1. Introduction
One of the most typical trademarks involving Fractional Calculus is the wide range of opinions about the notions of what is a natural fractional version of some integer order concept and what is not. On the one hand, this plurality leads interesting debates and fosters a critical thinking about whether research is going “in the right direction” or not. On the other hand, it is difficult to handle such an amount of different notions and ideas in the extant literature. In particular, it is common to find lots of generalized fractional versions of a single integer order concept, some of them not very accurate or incoherent between them. These debates are still very alive nowadays [
1].
In this frame, the task of this paper is to point out some relevant facts concerning the imposition of initial values for Riemann–Liouville fractional differential equations (FDE). Although the existence and study of FDE has been widely described, for instance in [
2,
3], in this paper, we provide strong reasons to reconsider the way of imposing initial values.
We have to highlight that our research has been conducted for the Riemann–Liouville fractional derivative, which is the most classical extension for the usual derivative. In addition, the results have been developed for the particular case of linear equations with constant coefficients. However, it seems natural that the ideas described here could be extended to much more general cases.
The main reason to study this issue for the Riemann–Liouville fractional derivative, and not for other fractional versions, is that it is the left inverse of the Riemann–Liouville fractional integral. In this sense, if we restrict the study of Fractional Calculus to functions defined on finite length intervals , it is a big consensus that Riemann–Liouville fractional integral with base point a is the unique reasonable extension for the integral operator . The previous asseveration is not a simple opinion, since the Riemann–Liouville fractional integral can be characterized axiomatically in very reasonable terms.
Theorem 1 (Cartwright-McMullen, [
4])
. Given a fixed , there is only one family of operators on satisfying the following conditions:- 1.
The operator of order 1 is the usual integral with base point a. (Interpolation property)
- 2.
The Index Law holds. That is, for all . (Index Law)
- 3.
The family is continuous with respect to the parameter. That is, the following map given by is continuous, where denotes the Banach space of bounded linear endomorphisms on . (Continuity)
This family is precisely given by the Riemann–Liouville fractional integrals, whose expression will be recalled during this paper.
Hence, it makes sense to study in detail fractional integral problems for the Riemann–Liouville fractional integral to derive consequences for the corresponding fractional equations afterwards. Finally, to draw the attention of curious readers, we mention again that one of the most interesting results that we have found out is that a FDE of order
with Riemann–Liouville derivatives can demand, in principle, less initial values than
to have a uniquely determined solution. This result differs from a widely held opinion (see Theorem 1, Section 5.5, in [
5]) which states that the necessary amount of initial values is
. The reason for this discrepancy is that the question involving the “fractional smoothness” of the solutions is often neglected, since many results are derived after a not totally rigorous usage of certain mathematical concepts or results. In other words, it is important to build first the space where solutions lie in, to later seek solutions in that space.
A complete range of highlighted results with their implications can be consulted in
Section 5, while the previous sections are devoted to the corresponding deductions.
Goal of the Work
The goal of this work is to study how should we impose initial values in fractional problems with a Riemann–Liouville derivative to ensure that they have a smooth and unique solution, where smooth simply means that the solution lies in a certain suitable space of fractional differentiability. To achieve this, we will depart from some known results involving the Riemann–Liouville fractional integral, since it arises as the natural generalization of the usual integral operator; recall Theorem 1.
First, we will recall some results that imply that fractional integral problems have always a unique solution. We also recall the fundamental notions concerning Fractional Calculus, and we pay special attention to the functional spaces where calculations are performed, and especially where fractional derivatives are well defined. Note that this point of “where are functions defined” is crucial to talk about existence or uniqueness of solution and is often neglected in the extant literature concerning Fractional Calculus. Indeed, to avoid this problem, much research has been conducted for Caputo derivatives instead of Riemann–Liouville, see, for instance, [
6,
7] or general comments in [
8]. The ideas of this paragraph are developed in the second section, and most of them are available in the extant literature, except (to the best of the author’s knowledge) Lemma 2, which plays a key role in the rest of the paper.
