Hi, the problem in your formulation is that F(0 appears in the right-hand-side of the differential equation. This is non standard. In case you replace F(0) by a constant k, the general form of the solution is the following:

f(s) = C1*exp(1/8/Pi*(k+(k^2+384*Pi^2)^(1/2))/T*s)+C2*exp(-1/8*(-k+(k^2+384*Pi^2)^(1/2))/Pi/T*s)

where C1 and C2 are constants.

Hi,

forget about F0 and the differential equation, I mentioned.

My main question concerns what I have asked in the attachments. I mean when T goes infinite, could we suppose 1/4/pi/T as epsilon or it should be 1/4/pi*epsilon? (when we regard this as coefficient for a second order differential equation).

Epsilon is a very small value as used in perturbation. Thank you.

Hi, is T the upper bound of s, i.e. do you want to integrate on [0,T]  or T is an independent conatant?

Hi,

Is that large or small ordo O(:) or o(:)?

T is an independent constant. T is very large and, epsilon=O(1/T)= very small.

I simplify the question. Is the differential equation

y"+a.epsilon.y'+b.epsilon^2.y=0

Equivalent to the ode:

y"+epsilon.y'+epsilon^2.y=0

?

Notice: a, b are finite constants.

In your first question, is F0 independent of T?

It appears to be a dependency between t and T, in the notations.

I repeat again, t is the independent coordinate, and T is a very large constant, like it approaches infinity. They are independent. I have simplified my question in my latest answer, just before your recent comment. That's the question, straightly. Thank you.

Then it's not equivalent.

(From the first pdf the variable was s and therefore I thought it was a Fourier transform, and a truncated expansion(?!)

Another method to construct an approximating function is to consider symmetry or anti-symmetry around 0.)

Yeah, I have not used a fixed variable and it brought controversy. So, your answer is, 'they are not equivalent'. Thank you to take your time to kindly answer this query. 

Referring to your pdf, the solutions for a given epsl

I do not understand what you mean.

Part I

"Is the differential equation y"+a.epsilon.y'+b.epsilon^2.y=0 Equivalent to the ode: y"+epsilon.y'+epsilon^2.y=0 ?"

It depends on the definition of equivalence:

1. When asking about equality of set of all solutions, the answer is no, unless  a=1  and  b=1

2. When asking about having at least one the same solution, it sufficies that at least one root of characteristic equations are common to both cases. e.g.  if  a = (1+ i \sqrt{3})/2.epsilon, b=0 (where  i  stands for the imaginary unit  i^2=-1).

3. When asking about possibility of obtaining the set of all solutions of one equation  from the set of all sollutions of the second equation, then it sufficies, that the root of the characteristic equation of the first equation are proportional to  roots corresponding to the second equation.

......  and many other choices are possibile

Part II

When returning to "infinitely small parameters", the subject should be concerned with problems like the following example:  "what is the limit of this and this, when the parameter  goes to  that and that.". Simply,

THERE IS NO REASON TO CONSIDER ONE EQUATION WITH A SO CALLED INFINITELY SMALL PARAMETER,  unless some approximation is involved

Regards, Joachim

Thank you professor Domsta. You are right, but I meant more about the order of magnitude. We know that mathematically O(epsilon) is similar to O (a× epsilon). Moreover, I meant how about generalizing this to the case where they stand for coefficients of a first order, second order or other higher order type of ordinary differential equation (?).

@ Saeb Amir Ahmadi Chomachar

Dear sir, again your posing the problem is unclear, since you are not saying what magnitude you are interested in. If  epsilon is very small, then the solution is close to a linear function    A.t+B, and the perturbation depends on epsilon.  Again, there are many types of definition of good approximation. This could be simply the difference of the exact solution and the approximate solution, or the ratio of the two solutions. Without defining what is good approximation and without any assumption of the compared solutions (say both with the same initial contidions), one cannot say anything about the order of magnitude (of what?)

