Wiener–Hopf technique for a fractional mixed boundary value problem in cylindrical layer

In this paper, we study the heat transfer modeling during freezing of a biological tissue and present an analytical approach for solving the heat transfer problem in cryosurgery. We consider a time-fractional bio-heat equation in the cylindrical coordinate and employ the Wiener–Hopf technique to find the temperature of tissue in two different domains by the factorization of associated Wiener–Hopf kernel. We discuss the fundamental roles of the Bessel and Wright functions in determining the analytical solution of fractional cryosurgery problem.


Introduction and mathematical formulation
The cryosurgery (cryotherapy or cryoablation) is a novel surgical technique that treats tumors or precancerous lesions by using the extreme cold to abnormal tissues.The region of tissue is usually induced by impacting a cryogenic probe.The probe consists of an insulated cylindrical tube with small diameter along with a tip, see Fig. 1.The tip is freezed by the liquid nitrogen (or other cryogenic agent such as carbon dioxide and argon) and the desired temperature usually tends to −196 • C to destroy the diseased tissue.The nature of this problem is fundamentally referred to the bio-heat transfer processes occurring in the living tissues and is described by the following classical Pennes's bio-heat transfer model in the cylindrical coordinates [12,13,16,20,32,44] 1 r with the initial and the mixed boundary conditions T (r, y, t = 0) = 0, (1-2) ∂ T ∂r (r = a, y, t) = 0, y > 0, (1-3) T (r = a, y, t) = f (t), y < 0.
(1-4) Analyzing and developing the cryosurgery problem and associated mathematical formulations with the different mixed boundary conditions have been studied in the literature, see for example [3,18,[25][26][27]39,40,43].The heat change mechanism and numerical analysis of the solutions for these problems are the basic motivations of the published works.
In this paper, we first intend to fractionalize the introduced problem (time-fractional cryosurgery problem) with respect to the Caputo fractional derivative, and in the second step we propose an analytical approach for finding the solution based on the Wiener-Hopf technique.We employ the Wiener-Hopf technique which is a very powerful method for solving the partial differential equations subject to the mixed boundary conditions on desired domains.The Wiener-Hopf technique is an applicable tool in the analyzing of diffraction theory, elastic and electromagnetic waves, crystal growth, fracture mechanics, flow problems, transient thermal problems, geophysical applications and mathematical finance [11,14,15,17,19,[21][22][23]28,30,31,33].
The main strategy of this method is implementation of the suitable integral transforms (here the Fourier transform) in the upper and lower half-planes as the two separate functions.The key step and mathematical challenge in obtaining the solution is decomposing the kernel into a product of two factors with the particular analyticity properties.The roles of types of the mixed boundary conditions (involving the Dirichlet, Neumann and Robin boundary conditions) are very fundamental and effective in this decomposition.
Despite the significant progress in fractional calculus and the partial fractional differential equations, the fractional mixed boundary value problems have been much less developed in the literature.The authors used this technique together with the Cagniard-Hoop technique for finding the solution of a fractional transient thermal mixed boundary value problem of distributed order [29].This is the our main motivation as establishing a bridge between the partial fractional differential equations and the mixed boundary value problems.Therefore, for the first time, we consider the following fractional problem with the initial and boundary conditions (1-2), (1-3) and (1-4) in the cylindrical coordinate and apply the Wiener-Hopf technique for solving this problem.Here, we assume that 0 < ν ≤ 1 and recall the Caputo fractional derivative as [24,34] We show that role of the Wright function in the fundamental solution of proposed problem (1-5) is determinative.The Wright function is introduced by [24,34] which for the negative parameter −1 < c < 0 is called as the Wright function of second kind (Mainardi function).In this sense, there is a great literature for the classical fractional heat equation without mixed boundary conditions which represents the fundamental solutions in terms of the Mainardi function, see [7][8][9][10] and references therein.Summarizing, we organize the paper as follows.In Sect.2, we use the joint Fourier-Laplace transforms to change the fractional cryosurgery problem into the modified Bessel differential equation and discuss the associated inverse transforms.In Sect.3, for the Wiener-Hopf kernel we find a suitable factorization in terms of the modified Bessel function of second kind.Section 4 is devoted to constructing an analytical solution of the fractional cryosurgery problem in terms of the Wright function for the different values of fractional order ν.

