How to get an expression for a Hypergeometric function with symbiotic variable

I want to calculate the inverse Laplace transform of ,

where G(r)= , s is the variable.

My code can't compute a Hypergeometric function function with symbolic variable, and it can't get a concrete inverse Laplace transformation.

here is my code

phi=4;alpha=2.2;beta=4.7e-05;rho=25;r1=500;r2=400;lambda=4.0e-05;
syms s r t
G_r=0.5*r^2*(hypergeom([phi,-2/alpha],(alpha-2)/alpha,-r^(-alpha)*beta*rho*s )-1);
L_IBd00_s=exp(2*pi*lambda*(  vpa(subs(G_r,r,r1),5) -vpa(subs(G_r,r,r2),5)  ));
L_IBd00_inv=ilaplace(1/s*L_IBd00_s);

the result is

>> vpa(subs(G_r,r,radius),5)

ans =

125000.0*hypergeom([-0.90909, 4.0], 0.090909, -1.342e-9*s) - 125000.0

>> vpa(L_IBd00_inv,5)

ans =

0.000012253*heaviside(t)*ilaplace(exp(10.0*pi*hypergeom([-0.90909, 4.0], 0.090909, -1.342e-9*s) - 6.4*pi*hypergeom([-0.90909, 4.0], 0.090909, -2.1926e-9*s))/s, s, t)

So how can i get an expression of L_IBd00_inv with respect to the variable t?

Answer 1

[Torsten]

MATLAB is not able to find the Inverse Laplace transform of such a complicated function as an analytic expression.

[Drawing]

Thank you for your response! But why can't we get an expression with respect to s in the hypergeom function, the result is "125000.0*hypergeom([-0.90909, 4.0], 0.090909, -1.342e-9*s) - 125000.0", which maybe difficult for MATLAB to perform Inverse Laplace transform.

[Torsten]

What kind of simplified representation of 125000.0*hypergeom([-0.90909, 4.0], 0.090909, -1.342e-9*s) - 125000.0 do you expect to get ? "hypergeom" is defined by an infinite series expression. Do you think this will be easier to handle for "ilaplace" ?

[Drawing]

Hello, Torsten! I thought i can get a simplified representation like " hypergeom([4.2,-1],2.2,1.1*s ) ans =1 - (21*s)/10", so that this will be easier to handle for "ilaplace" .

I read this opinion from a paper, the author mentioned "the inverse Laplace transform of 1/s*" can be computed directly in MATLAB.

[Muhammad Abdullah]

@Drawing Did you solve the problem, i have a similar problem related to hypergeom function (1F1) and then computing its inverse transform.

Following is the expression

z = 10.43*s/(s+1.0)

this will be passed to 1F1 and the result is

hypergeom(1.3, 14.3, -(10.43*s)/(s + 1.0))

need to evaluate this equation and then have to compute its inverse laplace transform...

Answer 2

[Muhammad Abdullah]

hello, I want to calculate the inverse laplace transform of a characteristic function

this is the fourier transform of h(x), which is converted to Laplace transform with p = -i*t

from here we make a characteristic function which is

this characteristic function is converted to hypergeometric function

I have to get the inverse laplace transform of this function, I have written the following code:

N = 3; x_bar = 2.5; a = 1.3; b = 13; c = a+b;

syms s t

p = -1*1j*s

z = -1*((N*x_bar*(c/a)*p)/(p+1))

h = hypergeom(a,c,z)

C_slow = h/((p+1)^N)

f(t) = ilaplace(C_slow)

output of code:

z =

-(s*165i)/(2*(- 1 + s*1i))

h =

hypergeom(13/10, 143/10, -(s*165i)/(2*(- 1 + s*1i)))

do we have to put the vlaue of ''s'' to evalute the hypergeom function? I don't know what i am missing here...any help would be appreciated