For being more concrete, let us compare two ode-s and their solutuions (a>0):

y''+3ay'+2a^2y= 0      y = A exp(-2ax) + B exp(-ax)

y''-3ay'+2a^2y= 0      y = A exp(2ax) + B exp(ax)

If  A=1  and  B=0 then the two solutions differ by   exp(-2ax) - exp(2ax) which is of order O(a.x)  for  a=o(1), x=O(1);  however, for  x \in [0,oo) the difference is  unbounded, independently of  positive  a .

Regards, Joachim

Somehow my answer got cut off.  Considering the equations y'' + a*epsilon*y' + b*epsilon*y=0, one need only consider the roots of the polynomial p(x)=x^2+a*epsilon*x+b*epsilon^2=0.  

The roots are x= ((-a*epsilon) +/- sqrt(a^2*epsilon^2 -4*a*b*epsilon^3))/2 =

(-a*epsilon +/- sqrt(a)*epsilon*sqrt(1-4b*epsilon))/2

When a infinity.  

When a>1, the solution is exponentially small or large depending on the value of 

sign of epsilon*(-a +/- sqrt(a)*sqrt(1-4*b*epsilon)). 

So depending on a, the solutions will not be equivalent in that the solutions form one condition on a can be mapped diffeomorphically onto the solutions of a different condition. 

I think I understand your question but let me phrase it my own way to see if my understanding is correct. We consider two different epsilons denoted e1 and e2. Consider the equation

F''(t) +e1*F'(t)+e2^2*F(t)=0

The solution will depend on both epsilons so we will write the solution as F(t,e1,e2) (the solution also depends on end-point conditions but we regard these as fixed so that dependence is not displayed in the notation). Now consider the limit of F(t,e1,e2) as e1 and e2 both go to zero. I think that you are asking if we get the same answer by taking the limit along different paths. In particular, one limit is obtained by letting e1 and e2 go to zero with e1 equal to one constant times e2, while another limit lets e1 equal a different constant times e2, and the question is whether both cases produce the same limiting function of t for F. Is that a correct interpretation of your question? I don't have an immediate answer but if the question is clear to everyone then somebody out there can probably help.

One more thought. The Helmholtz equation (or equivalent forms via changes in variables) has oscillatory solutions and nontrivial solutions satisfy homogeneous end-point conditions, while the modified Helmholtz equation (or equivalent forms) has non-oscillatory solutions and only the trivial (or null) solution satisfies homogeneous end-point conditions. If different kinds of limits that let e1 and e2 go to zero makes a difference as to whether the equation is Helmholtz-like or modified-Helmholtz-like, then the limiting function will depend on how the limits are taken.

Thank you, professor Domsta, professor Prevatt, and Mr. Edmonds. I have gotten some clues to my question.

@ L.D. Edmonds

Can you please explain, how  the Helmholtz partial differential equations is related to the current problem of ode?

you should simply get rid of the minute coefficients by rescaling the coordinates. Define a  nondimensional time coordinate as t_{0nonDimensional}=t_{real} * e2; then your equation looks like : y''+y+\xi*y'=0... with a non-dimensional resonance frequency 1. 

I personally never solve anything without rescaling it because of many many reasons such as saturation, ill conditioning, and useless complexities. if you need more info, just contact me. 

Good idea. Thank you.

Regarding the question of my reference to the Helmholtz equation in the context of ODE. It is simply the special case of a partial differential equation when a function depends on only one coordinate. It has the form y''+k^2*y=0. The modified Helmholtz equation has the form y''-k^2*y=0. If an equation that has a more general form can be converted into the Helmholtz equation via a suitable change in variables (in both dependent and independent variables) I call it Helmholtz-like. Similarly for the modified Helmholtz equation.

To be more specific, suppose the equation is

F"(x)+a*ep*F'(x)+ep^2*F(x)=0

where "a" and "ep" are constants. Let us make the change in dependent variables given by

H(x)=exp(a*ep*x/2)*F(x).

You might want to check my math because I did this in a hurry but if there are no mistakes in my math then the equation for H is

H''(x)+ep^2*(1-a^2/4)*H(x)=0.

If a^2 < 4 then the equation is the Helmholtz equation even in the limit as ep goes to zero. If a^2 > 4 then the equation is the modified Helmholtz equation even in the limit as ep goes to zero. Therefore, even when taking the limit as ep goes to zero, the solution can have different behaviors for different values of "a".

