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Eigenfunctions and Fundamental Solutions of the Fractional Two-Parameter Laplacian
We deal with the following fractional generalization of the Laplace equation for rectangular domains , which is associated with the Riemann-Liouville fractional derivatives , , where , . Reducing the left-hand side of this equation to the sum of fractional integrals by and , we then use the operational technique for the conventional right-sided Laplace transformation and its extension to generalized functions to describe a complete family of eigenfunctions and fundamental solutions of the operator in classes of functions represented by the left-sided fractional integral of a summable function or just admitting a summable fractional derivative. A symbolic operational form of the solutions in terms of the Mittag-Leffler functions is exhibited. The case of the separation of variables is also considered. An analog of the fractional logarithmic solution is presented. Classical particular cases of solutions are demonstrated.
where means the integer part of . Consider a class of the linear nonhomogeneous differential equations:
where , , , , is a prescribed function, and is to be determined. Denoting by
equation (1.3) can be written in the form , where is the identity operator. When , we come out with the classical Poisson equation. Therefore we call fractional partial differential equation (1.3) the fractional two-parameter Poisson equation (FPE). Its homogeneous analog is naturally called the fractional Laplace equation (FLE) or fractional two-parameter Laplacian.
In this paper we present a general operational approach  to describe eigenfunctions and fundamental solutions of the fractional two-parameter Laplacian based on the conventional right-sided Laplace transform 
of absolutely integrable functions with respect to the measure and its distributional analog
In particular, the function is entire of the order and type . The exponential function and trigonometric and hyperbolic functions are expressed through (1.7) as follows:
We will consider in the sequel the existence and uniqueness of general solutions of the fractional Laplacian and its particular cases. Possible applications and an investigation of the fractional two-parameter Poisson equation (1.3) are still out of the framework of this paper and will be done in forthcoming articles of the author.
2. Eigenfunctions and Fundamental Solutions of the Fractional Laplace Equation
We begin with the following.
Definition 2.1 (see ). By , one denotes the class of functions , which are continuously differentiable on the segment up to the order and is absolutely continuous on .
It is known  that the class contains only functions represented in the form
where and are arbitrary constants. It is not difficult to find that , Moreover, if then fractional derivative (1.1) exists almost everywhere and can be represented by the formula
A description of this class is given by the following.
Theorem 2.3 (see ). A function , if and only if , and , .
Definition 2.4 (see ). One will say that a function has a summable fractional derivative if , .
If exists in the ordinary sense, that is, is differentiable in each point up to the order , then evidently admits the derivative in the sense of Definition 2.4.
Nevertheless we note that for any summable function .
Consider now the eigenfunction problem for the fractional Laplace equation in the rectangular domain
where , in the following three cases:(i) belongs to classes , by and , respectively;(ii) admits a summable fractional derivative by and belongs to by or vice versa;(iii) admits summable fractional derivative , by and , respectively.
Theorem 2.5. In case (i) trivial solution of (2.4) is the only solution.
Proof. Indeed, taking the operator from both sides of (2.4) and using the identity it becomes
Hence, applying the operator to both sides of (2.5), we use the fact that due Fubini's theorem this operator commutes with . Then we obtain
Hence from conditions of the theorem we observe that fractional integrals of the equation (2.6) are Laplace-transformable functions. Therefore we may act on (2.6) by the conventional right-sided Laplace transform (1.5), let say, by with . Taking into account its convolution and operational properties  after straightforward calculations we arrive at the following second kind homogeneous integral equation of the Volterra type:
where , , , and
Appealing to [5, Chapter ] we find that (2.7) has the only trivial solution in the space of summable functions and because for each . Cancelling the Laplace transform and using its uniqueness property for summable functions we get . Theorem 2.5 is proved.
In case (ii), (2.6) should be substituted by the following equalities (see (2.3)):
or where we denoted by Cauchy's fractional initial conditions. Treating, for instance, (2.9) we take the Laplace transform from its both sides and arrive at the following integral equation: where It is known  that a unique solution of (2.15) in the class of summable functions is which involves as the kernel the generalized Mittag-Leffler function (1.7). Next, substituting (2.16) into (2.15), using (1.7), (2.17), index laws for fractional operators , and the estimates we write solution (2.18) of the Volterra type equation (2.15) in terms of the Mittag-Leffler functions: In order to cancel the Laplace transformation by in (2.20) we will appeal to its distributional form (1.6) in Zemanian's space (see ), which is dual of the countable-union space of test functions defined by where is a sequence of real numbers , which converges monotonically to as and each is a testing-function space of smooth functions and for each nonnegative integer it satisfies According to [4, Chapter III] we assign a topology generated by the multinorm (2.22). Consequently, is a countably multinormed space and the kernel of the Laplace transform is a member of if and only if . Taking the space we have an advantage that the space of smooth functions with compact support is dense in and the members of the dual are distributions. Moreover, any is a right-sided Laplace-transformable generalized function via the formula (1.6) with the right half-plane as a region of definition. Meanwhile, any analytic function on the half-plane , which satisfies the estimate where is a polynomial, may be identified as the Laplace transform (1.6) of a right-sided Laplace-transformable generalized function which is concentrated on . Finally, the uniqueness and inversion properties are true and the inversion formula has the form in the sense of convergence in for any .
