@Markiyan Hirnyk 

Thanks. It's fixed now. Somehow two cut-and-pastes got merged.

@Preben Alsholm 

Thanks, Preben. It's a lot to absorb. To further my understanding of these steps, would you mind completing these steps just for the epsilon = 0 case, for which we know there's a nice solution? That is, take the ODESOLStruc (with epsilon=0), solve the first-order ODE with dsolve, do the integration, and invert the function to obtain something akin to the nice solution? I can do the dsolve and the int, but not the inversion.

@Carl Love 

I don't know the small parameter method, but I guess that you start with a solution for epsilon = 0. For what it's worth, Maple has a nice compact symbolic solution for that:

dsolve(eval(eq, epsilon= 0));

     x(t) = _C2*sqrt(2)*sqrt(1/(_C2^2+1))*JacobiSN(((1/2*I)*sqrt(2)*t+_C1)*sqrt(2)*sqrt(1/(_C2^2+1)), _C2)

And Maple even has a sort of symbolic solution for the fully parameterized equation:

dsolve(eq);

     x(t) = _a &where
           [{(diff(_b(_a), _a))*_b(_a)-_b(_a)*_a^4*alpha*epsilon
                +_b(_a)*beta*epsilon+_a^3-_a = 0},
            {_a = x(t), _b(_a) = diff(x(t), t)},
            {t = Int(1/_b(_a), _a)+_C1, x(t) = _a}]

I don't know what this means, and I was hoping that someone here could explain it to me. In particular, What does it mean that _a appears as both the dependent variable and as an independent variable? Why is there no _C2? How can this solution be used? Why is it better than just giving no solution?

@Rouben Rostamian  

I don't understand the mathematics here, so this comment may be irrelevant: Before you get too far involved with this problem in a Mapleish way, remember that Maple currently will only numerically solve PDEs with exactly two independent variables. There's no such hard limit for symbolic solutions.

You were using Maple's 2D input and you typed the epsilon in the first line too close to the following left parenthesis, so it was interpretted as if epsilon were a function rather than as a coefficient. If you use Maple's 1D input, this and numerous other problems will be avoided. In the second line, you will get better results using eval rather than subs. The eval forces the evaluation of the derivatives. So, change these two lines to

eq:= diff(x(t),t$2) - epsilon*(alpha*x(t)^4 - beta)*diff(x(t),t) - x(t) + x(t)^3;
eq2:= eval(eq, x(t)= 1 + epsilon*x[1](t) + epsilon^2*x[2](t));

@Christopher2222 How does one hide a code edit region? I see the option to collapse it, but that doesn't completely hide it.

@MDD 

If you need for your procedure to be able to detect empty Matrices, then I recommend looking at the ColumnDimension rather than the RowDimension, because, like I said, any reasonable attempt to create an empty Matrix out of square brackets leads to a Matrix with one row and zero columns. (<<[][]>> creates a Matrix with zero rows and one column.)

If you need for your procedure to create empty Matrices, then use Matrix() or, if speed is critical, rtable(1..0, 1..0, subtype= Matrix).

If you post your procedure, I can make further suggestions.

@Haley_VSRC 

The corresponding mathematical reason is that b must be less than or equal to the minimal value of t in order to avoid nonreal values. The graphical result is that the omitted points are very close to the regression curve horizontally.

@Markiyan Hirnyk 

No, the {} indicates that the set of elements is the empty set, which is different from the empty set being an element.

@MDD 

NonzeroRowsAndColumns:= (M::Matrix)-> [
     add(`if`(ArrayTools:-IsZero(M[j,..]), 0, 1), j=1..op([1,1], M)),
     add(`if`(ArrayTools:-IsZero(M[..,j]), 0, 1), j=1..op([1,2], M))
]:

@Markiyan Hirnyk 

When I said that my fit had lower residuals, I was using the fit that treated the first four points as outliers (my Sol_Carl5), and I compared it to doing the equivalent with DirectSearch:-DataFit, using all nine fitmethods and taking the minimal residuals. Done this way, my method has less than half the residuals.

But, if I use all four methods crossed with all nine fitmethods (for a total of 36 calls to DataFit), I can make a small improvement over my method. Here's the worksheet:

Download RecenteredPowerFit_vs_DirectSeach.mw

@Markiyan Hirnyk I'm not trying to blame anybody! I was just trying to tell you that both issues were caused by the same mistake. Geez, I can't even say that you're right without you finding some way to criticize!

@capea 

You didn't execute the expandoff command! That is nescessary to prevent simplification of residues. Also, don't forget to restart before the whole process.

@Markiyan Hirnyk You're absolutely right. I made a mistake. I will correct the Answer now.

Regarding the expand portion, you're experiencing the same mistake. After I make the correction to the Answer, try expand(Psi(3)) before and after the expandoff commands.

@MDD Try using the Maple help: Enter ?Psi at the Maple prompt.

Psi(z) = diff(ln(GAMMA(z)), z)