbot method.

Edited: 2017.11.10.

I'm adding solution with series approximation.

Good Luck.

Fractional_differential_equations.mw

Fractional_differential_equations_with_series_approximations.mw

Numerical_Inverse_Laplace_Transform_Talbot_method.mw

for Order =10.

From_MMA.mw

integral.mw

Mariusz Iwaniuk

eq:=(1/6)*7^(1/2)*6^(1/2)*hypergeom([1/14], [15/14], 1/(1/y0)^(7/3))/(y0^(1/6)*(1/y0)^(1/6))-hypergeom([1/14], [15/14], 1)-1

fsolve(eq, y0 = 1 .. 2)

1.7417148480262249059

answer.mw

curve := [2*t*(3*t^4+50*t^2-33)/(t^2+1)^3, (2*(7*t^6-60*t^4+15*t^2+2))/(t^2+1)^3]

eq := [x = op(1, curve), y = op(2, curve)]

sort(op([2, 1], eliminate(eq, t)), [x, y], ascending)

 For R>0 and a>0.

((a^(1/2)*ln(1-a+2*(1-R)^(1/2)*(-a-R)^(1/2)-2*R)-I*ln(-(2*I)*a^(1/2)*(1-R)^(1/2)*(-a-R)^(1/2)+(a-1)*R-2*a)+((1/2)*I)*ln(a))*(-1+R)^(1/2)*(-a-R)^(1/2)+(a+R)^(1/2)*(I*ln((2*I)*a^(1/2)*(1+R)^(1/2)*(-a+R)^(1/2)+(a-1)*R+2*a)-a^(1/2)*ln(1-a+2*R+2*(1+R)^(1/2)*(-a+R)^(1/2))-((1/2)*I)*ln(a))*(1-R)^(1/2))/((a+R)^(1/2)*(1-R)^(1/2)*a^(1/2))

Copy code and paste to Maple.It should work.

integral.mw

Mariusz Iwaniuk

Hi

You can use a "prime" from palletes. :)

2D_is_ok..mw

M_Iwaniuk

Use a MultiSeries package.

I_Mariusz

Read this :http://www.mapleprimes.com/questions/200812-Error-in-Dsolvenumericbvp-Initial#

Only_5.mw

I_Mariusz

Numerically this integral can be calculated.


See atached file.Numeric.mw

I_Mariusz

f(2)=-pi/2 is a correct answer.See atached file.

Maybe You want f(x)=abs(arcsin(1-x)), then f(2)=Pi/2  ,f>0  ?

Answer.mw

I_Mariusz

Yours homework.

PDE.mw