So in order to find eigenfunctions and general fundamental solutions of the fractional Laplace equation (2.4) we will invert the Laplace transform in (2.20) by using formula (2.24). Of course, we understand that the conventional right-sided Laplace transform (1.5) is a particular case of (1.6) being applied to a regular generalized function ,
Further, we have
where is a fractional part of the number, the convergence is in , and we assume that for any . Therefore, canceling the Laplace transformation in (2.20) and taking into account (2.16) after straightforward calculations we get the expression for a family of eigenfunctions of (2.4): where is the generalized Wright function : and the convergence of series in (2.26) is in . Letting in (2.26) we immediately come out with a classical fundamental solution of (2.4): Taking into account definition (1.7) of the Mittag-Leffler function, solution (2.28) may be written in the operational form Analogously, in the case of (2.10) we show that are also correspondingly eigenfunctions and a fundamental solution of (2.4).
On the other hand we may write solutions (2.29) and (2.31) in the form of the generalized Neumann series. Namely, we find
taking in mind the analyticity of series in (2.32) by in the interval and by . In the same manner, we represent (2.31) by the expression with arbitrary assuming analyticity of the corresponding series by in the interval and by . Now taking into account zero values , , , it is not difficult to verify that (2.32) and (2.33) are classical fundamental solutions of the fractional Laplacian subject to conditions (2.11), (2.12), (2.13), and (2.14) respectively. Thus we have proved
Theorem 2.6. In case (ii) functions (2.26) and (2.30) represent eigenfunctions of the fractional Laplacian (2.4) and expressions (2.28) and (2.31) are unique classical fundamental solutions subject to conditions (2.11), (2.12), (2.13), and (2.14) respectively. These solutions can be written in the corresponding form of generalized Neumann series (2.32) and (2.33) under additional conditions of analyticity.
Consequently, in the right-hand side of Volterra's equation (2.15) we get an additional term
which will give a source for generalized eigenfunctions and fundamental solutions of the fractional Laplacian (2.4). In fact, owing to the estimates
we write solution of the Volterra type equation (2.35) in terms of the Mittag-Leffler functions and generalized Neumann series:
Cancelling the Laplace transformation we take in mind the relations
where is Dirac's delta-function and denotes the convolution product. Therefore, after straightforward calculations we get the expression for a family of eigenfunctions of (2.4):
which may be written in the operational form
are also correspondingly eigenfunctions and generalized fundamental solutions of (2.4).
Example 2.8. As a particular case, it is not difficult to obtain from (2.28), (2.31) the classical fundamental solution of the Laplace equation . Indeed, putting , , we assume, correspondingly, , in (2.28), (2.31), and for instance, solution becomes Analogously we treat solution (2.31).
3. Separation of Variables: Analytic Solutions
The method of separation of variables allows us to simplify eigenfunctions and fundamental solutions of the fractional Laplacian. Indeed, putting , substituting in (2.34), and taking into account initial conditions (2.11), (2.12), (2.13), and (2.14) it becomes
where , are arbitrary constants. If , we divide (3.1) by this product and separate variables, getting two Abel's type second kind integral equations to define , , namely,
where are constants. We note that the equality for at least one point agrees with (3.1) and (3.2). So we solve the latter equations similarly to (2.18), arriving at the following family of eigenfunctions , where
On the other hand, we may write these solutions in terms of the generalized Neumann series. Precisely, denoting by
Therefore the resolvent functions , are analytic in the open discs:
respectively. Thus we write a family of eigenfunctions for (2.4) in the resolvent form
Indeed, substituting (3.9) into (2.4) taking into account the values after a simple change of the summation index into the series we easily satisfy (2.4). But we will extend our family of eigenfunctions considering
with arbitrary such that , and the corresponding resolvent function (3.5) are analytic by and . So substituting (3.10) into (2.4) and ignoring trivial cases which drive immediately to (3.10) with , (see (2.3)), after separation of variables we obtain fractional differential equations to define :