Second, we see how each FDE of order is linked with a family of fractional integral problems, whose source term lives in a dimensional affine subspace of . This means that each solution to the FDE is a solution to one (and only one!) fractional integral problem of the dimensional family. Conversely, any solution to a fractional integral problem of the family solves the FDE, provided that the solution is smooth enough. In general, the set of source terms of the family of fractional integral problems that provide a smooth solution will consist in an affine subspace of of a dimension lower than . This is done in the third section.
Third, we characterize when a source term of the dimensional family induces a smooth solution, and thus a solution to the associated FDE. In particular, the affine space of such source terms is shown to have dimension , where is the highest order of differentiability in the FDE such that . This characterization induces a natural correspondence between each source term inducing a smooth solution for the integral problem and a vector of initial values fulfilled by the solution. More specifically, if we denote the solution to the FDE by , the initial values ensuring existence and uniqueness of solution are for . The content of this paragraph is discussed in the fourth section.
Finally, we establish a section of conclusions to highlight the most relevant obtained results, and to point out to some relevant work that should be performed in the future to continue with this approach.
2. Fundamental Notions
In this section, we will introduce the fundamental notions of Fractional Calculus that we are going to use, together with their more relevant properties and some results of convolution theory that will be useful for our purposes. We assume that the reader is familiar with the basic theory of Banach spaces, Special Functions, and Integration Theory, especially the main facts involving the space of integrable functions over a finite length interval, denoted by , and the main properties of the function.
2.1. The Riemann–Liouville Fractional Integral
We briefly introduce the Riemann–Liouville fractional integral, together with its most relevant properties. We make this introduction from the perspective of convolutions, since it will be relevant to notice that the Riemann–Liouville fractional integral is no more than a particular convolution operator, to apply later some adequate results of convolution theory.
Definition 1. Given , we defined its associated convolution operator as defined as for and . Under the previous notation, we say that f is the kernel of the convolution operator . Definition 2. We define the left Riemann–Liouville fractional integral of order of a function with base point a asfor almost every . In case that , we just define Without loss of generality, we will assume that , since the results for a generic value of a can be achieved after developing them for and applying a suitable translation. Moreover, when using the expression “Riemann–Liouville fractional integral”, we will be referring to the left Riemann–Liouville fractional integral with base point .
Remark 1. We observe that, for , the Riemann–Liouville fractional integral operator can be written as a convolution operator , with kernel It is well known that the Riemann–Liouville fractional integral fulfills the following properties, see [
9].
Proposition 1. For every :
is well defined, meaning that .
is a continuous operator (equivalently, a bounded operator) from the Banach space to itself.
is an injective operator.
preserves continuity, meaning that .
We have the Index Law for . In particular, .
Given and , we have that is absolutely continuous and, moreover, .
Furthermore, we will also use several times the following well known and straightforward remark, see [
9].
Remark 2. We have that, for and ,Indeed, if and only if . 2.2. The Riemann–Liouville Fractional Derivative
In this subsection, we will indicate the most relevant points when constructing the Riemann–Liouville fractional derivative. We will begin with a short introduction to absolutely continuous functions of order
n, since the spaces where Riemann–Liouville differentiability is well defined can be understood as their natural generalization for the fractional case, see [
9].
2.2.1. A Short Reminder Involving Absolutely Continuous Functions and the Fundamental Theorem of Calculus
It is widely known that absolutely continuous functions play a key role in several theories of Mathematical Analysis. These functions can be characterized via a “” definition, but we only recall that absolutely continuous functions are, essentially, antiderivatives of functions in up to addition with a constant. We will see later how this idea is highly relevant to construct the maximal spaces where Riemann–Liouville fractional derivatives are well defined.
Theorem 2 (Fundamental Theorem of Calculus)
. Consider a real function f defined on an interval . Then, if and only if there exists such that This last result allows us to define the derivative of an absolutely continuous function on
as a certain function in
. If
, we define its derivative
as the unique function
that makes (
1) hold.