Dear friend. As I see, Helmholtz equation describes wave and has a harmonic solution. Also notice that it has no first order derivative term in its ode format.

That is correct. There is no first derivative in the ode for H, but F is exp(-a*ep*x/2) multiplied by H and there is a first derivative in the ode for F. When a^2 < 4, H has a harmonic solution, so F is exp(-a*ep*x/2) multiplied by a harmonic solution. But when a^2 > 4,  F is exp(-a*ep*x/2) multiplied by a non-harmonic solution, which has very different properties than the former case in terms of being able to satisfy end-point conditions.

Honestly, it is unnecessary to  introduce the Helmhotz equations to the discussion. General solutions for second order ode with constant coefficients are well discribed in any book on ode in first chapters.  The problem is on interpretation of the meaning of "equivalent equations", which is not given yet by the Author of the question.

There is a very important distinction between equation types in terms of uniqueness of solutions when end point conditions are specified. But I agree that the meaning of "equivalent equations" has not yet been explained.

Yes the OP needs to define what he means by "equivalent." Normally it would be that the solutions in the phase space (R^2) would be homeomorphic. However, I'm not sure that is what he means.  It also might be beneficial to rewrite the equations in the "standard" form (p(t)*y')'+q(t)y = 0.  After doing that the question might be a little clearer and the wealth of results concerning both oscillatory and non-oscillatory equations that one could find in say the classic text by Phillip Hartman, "Ordinary Differential Equations." 

https://www.researchgate.net/project/Analytic-solution-to-the-Blasius-equation

@ Saeb AmirAhmadi Chomachar

1. What is the definition of equivalent equations?

2. In the manuscript you are letting  T  to go to infinity, but in the resulting formula (13) T appears again. This is highly unreadable.

Thus, can we continue the discussion?

Just a comment on the posted manuscript. There are a lot of claims about ignoring terms but no estimates on the asymptotic bounds to validate the claims.  Even before we get to simplifying the models those details need to be developed.  Secondly as Professor Domsta states we need to know the type of "equivalences" for the equations.  The most common type is to  topological conjugacy for the flow of the dynamical system generated by the equation.  Away from an equilibrium point of the equation, conjugacy is simple to analyze via the flow box theorem.  For an equilibrium point, the results are the topic of the famous Hartman-Grobeman theorem.

www.math.isa.umich.edu/~rauch/558/conjugacyoutline.pdf

One could also consult "Ordinary Differential Equations" by Philip Hartman

Mr professor Domsta and Mr professor Prevatt. I should say that, your formal math terminology is barely understood by me, because I have had no advanced pure mathematical background in my academic career.  I have been only an engineering student , but I defend my research attached there.

I took a guess as to what the original question was and this is how I would have asked the question if my understanding of the question is correct. We are given an equation for F that can be expressed in the form

F"(x)+a*ep*F'(x)+ep^2*F(x)=0

where "a" and "ep" are constants. One thing that was never stated but is crucial for the question to be unambiguous is a statement of end-point conditions. To make the question less ambiguous (but some ambiguity will still remain), suppose it is given that, for two given points x1 and x2, we have F(x1)=y1 and F(x2)=y2 where y1 and y2 are also given. While the solution to this boundary-value problem is customarily written as F(x), the present discussion displays the dependence on everything so we write the solution as F(x,a,ep,x1,x2,y1,y2). The question, as I understand it, is whether or not the limit of F(x,a,ep,x1,x2,y1,y2) as ep goes to zero is equal to the limit of F(x,1,ep,x1,x2,y1,y2) as ep goes to zero. Ambiguity that still remains is that we are not told if this equality needs to apply to all values of (x,a,x1,x2,y1,y2) in order to get a "yes" answer to the question, or are there constraints imposed on these parameters. If there are no constraints imposed, so "yes" requires equal limits for all imagined parameters, then I say that the answer is "no" because I can come up with counter-examples in which one limit has a family of solutions while the other limit has a unique solution.

@ Mr. Edmonds. good ponder. 

best regards

we can not rewrite the second order ordinary differential equation as it raises because the left lateral boundary does not exist