@9009134 This is not a fractional DE; it's an ODE with a derivative raised to a fractional power. Because of the particularly simple form, that power can be moved to the other side of the equation. Because the dependent variable doesn't appear on the right side of the equation, it's simply an integration problem. Solving your problem is equivalent to integrating

(A*cosh(y) + B*sinh(y) + C*y + D)^E

for noninteger E.

@marians Sorry, I was using the wrong circle. I was using the circle of radius 1 centered at sqrt(11/2). It's supposed to be the circle of radius sqrt(11/2) centered at 1.

@9009134 I'm not sure, but I doubt it.

The exp at the end of your integrand is missing its argument.

@9009134 I searched extensively for a real value of A3 that worked, and I couldn't find one. The graphical evidence that I collected leads me to strongly believe that no real A3 exists. For real A3, the minimal value of u(1) is approximately 1.4 and is attained at A3 approximately 2.4.

@azwa saleh From the article's abstract, it seems interesting. However, I can't read any more than the abstract because the article itself is behind a pay wall.

Could you provide some reference materials, some web pages perhaps, that define what you're talking about? I don't understand "stability region" or "diagonally implicit".

@Al86 I don't have enough knowledge of the underlying science to answer your question directly without further reference material. If you get a Maple error, I can very likely figure out the cause. If you supply an algorithm and want to ascertain whether some Maple code correctly implements it, I can very likely do that. I presume that the algorithm is in the article that you linked in your OP, but I can't access that article because it is behind a pay wall.

@macmp91 You can't effectively make an assignment to both B and B[b] in the same code. One assignment will overwrite the other. If you want to have B subscripted by b for stylistic reasons (i.e., because you like the way that it appears in the prettyprinted code), then use B__b (double underscore).

@Preben Alsholm I'm a bit skeptical about the sum of the squares of the residuals (i.e., the vertical distance between point and curve) being the objective function. Kitonum's function is clearly a better fit of the amplitude of the data. DirectSeach:-DataFit comes with 9 stock objective functions, including sum of squares of residuals. I haven't tried any of these yet.

Just eyeballing your graph, it looks like A*x^2*sin(B*x) where A and B are positive constants to be determined from the data. A is approximately 0.015 and B is approximately 280.

@acer I find it curious that all three solutions presented so far converge at a midpoint deciding that the desired value is a root of cos(16*x). This was not a goal of my solution, which I produced independently of the other solutions.

@acer What does _S000002, which is the argument to arccos in two of the cache entries, represent?

Your error isn't "Newton iteration is not converging"; it's "Initial Newton iteration is not converging." I changed the title of this Question to refect this. It may seem like a trivial distinction until you know that both are valid error messages from the numeric solution of BVPs but the steps required to overcome them can be vastly different. In particular the "continuation parameter" technique, discussed below, can only be used to overcome the "initial" error.

@Ronan The commands that both Kitonum and I gave aren't meant to be executed in isolation. They are meant to be executed only after ra has been defined in terms of x by the code given in the original Question.