Remark 3. It is relevant to have in mind that the previous definition makes sense because the antiderivative operator is injective, recall Proposition 1. In particular,where “1
” denotes the constant function with value 1
. Of course, it is possible to talk about absolutely functions of order n, for . In this case, for any , we say that if and only if and .
Thus,
consists of functions that can be differentiated
n times, but the last derivative might be computable only in the weak sense of Fundamental Theorem of Calculus. Analogously to the previous remark, we have the following result, see page 3 of [
9].
Remark 4. We have thatafter applying n times the Fundamental Theorem of Calculus 2. Moreover, the sum is direct since the property implies that , and the only polynomial of degree at most satisfying those conditions is the zero one. The relevant observation is that the vector space of functions that can be differentiated n times, in the sense of Fundamental Theorem of Calculus 2, has two distinct parts that only share the zero function.
The left part
consists of polynomials of degree strictly lower than
n. These functions describe, indeed, the kernel of the operator
and, thus,
The right part consists of functions that are obtained after integrating n times an element of , and hence it contains functions of trivial initial values until the derivative of order .
At this point, we recall the following result, which is widely known and can be proved immediately with the previous definition.
Proposition 2. If , we have that Although Proposition 2 seems irrelevant, it hides the key for a successful treatment of the fractional case. In the next part of the paper, we will reproduce a natural construction for the fractional analogue of the spaces .
We will arrive to the same definition that was already presented in [
9]. However, we will emphasize that the space of fractional differentiability of order
will never be contained in the space of order
, except if
. This fact will cause FDEs to have less solutions than expected.
2.2.2. The Fractional Abstraction of the Spaces
It is reasonable to define the Riemann–Liouville fractional derivative in such a way that it is the left inverse operator for the Riemann–Liouville fractional integral of the same order. After doing this, an easy analytical expression for its computation follows. Moreover, this explicit description can be extended to a bigger space, and it coincides with the definition available in the classical literature [
9].
Property 1. Consider . The Riemann–Liouville fractional derivative of order α (and base point 0) fulfils that it is the left inverse of the Riemann–Liouville fractional integral of order α, meaningfor every . We should note that, due to the injectivity of the fractional integral, Property 1 defines the Riemann–Liouville fractional derivative on the space . Moreover, it will be a surjective operator from to .
However, it is clear that we are ignoring something if we pretend that matches perfectly the usual derivative when is an integer. In particular, we observe that, for an integer value of , Property 1 only describes the behavior of over the space . Nevertheless, we are missing its definition over the supplementary part of in , which is .
This problem is easily solved, since it is possible to describe Property 1 more explicitly. We just observe that the left inverse for is given by the expression , due to the Fundamental Theorem of Calculus 2 and the Index Law in Proposition 1. Thus, one could define in a more general space than , since the only necessary condition to define is to ensure that . Hence, the following definition is natural.
Definition 3. For each , we construct the following spacewhich will be called the space of functions with summable fractional derivative of order α. If , we define . Remark 5. Therefore, functions of are defined as the ones producing a function in after being integrated times. This new function can be differentiated times in the weak sense of Fundamental Theorem of Calculus 2.
It is relevant to point out that the previous definition, although sometimes related, is different from other notions of “fractional spaces” available in the literature like, for instance, the Fractional Sobolev Spaces in Gagliardo’s sense [
10]. In our case, Definition 3 coincides with the one already presented in [
9], and we can make it totally explicit.
Lemma 1. For any , we have that Proof. First, we check . If there is a function f in both summands, then will be simultaneously a polynomial of degree at most , and a function in . Therefore, has to be the zero function after repeating the argument in Remark 4 and, since fractional integrals are injective (Proposition 1), .
It is clear that, applying to the right-hand side, we will produce a function . Moreover, it is trivial that any function in can be obtained in this way by virtue of Remark 2. Since the operator is injective, the result follows. □
From the previous lemma, we get this immediate corollary.