Implementation of integral transforms
2.1 Applying joint Fourier-Laplace transforms For finding T in (1)(2)(3)(4)(5), it is convenient to introduce an alternative problem for T ε depending on a small parameter ε and converging to T as ε → 0. This problem is stated as follows 1 r and it is obvious that we can recover the solution of original problem as T ε (r, y, t). (2-5) In order to determine the solution of problem by means of the integral transforms, we first apply the Laplace transform on the function and use the following fact [34] to get the following transformed equation with the associated boundary conditions where F(s) = L{ f (t); s}.Here, in order to eliminate s ν from Eq. (2-8) we rescale the variables r and y as and set At this point, we apply the Fourier transform for the variable y and define such that = + + − .The superscripts "+" and "−" denote that + is regular in the upper half-plane and − is regular and nonzero in the lower half-plane (α) < 0, see Fig. 2. Applying the Fourier transform on Eq. (2-13) which coincides to modified Bessel differential equation gives the general solution as Considering the boundedness of solution as r → ∞, we find the solution as follows where K 0 is the modified Bessel function of the second kind given by The branch of the square root function in K 0 is chosen such that its real part satisfies ( √ α 2 + 1) ≥ 0 and they do not cross S. Let now us introduce the functions + and and (α) < 0, respectively, and apply the Fourier transform to the boundary conditions to get Eliminating A(α, s) between the two obtained equations, we arrive at where and In view of a factorization for and G − (α) is regular and nonzero in the lower half-plane (α) < 0, Eq. (2-25) can be rewritten as . (2-28) The right-hand side of Eq. (2-28) is regular in the lower half-plane and the second term in the left side is regular in the upper half-plane.But the first term is not regular in each of the half-planes.In this sense, we use the pole removal method [37,38] and note that (2-29) By substituting Eq. (2-29) into Eq.(2-28) and rearranging terms, we obtain In this form, the left-hand side of Eq. (2-30) is regular in the upper half-plane, while the right-hand side is regular in the lower half-plane.We now use the Liouville's theorem and take the limit as |α| → ∞ to deduce the both sides of Eq. (2-30) must be zero.Therefore, and the coefficient A(α, s) is obtained as . (2-32) Referring to (2-20), we finally get the solution in the following form Using the fact G − (α) = G + (−α) (see the next section), the above relation is presented in the following form.
In this representation, the first integral is suitable for computation when y < 0 and the second integral is suitable for y > 0

Inverse Fourier transform
Here, we use a suitable contour (shown in Fig. 3) and deform the original contour to the contour AG F E DC B.
We encounter with the simple pole at α = ε √ s ν i and two branch points.Therefore, for the case y < 0 after applying a little algebra and tending ε to zero, we gain It is obvious that the associated integrals on the arcs G A and BC as the radius of semi-circle in the upper half-plane vanish.We now employ the relations (5-6) and (5-7) in Appendix to get (2-36) In the case y > 0, we deform the original contour to the contour A G F E D C B and it is clear that it does not cross any singularity.Therefore, we have or equivalently (2-37)

Factorization of Wiener-Hopf kernel
In this section, we intend to present a factorization formula for the function G(α).We state the following theorems and show the associated representation of our factorization.
Theorem 3.1 [30] Let G(α) be a analytical function and satisfy the following conditions. where B(w) + B(we iπ ) .In addition, if G(α) has not any simple zeros and poles, and B(w) has not any poles on the positive real axis, then (3-1) is rewritten as Theorem 3.2 [30] For τ > −τ + , the asymptotic behavior of the factor G In order to apply the above theorem for the factorization of G(α), we set where For this case, we can write where D + (α) is regular in the upper half-plane and D − (α) is regular in the lower half-plane.We now apply Theorem 3.2 and use the asymptotic expansion for the Bessel function (see the relation  in Appendix as Therefore the associated residue is obtained in terms of the Hankel functions as H (1)  0 (w Recurrence formulas for the Hankel functions (see the relations (5-10) and (5)(6)(7)(8)(9)(10)(11) in Appendix) gives and consequently 1 (w + H (1) H (1)  0 (w At this point, we apply the relation (5-7) to rewrite the above relation as 1 (w + H (1) and we use the Wronskian of Hankel functions (see the relation (5)(6)(7)(8)(9)(10)(11)(12) in Appendix) to get Also, by using the relations (5-8) and (5-9) we find and consequently Since D(α) has not any simple zeros and poles, and B(w) has not any poles on the positive real axis, we get the following relations using (3-4) where G + (iη) = G + (0)e q(iη) and q(η, s) In order to compute the above expression we consider and employ the Titchmarsh's theorem [2,[4][5][6]17] for the inverse Laplace transform of functions involving the zero branch point If we use the relations (5-13), (5)(6)(7)(8)(9)(10)(11)(12)(13)(14) and (5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15) in Appendix and set q(η, ze then, the imaginary part of the function N (ze −iπ ) is obtained as Finally, by using the following fact for the inverse Laplace transform of Bessel function [35,36] L and applying the convolution theorem for y < 0 we get In this case, by applying the Schouten-Van der Pol theorem [17] L the analytical solutions (given by (4-11) and (4-22)) give to the following relations.For y < 0 we get and for y > 0 we obtain where W is the Wright function.

Fig. 1 A
Fig. 1 A graph of tissue in cryosurgery problem