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Axioms 2018, 7(3), 66; doi:10.3390/axioms7030066
Solutions to Abel’s Integral Equations in Distributions
Department of Mathematics and Computer Science, Brandon University, Brandon, MB R7A 6A9, Canada
Author to whom correspondence should be addressed.
Received: 10 August 2018 / Accepted: 31 August 2018 / Published: 2 September 2018
The goal of this paper is to study fractional calculus of distributions, the generalized Abel’s integral equations, as well as fractional differential equations in the distributional space based on inverse convolutional operators and Babenko’s approach. Furthermore, we provide interesting applications of Abel’s integral equations in viscoelastic systems, as well as solving other integral equations, such as , and .
Keywords:distribution; fractional calculus; Mittag–Leffler function; Abel’s integral equation; convolution
Fractional modeling is an emergent tool which uses fractional differential and integral equations to describe non-local dynamic processes associated with complex systems [1,2,3,4,5,6,7,8]. Integral and fractional differential equations arise in numerous physical problems [9,10,11,12], in the fields of chemistry, biology, electronics, noncommutative quantum field theories , and quantum mechanics . Mathematical models of systems and processes in the mentioned areas of engineering  and scientific disciplines involve integrals of unknown functions and derivatives of fractional order. As far as we know, fractional calculus provides an excellent tool to construct certain electro-chemical problems and characterizes long-term behaviors [16,17], allometric scaling laws, hereditary properties of various materials and so on . This is the main advantage of fractional differential equations, in comparison with classical integer-order models in practice. Recently, Srivastava et al. presented the model under-actuated mechanical system with fractional order derivative . Many initial and boundary value problems associated with ordinary (or partial) differential equations, can be converted into Volterra integral equations [1,20]. The Volterra’s population growth model, biological species living together, and the heat change can all be characterized by integral equations. For example, Gorenflo and Mainardi  provided applications of Abel’s integral equations, of the first and second kind, in solving the partial differential equation which describes the problem of the heating (or cooling) of a semi-infinite rod by influx (or efflux) of heat across the boundary into (or from) it’s interior. In 1985, Hatcher  worked on a nonlinear Hilbert problem of a power type, solved in closed form by representing a sectionally holomorphic function by means of an integral with power kernel, and transformed the problem to one of solving a generalized Abel’s integral equation. The development of integral equations has led to the construction of many real world problems, such as mathematical physics models [23,24], scattering in quantum mechanics and water waves. There have been lots of techniques, such as numerical analysis and integral transforms [25,26,27], thus far to studying fractional differential and integral equations, including Abel’s equations, with many applications [1,20,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42].
Kilbas et al.  presented a solution in a closed form of multi-dimensional integral equations of the first kind with the Gauss hypergeometric function in the kernel over special pyramidal domains.
Raina et al.  later on investigated the solvability of the one-dimensional Abel-type hypergeometric integral equation, given bywhere with and , as well as the multidimensional Abel-type hypergeometric integral equation over a pyramidal domain in . The generalized fractional integral and differential operators are introduced and their properties are investigated systematically based on the results obtained.
Srivastava and Buschman  presented the comprehensive theory and numerous applications of the integral equations of convolution type, and of certain classes of integro-differential and non-linear integral equations, including Abel’s integral equations, in the classical sense.
We start with the necessary concepts and definitions of fractional calculus of distributions in based on the generalized convolution in the Schwartz space. Using inverse convolutional operators and Babenko’s approach, we study and solve several Abel’s integral (for all ) and fractional differential equations, as convergent series or in terms of the Mittag–Leffler functions. Many of the results derived can not be archived in the classical sense including numerical analysis methods, or by the Laplace transform. Applications are presented at the end in viscoelastic systems, and for solving other types of integral equations which can be converted into Abel’s ones.
2. Abel’s Integral Equations in Distribution
In order to study Abel’s integral and fractional differential equations distributionally, we briefly introduce the following basic concepts in distribution. Let be the Schwartz space (testing function space)  of infinitely differentiable functions with compact support in R, and the (dual) space of distributions defined on . A sequence goes to zero in if and only if these functions vanish outside a certain fixed bounded set, and converge to zero uniformly together with their derivatives of any order. We further assume that is the subspace of with support contained in .
The functional for is defined aswhere . Clearly, is a linear and continuous functional on , and hence .
Then, is a locally integrable function on R (clearly not continuous) anddefines a regular distribution .
