Try entering any computation, like "2+2". If you then lose the kernel, I'd suspect a firewall issue (like the error message says). Have you changed your firewall or Windows version recently? Are you on an institutional network that updates the firewall automatically?

@rahinui You must be using a very old version of Maple if the set order is session dependent.

@vv The straightforward dsolve/rkf45 solution that I got had a major periodic component with period approximately 10. I don't see how a series solution could possibly approximate that over the range 0..70 after only 3 iterations.

What difference would it make in the plot if the length were simply a positive number like 1 instead of your L?

@Ritti

To work on your version of Maple, try replacing the getdata command with

Smat:= op([1,3],sPlt);

and likewise for W.

Your ranges for x and t are much too large. Try using x= 0..1, t= 0..0.1. It seems to me that the x-range is restricted by the boundary conditions to be at most 0..1. I don't see how it can solve outside that boundary and why it doesn't raise an error for your x= 0..10. There's no upper limit for t, but the solution is boring if t gets much larger than 0.1.

The program's name is Maple, M-a-p-l-e. It's offensive to me when you people, Mr. riti, repeatedly misspell as "mapple" the name of the program that I love.

Could you post the original system of differential equations? I mean the equations from before the "homotopy analysis method" or any other partially applied solution technique has been used.

@ivanfanthony Post them as a separate Question. If they're substantially the same as this system, I'll try to get you to make changes to the code that I gave you. If they're substantially different, I'll probably do it. Either way, post everything that you know about the system right in the Question so that I don't need to ask a bunch of followup questions.

@ivanfanthony 

I updated the code, and put in the parameter values that you gave. Please get the new code from the code window in the Answer above.

I think that there's an error either in your equations or in the parameter values, for these reasons:

1. The plots show extreme exponential growth. At k = 100, the values are up to 10^122.
2. The equations effectively contain (mu+sigma) and (mu+gamma), but using the values that you gave for those parameters, we have mu << sigma and mu << gamma, so the value of mu effectively contributes nothing to those computations.

@ivanfanthony I just posted my code with my ridiculuous values for the constants. You just need to change them in my code. The 11 lines where you change the values are all in one place.

Ideally, yes, we'd like K to be infinity. But the reality is that Maple cannot symbolically solve these equations. Nonetheless, the code I gave can theoretically compute any finite number of values. If you use a very large K and the computation is taking too long, I can easily make it run much faster (while sacrificing a little bit of the readability of the code).

@ivanfanthony 

Okay, give me some sample parameter values, and I'll give you code to plot them. I'll set it up so that you can easily change the parameter values. So, I need numeric values for these: N, S(0), R(0), I(0), E(0), beta, delta, gamma, mu, sigma, and the highest value of k.

@ivanfanthony So does that mean that I can just replace delta(k) with N?

@ivanfanthony You say that delta(k) and delta are both constants, but are they the same constant? That is, are they equal?

@ivanfanthony de

The S(k+1) equation has delta(k) while the I(k+1) equation has simply delta, which is inconsistent. So, is delta a function of k or a constant?

You say that you "can do the I(k+1) and R(k+1)". It's not possible to solve them separately. You either solve all 4 or none.

You need another equation, of the form S(k+1) = ....

It's unclear to me whether I is a function, a constant, or the specific constant sqrt(-1). If it's a function, then you also need an equation of the form I(k+1) = ....

Regarding your title: These are called difference equations. Differential equations are something else, containing derivatives of unknown functions.

@JoshuaLeiter Note that when Explore is used with parameters that cause an axis's nominal length to change (as the A parameter does to the y-axis in this example), then it's usually necessary (that is, if you want the plots to make sense as the parameters vary) to modify the plot command so that that axis's nominal length is the same for all values of the parameters. That's why my plot has the view option. The old-style animation commands (such as plots:-animate and plots:-display(..., insequence)) don't require this nuance.