Corollary 1. Given , we have that if, and only if, and also , for each .
Hence, we can use Property 1 and Corollary 1 to define the Riemann–Liouville fractional derivative, coinciding with Definition 2.4 in [
9].
Definition 4. Consider and . We define the Riemann–Liouville fractional derivative of order α (and base point 0) aswhere the last derivative might be only computed in the weak sense of the Fundamental Theorem of Calculus. 2.2.3. Properties of the Space
We want to fully understand how
works over
, and the most natural way is to split the problem into two parts, as suggested by Lemma 1. We already know that
is the left inverse for
, so we should study how
behaves when applied to
. It is a well known and straightforward computation that
Hence, the kernel of
has dimension
and is given by
Moreover, we should note that, if
with
for each
, it is immediately necessary to do the following calculations from Remark 2, where
,
The previous formula generalizes the obtention of the Taylor coefficients for a fractional case and it can be used to codify functions in
modulo
, since
for
, due to Proposition 1.
2.2.4. Intersection of Fractional Summable Spaces
In general, fractional differentiation presents some extra problems that do not exist when dealing with fractional integrals. One of the most famous ones is that there is no Index Law for fractional differentiation. One underlying reason for all these complications is the following one.
Remark 6. The condition does not ensure , although the condition trivially does. This makes Riemann–Liouville fractional derivatives somehow tricky, since the differentiability for a higher order does not imply, necessarily, the differentiability for a lower order with a different decimal part. In particular, this fact has critical implications when considering fractional differential equations, as we shall see in the paper, since the unknown function has to be differentiable for each order involved in the equation. These problems give an idea of why it can be a reasonable approach to work with fractional integrals instead, and try to inherit the obtained results for the case of fractional derivatives, instead of proving them for fractional derivatives directly.
Consequently, it is interesting to compute the exact structure of a finite intersection of such spaces of fractional differentiability of different orders. To the best of our knowledge, this result is not available in the extant literature.
Lemma 2. Considering , we have thatwhere is the maximum such that . If such a does not exist, the result still holds after defining . In particular, and it has codimension .
Proof. It is obvious that
. Hence,
It is clear that
lies in
, for any
. This is simply because, due to the Index Law,
, since
.
Thus, the remaining question is to see when a linear combination of the
, where
, lies in
. The key remark is to realize that, for any finite set
if and only if
or
, for every
with
. Consequently, it is enough study when
lies in
, and there are two options:
If , we know that always. This happens because either or .
In other case, and do not share decimal parts and we need to have that can be rewritten as . If we want this to happen for every j such that , the condition is equivalent to , where is the greatest such that . Indeed, it can be rewritten as .
Therefore, the coefficients which are not necessarily null are the ones associated with , where . □
Remark 7. Due to Lemma 2, any affine subspace of with dimension strictly higher than contains two distinct functions whose difference lies in . Thus, in any vector subspace of with dimension strictly higher than , there are infinitely many functions that lie in .
2.3. Fractional Integral Equations
Consider the fractional integral equation
where
,
and assume that it has a solution
. Since
, and the left-hand side lies in
, the condition
is mandatory to ensure the existence of solution. In that case, we can apply the operator
to the previous equation and we obtain
where
and
for
. If we use the notation
Equation (
5) can be rewritten as
Therefore, it is relevant to study the properties of the operator
from
to itself, in order to understand Equation (
6).
2.3.1. Is Bounded
This claim is a very well-known result, since each summand in
is a bounded operator, and Id too, see [
2]. It is also possible to prove this, just recalling that
is a convolution operator with kernel in
and, thus, a bounded operator.
2.3.2. Is Injective
To prove that is injective, we will need a result concerning the annulation of a convolution. In a few words, we need to know what are the possibilities for the factors of a convolution, provided that the obtained result is the zero function. Roughly speaking, the classical result in this direction, known as the Titchmarsh Theorem, states that the integrand of the convolution from 0 to t is always zero, independently of t.