Let . The distributional derivative of f, denoted by or , is defined asfor .
Assume f is a distribution in and g is a function in . Then the product is well defined byfor all functions as .
Clearly, and every distribution has a derivative.
It can be shown that the ordinary rules of differentiation also apply to distributions. For instance, the derivative of a sum, is the sum of the derivatives, and a constant can be commuted with the derivative operator.
It follows from  that is an entire function of on the complex plane, and
Clearly, the Laplace transform of is given bywhich plays an important role in solving integral equations .
For the functional , the (distributional) derivative formula is simpler than that for . In fact,
The convolution of certain pairs of distributions is usually defined as follows, see Gel’fand and Shilov  for example.
Let f and g be distributions in satisfying either of the following conditions:
- either f or g has bounded support (set of all essential points), or
- the supports of f and g are bounded on the same side.
Then the convolution is defined by the equationfor .
The classical definition of the convolution is as follows:
If f and g are locally integrable functions, then the convolution is defined byfor all x for which the integrals exist.
Note that if f and g are locally integrable functions satisfying either of the conditions in (a) or (b) in Definition 1, then Definition 1 is in agreement with Definition 2. It also follows that if the convolution exists by Definitions 1 or 2, then the following equations hold:where all the derivatives above are in the distributional sense.
Let and be arbitrary complex numbers. Then it is easy to showby Equation (3), without any help of analytic continuation mentioned in all current books.
Let be an arbitrary complex number and be the distribution concentrated on . We define the primitive of order of g as convolution in the distributional sense
Note that the convolution on the right-hand side is well defined since supports of g and are bounded on the same side.
Thus Equation (7) with various will not only give the fractional derivatives, but also the fractional integrals of when , and it reduces to integer-order derivatives or integrals when . We shall define the convolutionas the fractional derivative of the distribution with order , writing it asfor Re. Similarly, is interpreted as the fractional integral if Re.
In 1996, Matignon  also studied fractional derivatives in the distributional sense using the kernel distribution , and defined the fractional derivative of order of a continuous (in the normal sense) causal (zero for ) function g, as , and further obtained a relation between the distributional derivative and the classical one for a smooth function. Mainardi  extended Matignon’s work and formally defined the fractional derivative of order of a causal function (not necessarily continuous) as
The limit case is defined as
In addition, Podlubny  investigated fractional calculus of generalized functions by the distributional convolution and derived the following identities of fractional derivatives and integralswhere and k is a nonnegative integer.
The following theorem can be obtained from  with a minor change in the proof.
Let be a given distribution and y be an unknown distribution in . Then the generalized Abel’s integral equation of the first kindhas the solutionwhere α is any real number in R. In particular, if , where is an infinitely differentiable function on and , then we have four different cases depending on the value of α.
- If for , then
- If , then
- If , then .
- If , then for
Let k be a nonnegative integer, and with . Then, the integral equationhas the solution in the space
Equation (9) is equivalent towhich gives
Theorem 1 implieswhich simplifies to
Using the formulawe infer that
This completes the proof of Example 1. ☐
In particular, the integral equationhas the solution in the spaceusing
We must mention that Equation (10) cannot be solved by the Laplace transform since the distribution is not locally integrable and its Laplace transform does not exist.
Similarly, the integral equationhas the solution in the spaceby Equation (2).
Let and . Then, the integral equationhas the solution in the spacewhere
Equation (11) can be written aswhich is equal to
Applying Theorem 1, we getwhich distributes to
The Taylor expansiongives
We note that the seriesis absolutely convergent by the ratio test. Indeed,
This completes the proof of Example 2. ☐
We should point out that the series is the sum of singular and regular distributions. Indeed, let j be the largest non-negative integer, such that . Thenwhere the termis a singular distribution, whileis regular.
In the special case of , we get
Note that the function is locally integrable on R.
In particular, we have for thatusing
This also can be derived directly from Equation (11). In fact, it becomes forwhich claims thatby noting that
We further note that Equation (11) becomesfor .
Similarly, the integral equationhas the solution in the spacewhere and
Indeed, we apply Theorem 1 to get
Applying the following Taylor’s expansionswe arrive at
This completes the proof by noting that both and are absolutely convergent by the ratio test.
