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Pretty new to Mathematica here. I'm looking to verify that $$P(s,\tilde{t}|_{s_0}) = 2\tilde{b}_{\rho} \frac{s^{\alpha+1}}{\check{s_0}^{\alpha}}I_{\alpha}(2\tilde{b}_{\rho}s\check{s}_0)exp[-\tilde{b}_{\rho}(s^2+\check{s_0}^2)]$$

Is a solution to

$$\frac{\partial}{\partial{\tilde{t}}}P(s, \tilde{t}) = \frac{\partial}{\partial s}[(2b_{\rho}s - \frac{\rho}{s})P(s,\tilde{t})] + \frac{\partial^2}{\partial s^2}[P(s,\tilde{t})]$$

Where $$\alpha = \frac{\rho - 1}{2}$$ $$\tilde{b}_{\rho} = \frac{b_{\rho}}{1-e^{-\tilde{t}}}$$ $$\check{s}_0 = s_0 exp(\frac{-\tilde{t}}{2})$$ and $$I_{\alpha}$$ is a modified Bessel function of the first kind.

Following with this link, I wrote the following into Mathematica:

First I defined the variables in the solution as given above:

a = (r - 1)/2

b = b1/(1 - Exp[-t])

s0 = s01*Exp[-t/2]

i = BesselI[a, 2*b*s*s0]

I entered the solution like such:

q[s, t] = 2 b s^(a + 1)/(s0)*i*Exp[-b (s^2 + s0^2)]

I entered the general form of the equation:

E3 = D[p[s, t], t] == D[(2 b1 s - r/s) p[s, t], s] + D[p[s, t], {s, 2}]

Now I'm looking to "replace" the proposed solution q into E3 as p[s,t] and hope to get {True} as the output:

Simplify[E3 /. q[s, t]]

But the output is says: "___[contents of q]__ is neither a list of replacement rules nor a valid dispatch table and so cannot be used for replacing."

So I must be assigning something incorrectly... Does you see something wrong with this or know of an easier way to verify PDE solutions using Mathematica?

Thanks!