Theorem 3 (Titchmarsh, [
11])
. Suppose that are such that . Then, there exist such that the following three conditions hold: in the interval ,
in the interval ,
.
Remark 8. In particular, the Titchmarsh Theorem states that the operator is injective, provided that and that f is not null at any interval for .
In particular, we need the following result.
Corollary 2. The operator described in (6) is injective. Proof. Note that we can not apply Theorem 3 directly to , since it is not a convolution operator due to the “Id” term. However, is a convolution operator and we conclude, following Remark 8, that is injective. If the previous composition is injective, the right factor has to be injective. □
2.3.3. Is Surjective
In this case, we will use the following result, concerning Volterra integral equations of the second kind. This result essentially states that some family of integral equations do always have a continuous solution, provided that the source term is continuous.
Theorem 4 (Rust, [
12])
. Given , the Volterra integral equationhas exactly one continuous solution , provided that and that the following two conditions hold:If , then .
If is big enough, then for some ,
Remark 9. We know that the image of will contain since fractional integrals map continuous functions into continuous functions (Proposition 1). Moreover, will be defined by a continuous kernel when ; recall that was the least integral order in Υ.
We need to conclude that, indeed, the image of is .
Corollary 3. The operator described in (6) is surjective. Proof. Consider
and the equation
We need to show that there is
solving this problem. Observe that
x solves the previous equation if and only if it solves
but now the source term is in
. If we repeat this idea inductively, we see that
x solves the original equation if and only if
The right-hand side will be continuous for and, by Remark 9, it will have a solution. □
2.3.4. Is a Bounded Automorphism in
We have already seen that is bounded and bijective, and hence the inverse is also bounded due to the Bounded Inverse Theorem for Banach spaces. Therefore, we have the following result.
Theorem 5. The operator , described in (6) is an invertible bounded linear map from the Banach space to itself, whose inverse is also bounded. In particular, we get the following corollary
Theorem 6. Given, the equationhas exactly one solution .
Although it is not the scope of this paper, we highlight that such an equation can be solved using classical techniques for integral equations or specifical tools for the particular case of fractional integral equations, like the one exposed in [
13].
3. Implications of Fractional Integral Equations in Fractional Differential Equations
It would be desirable to have a similar result to the previous one for the case of FDE, ensuring existence and uniqueness of solution. Of course, in order to ensure uniqueness of solution, it is necessary to add some “extra conditions” to the differential version of the equation. Namely, one possibility is to impose initial values. In particular, given an ODE of order with unknown function u, we know that, after fixing the values , we can ensure uniqueness of solution under general hypotheses. The question is: what is the reasonable “fractional analogue” of the previous idea?
Before answering the previous question, we study the solutions of this general linear problem with constant coefficients
where
and
. Of course, the first point to answer is where should we look for the solution to (
7). It is very relevant to clarify completely this issue, since there are classical references; for instance, see [
5] (Theorem 1, Section 5.5), which state the following theorem or equivalent versions, which are not totally accurate as we shall comment on in the follow-up.
Theorem 7. Consider a linear homogeneous fractional differential equation (for Riemann–Liouville derivatives) with constant coefficients and rational orders. If the highest order of differentiation is α, then the equation has linearly independent solutions.
It is important to note that several references are not clear enough about the notion of solution to a fractional differential equation. With the previous sentence, we mean that it is desirable to introduce a suitable space of differentiable functions first, to later discuss about the solvability of the fractional differential equation. We devote the rest of the paper to show that the previous theorem is only true in some weak sense. Indeed, after defining formally the notion of “strong solution”, we will see that, in general, there are less than linearly independent solutions. Indeed, only for those “strong” solutions it will be coherent to talk about initial values.
The inaccuracy involving Theorem 7, and similar results, relies in the fact that the notion of solution is not completely specified. Moreover, it is common to find “proofs” that use Laplace Transform techniques without enough mathematical rigor (the final step of inverting the transform is neglected), the order of infinite sums and linear operators is interchanged (without regarding if there are sufficient hypotheses that make it legit),...