In the case of we get
In particular when
We shall extend the techniques used by Yu. I. Babenko in his book , for solving various types of fractional differential and integral equations in the classical sense, to generalized functions. The method itself is close to the Laplace transform method in the ordinary sense, but it can be used in more cases , such as solving integral or fractional differential equations with distributions whose Laplace transforms do not exist in the classical sense as indicated below. Clearly, it is always necessary to show convergence of the series obtained as solutions. In , Podlubny also provided interesting applications to solving certain partial differential equations for heat and mass transfer by Babenko’s method.
To illustrate Babenko’s approach in detail, we solve the following Abel’s integral equation of the second kind in the spacefor . Note that the Laplace transform does not work for this equation, since the Laplace transform of does not exist. However, this equation can be converted into
This implies by Babenko’s method thatusing
Let , and . Then the fractional differential equationhas the solution in the spacewhere and with .
We see that Equation (13) is equivalent to
Applying to both sides, we get
This implies, by Babenko’s approach
This completes the proof of Example 3. ☐
In particular, the ordinary differential equationhas the solution in the spacewhere
On the other hand, we derive that the fractional differential equationhas the solution in the spaceby usingwhere erfc is the complement to the error function (erf),
Clearly, this example can also be solved using the Laplace transform. Applying the Laplace transform to the equationwe come to
Using the inverse transform, we haveby the formula 
We must add that the following fractional differential equationcan also be solved by the same technique used in Example 3. Though it fails to do so by the Laplace transform, as the distribution is singular.
Many applied problems from physical, engineering and chemical processes lead to integral equations, which at first glance have nothing in common with Abel’s integral equations. Due to this perception, additional efforts are undertaken for the development of analytical or numerical procedure for solving these equations. However, their transformations to the form of Abel’s integral equations will speed up the solution process , or, more significantly, lead to distributional solutions in cases where classical ones do not exist [40,41].
Let and . Then the following integral equationhas the solutionwhere f is a differential function in .
Making the variable changes and . Then Equation (14) becomeswhich is Abel’s integral equation of the first kind. Therefore, we arrive atwhich implies that
This completes the proof of Example 4. ☐
In particular, we have that for and
Using the Taylor seriesif , andwe come towhich is obviously convergent. Furthermore, setting we finally infer that
Clearly, Equation (14) can be converted into
Settingthen the integral equationhas the solutionsince
Further, settingthen the integral equationhas the solution
Assume that the functions g and f are given and g is a nonzero function satisfying the conditionfor all . Then the integral equationhas the solution
Making the substitution
Equation (15) becomeswhich infers thatsince g is a nonzero function. Further, setting we come towhich is Abel’s integral equation. Hence, we get the solutionusing
This completes the proof of Example 5. ☐
A particular example can be derived from setting . We leave this to interested readers. We should point out that the termis in the distributional sense. Otherwise, it is undefined if we let and f be chosen in such that .
3. The Applications in Viscoelastic Systems
A modeling is a cognitive activity which we use to describe how devices, or objects of interest, behave.
Elasticity is the ability of a material to resist a distortion or a deforming force and return to its original form when the force is removed. According to the classical theory in the infinitesimal deformation, the most elastic materials, based on Hooke’s Law, can be described by a linear relation between the strain and stress andwhere E is a constant, known as the elastic or Young’s modulus.
However, in a more complicated fractional viscoelastic model, one [49,50] constructs the following integral equationwhereand , being the shear modulus.
According to the kernel function in Equation (16), and are equal with a memory-less system and full-memory system in creeping state respectively owing to . Clearly, we can derive that forusing Equation (2) in distribution.
When , we convert Equation (16) into
By Babenko’s approach, we imply thatwhich is the relation between the stress and strain . In particular, we derive thatif the strain .
With Babenko’s approach, we have studied and solved several Abel’s integral and fractional differential equations based on fractional calculus and convolutions of distributions in the space . Some of the results obtained are not achievable in the classical sense, such as numerical analysis methods or the Laplace transform, since the equations involve generalized functions which are not locally integrable, and undefined at points in R. Generally speaking, these equations can be expressed in terms of series or the Mittag–Leffer functions using inverse convolution operators in distribution. At the end, we demonstrate applications of Abel’s integral equations in viscoelastic systems, and for solving other different types of integral equations with potential demands in physical problems.
The order of the author list reflects contributions to the paper.
This research was funded by NSERC (Canada) under grant number 2017-00001.
The authors are grateful to the reviewers and academic editor for their careful reading of the paper with very productive suggestions and corrections, which certainly improved its quality.
Conflicts of Interest
The authors declare no conflict of interest.
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