Applications of analysis to fractional differential equations

I show how to apply theorems from analysis to fractional differential equations. The ideas feature the Arzela-Ascoli theorem and Weierstrass' approximation theorem, leading to a new approach for solvability of certain fractional differential equations.

When do fractional differential equations have maximal solutions? This video discusses this question in the following way. Firstly, a comparison theorem is formulated that involves fractional differential inequalities. Secondly, a sequence of approximative problems involving polynomials is analyzed and it is shown that there is a subsequence of solutions whose limit is the maximal solution to the original problem of interest. In particular, the interval of existence for the maximal solution is the optimal length, aligning with best practice in the local theory of existence of solutions. We achieve this through an application of the Arzela--Ascoli Theorem and our aforementioned comparison result.

Download the journal article from "Fractional Differential Calculus".
Christopher C. Tisdell, "MAXIMAL SOLUTIONS TO FRACTIONAL DIFFERENTIAL EQUATIONS"
Volume 5, Number 1 (2015), 79–85,
doi:10.7153/fdc-05-07

Free ebook

[Matchbox]

Hi Dr. Tisdell I watched this video of applications of analysis to fractional differential equations, it's amazing how you break down a hard level of mathematics into a way where it's very easy to comprehend it. Thank you for the upload of these video's

[MrJohnsurf]

Dear Chris,

Regarding fractional derivatives, what restrictions are placed on them vis-a-vis ordinary derivatives? Do they commute with ordinary derivatives? For example, let D stand for the derivative operator and consider the following properties that ordinary derivatives have.

(1) D^m (f) is single-valued and exists for polynomials for integer m.
(2) D^m(kf) = kD^m(f) (where k is a constant -- Constant Multiple Rule)
(3) D^m(D^n(f)) = D^(m+n)(f) Law of Exponents)

Now, if we assume that the above properties also hold for fractional derivatives, then we will have the following sequence of consequences in the case of the half-derivative:

Lemma 1: D^1/2(0) = 0
proof: D^1/2(0) = D^1/2(0*0) = 0*D^1/2(0) = 0 by (2) and assuming that D^1/2(0) is finite.

Lemma 2: D^1/2(1) = k for some constant k
proof: D(D^1/2(1)) = D^(1+1/2)(1) = D^1/2(D(1)) = D^1/2(0) = 0,
so D^1/2(1) is a constant -- call it k.

Lemma 3: D^1/2(1) = 0 (the above k is 0)
proof: D^1/2(1) = k, so
D^1/2(k) = D^1/2(D^1/2(1)) = D(1) = 0, but
D^1/2(k) = D^1/2(k*1) = k*D^1/2(1) = k*k by (2)
therefore k*k = 0, so k = 0

Lemma 4: D^1/2(x) = constant
proof: D(D^1/2(x)) = D^(1+1/2)(x) by (3) = D^1/2(D(x)) = D^1/2(1) = 0

Lemma 5: 1 = 0
1 = D(x) = D^1/2(D^1/2(x)) = D^1/2(constant) = D^1/2(constant*1) =
constant*D^1/2(1) = constant*0 = 0

Therefore we can't have both the Constant Multiple Rule and the Law of Exponents holding for half-derivatives and ordinary derivatives.

My questions are:

1) What restrictions must you apply on the set of functions you are trying to define half-derivatives on so that you don't run into the above contradiction? Polynomials have problems as shown above in the presence of the constant multiple property and the law of exponents.

2) If you want to keep the set of functions that half-derivatives can be applied to include polynomials, then which of the other laws are you willing to give up -- the Constant Multiple Rule or the Law of Exponents? We can't keep both rules... yet they seem like very natural rules you would want to keep and they hold for ordinary derivatives. Maybe D(D^1/2(f)) does not equal D^1/2(D(f)) and the half-derivative operation does not commute with ordinary derivatives implying that the Law of Exponents doesn't hold without restrictions on the range of values of the exponents?

It appears that you must give up the Law of Exponents or the Constant Multiple Rule or find some clever way to restrict the set of functions that fractional calculus is applied to. Which way from here for the fractional calculus? Do we give up commutation of fractional derivatives with ordinary derivatives and work without the Law of Exponents, or maybe put restrictions on the range of exponents that can be added?

What does functional analysis say about commutation of these fractional operators with ordinary differential operators?

[ayyubaibrahim]

Thank you so much, I love your presentations

[Dr.Hesham_Ghoneim]

Hi dr. krisdell can you upload FDDEs fractional delay deffirential equations 

[prahalatha]

Enjoying Mathematics rather than studying while listening your class. Sir, I want to do research in Fractional Differential Equations(FDE). Mayl you tell me the best reference book to prepare the basics to higher level in FDE?

[saranyarayappan]

Thank u so much sir. I've benefitted a lot from your videos.
I have one doubt, how did u choose \alpha in that sense.

[myblog]

Sir, pls upload geometrical interpretation of FDE

[johnsalkeld]

Are they related to stoichastic differential Equms

[madihagohar]

Dr Chris can you please upload the existence and uniqueness of Hadamard fractional derivative

[duraikalapandi]

Thank you sir. What is the real life application of fractional differential equation? Can u say one example for it?