If we go back to (
7), we can make the following vital remark.
Remark 10. We recall that, in the usual case of integer orders, we look for the solutions in . Although it is quite common to forget it, the underlying reason to do this is that when every is a non-negative integer. However, in general, this does not necessarily happen when the involved orders are non-integers. Thus, we may have and, of course, a solution to Equation (7) has to lie in . Consequently, it is convenient to know the structure of the set
, which has already been described in Lemma 2, to study existence and uniqueness of solution. Of course, to expect uniqueness of solution, some initial conditions have to be added to Equation (
7), but this will be discussed in the next section. The fundamental remark is that solutions to Equation (
7) fulfill
Consequently, it is quite natural to make the following reflection. If
solves (
7), it is because
We will refer to the set of solutions to Equation (
9), as the set of weak solutions. The previous terminology obeys the following reason: although a solution to (
7) solves (
9), the converse does not hold in general. Namely, a solution to (
9) may not lie in
. Of course, if the weak solution lies in
, then it solves (
7). The set of solutions to (
7) will be called the set of strong solutions.
At this moment, we know two vital things:
We have already described that is a vector space of dimension . Therefore, the set of weak solutions has dimension too, since it is the image of the affine space via the automorphism .
The dimension of the set of strong solutions is bounded from above by
. If the dimension were higher, due to Remark 7, we could find two different solutions to (
7) whose difference would lie in
. After writing their difference as
, where
, it would trivially fulfill
which is not possible since the linear operator on the left-hand side is injective.
From these remarks, there are some remaining points that need to be studied in detail. First, we prove that the bound is sharp by inspecting which of elements in guarantee that the weak solution associated with those elements is, indeed, a strong one.
Remark 11. We have for every increment , but not for . Moreover, if is chosen as the right summand on the right-hand side in (9), we have that, for , if and only if . Therefore, after putting together the two previous ideas, we arrive to the following conclusion. In order to have a strong solution in (9), it is mandatory to select a source term . We see that the converse also holds, since can be shown to be sufficient, in order to have a strong solution.
Lemma 3. If solves for , then .
Proof. If we use the notation
, we deduce from Equation (
10) that
Observe now that the summand
can be decomposed into two parts, since two different situations can happen:
If , we see that will be always in the space for every . This simply occurs because the worst choice is and . Indeed, for small enough, the previous space is contained in , since is strictly positive.
If , there are two options:
- -
If , we have that lies again in , since the maximum value admitted for k is .
- -
If , we have that for some .
Thus, we can write
, and arrive to the equation
Note that f lived in a dimensional vector space, but lives in a (at most) dimensional vector space.
If we repeat the process, we obtain
with
lying in a (at most)
dimensional vector space. After enough iterations, the vector space has to be zero dimensional and we will have the situation
Therefore,
. Finally, if we use that
it follows
. □
4. Smooth Solutions for Fractional Differential Equations
Until this point, we have checked that, in general, there are more weak solutions (a dimensional space) than strong solutions (a dimensional space). We have also seen how weak solutions are codified depending on the source term, more specifically depending on the element chosen in . Moreover, we know that, if the choice is made in a certain subspace of , then the obtained solution is a strong one. However, one could think about codifying strong solutions directly in the fractional differential equation via initial conditions, instead of using fractional integral problems and selecting a source term linked to a strong solution. Therefore, the last task should consist of relating the choices for that give a strong solution with the corresponding initial conditions for the strong problem.
First, to simplify the notation, we reconsider Equation (
7) with the additional hypotheses that each positive integer less then or equal to
can be written as
for some
j.
This does not imply a loss of generality, since we can assume some , if needed. The only purpose of this assumption is to ease the notation in this proof, in the way that is described in the following paragraph.
If
, then
is the least possible non-integer difference
. Thus, we can use the previous notational assumption to check that
for
and
. Thus,
, and
are
constants multiplying integrals of integer order in (
10).
Now, we provide the main result of this section.
Lemma 4. Under the previous notation, Equation (7) with given initial values has a unique solution in . This solution coincides with the unique solution to (10), where the source term is the unique function fulfilling Proof. Consider again Equation (
10)
Recall that we look for strong solutions to (
10) that lie in the functional space
, so we write
Moreover, take into account that a strong choice for
allows us to write
Now, we will just derive the initial conditions after applying
, for every
, and substituting
in (
10).
On the right-hand side, this is easy, since
and, thus, the substitution at
gives zero. The function
can be computed trivially, due to the expression of
f, obtaining
On the left-hand side, on the one hand, we have again a similar situation to the previous one, since for any subindex and, thus, the substitution at gives zero. On the other hand, has three possibilities:
If , then is a scalar multiple of a power of t with positive exponent. Thus, when we make the substitution at , we get 0.
If , then is constant, and it is obviously defined for .
If , then is an integer and the computation gives the zero function.
The interest of the previous trichotomy is that we never obtain some
with
. In other cases, we would have huge trouble, since we could not evaluate the expression for
. Fortunately, we can always apply
to Equation (
10), for every value
, and substitute at
. We arrive to the following linear system of equations:
Note that all the involved derivatives in the initial conditions have the same decimal part, since only coefficients appear in the system. We also highlight that the system always has a unique solution, since it is triangular and it has no zero element in the diagonal. Therefore, a choice for f linked to a strong solution determines a vector of initial values and vice versa in a bijective way. □
We shall give two examples summarizing how to apply all the previous results.
Example 1. Consider the following fractional differential equation (strong problem):and define . In this case, note that , since all the differences are integers. The strong solutions for the example will lie in . The dimension of the affine space of strong solutions will be , and the initial conditions that ensure existence and uniqueness of solution will be , and . Moreover, after left-factoring , we find that the associated family of weak problems is where , which lives in a three-dimensional space. The a priori weak, obtained solution is always strong since we have that .
Finally, the relation between a choice for providing a strong solution and the initial conditions , and is Example 2. Consider the following fractional differential equation (strong problem):and define . In this case, note that , since it fulfills the property that is the least possible non-integer difference . The strong solutions for the example will lie in . The dimension of the affine space of strong solutions will be and the initial conditions that ensure existence and uniqueness of solution will be and . Moreover, after left-factoring , we find that the associated family of weak problems iswhere , which lives in a four-dimensional space. The a priori weak, obtained solution will be strong if . Finally, the relation between a choice for providing a strong solution and the initial conditions and is 5. Conclusions
We summarize the conclusions obtained in this paper.
We have recalled the main results involving existence and uniqueness of solution for linear fractional integral equations with constant coefficients.
We have seen that, from each linear FDE with constant coefficients of order , it is possible to derive a dimensional family of associated fractional integral equations, in a natural way. Moreover, each solution to the fractional differential equation fulfills exactly one of these fractional integral equations.
We have shown that there exists a dimensional subfamily (of the dimensional family of associated fractional integral equations) such that each solution to a problem of the subfamily gives a solution to the original linear fractional differential equation of order . This value is obtained as the greatest differentiation order in the FDE such that is not an integer. If such a value does not exist, the same result holds after defining .
We have seen how initial values at for the derivatives of orders guarantee existence and uniqueness of solution to a linear fractional differential equation with constant coefficients of order . We have described the correspondence between such initial values for the FDE and the selection of a source term in the dimensional subfamily of integral equations, in such a way that both problems have the same unique solution. If , this first initial value is imposed, indeed, for the fractional integral of order .
We expect that this idea can be extended to different types of fractional differential problems. It would be nice to amplify the scope of this work to a more general case than the one of constant coefficients. Furthermore, the same philosophy could be applied to other type of problems such as, for instance, periodic ones that have relevant applications [
14].