A Aghili

A Ansari

AA Kilbas

Alireza Ansari

D Babusci

DG Duffy

E Orsingher

G Dattoli

J Chen

O Vallee

VA Ditkin

Boundary Value Problems

English

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Ansari Boundary Value Problems 2012, 2012:125http://www.boundaryvalueproblems.com/content/2012/1/125RESEARCH Open AccessFractional exponential operators andtime-fractional telegraph equationAlireza Ansari**Correspondence:alireza_1038@yahoo.comDepartment of AppliedMathematics, Faculty ofMathematical Sciences, ShahrekordUniversity, P.O. Box 115, Shahrekord,IranAbstractIn this paper, the Bromwich integral for the inverse Mellin transform is used for ﬁndingan integral representation for a fractional exponential operator. This operator can beconsidered as an approach for solving partial fractional diﬀerential equations. Also,application of this operator for obtaining a formal solution of the time-fractionaltelegraph equation is discussed.MSC: 26A33; 35A22; 44A10Keywords: Laplace transform; Mellin transform; partial fractional diﬀerentialequation; Wright function1 Introduction and problemWe consider the exponential operatoreλ[q(x) ddx+v(x)]f (x) = f(x(λ))g(λ), (.)where x(λ), g(λ) are speciﬁed by the system of ﬁrst-order diﬀerential equations []⎧⎨⎩ddλx(λ) = q(x(λ)), x() = xddλg(λ) = v(x(λ))g(λ), g() = .(.)By the above exponential operator, Dattoli et al. found solutions of some boundary valueproblems arising in mathematical physics in terms of integral transforms type; see [, ]and references therein. Also, they used this operational technique to describe propertiesof some special polynomials and functions [–]; also see [].When we encounter an exponential operator of higher order eλα sα , where α is integeror non-integer and s = q(x) ddx + v(x), it is of interest to have an integral representation toreduce the order and apply the relation (.). For example, for exponential operators oforders two and three, we can write the Gauss-Weierstrass and the Airy integrals [, ]eλs = √π∫ ∞–∞e–ξ+λsξ dξ , (.)eλs =∫ ∞–∞e √λsξ Ai(ξ )dξ , (.)© 2012 Ansari; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu-tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in anymedium, provided the original work is properly cited.Ansari Boundary Value Problems 2012, 2012:125 Page 2 of 6http://www.boundaryvalueproblems.com/content/2012/1/125where Ai(ξ ) is the Airy function of the ﬁrst kind given byAi(ξ ) = π∫ ∞cos( t + ξ t)dt. (.)For the fractional exponential operator e–λα sα ,  < α < , it may occur that this operatorcan be written as the Laplace transform of the Wright function [–]e–λαsα =∫ ∞e–st tW(–α, ; –λαt–α)dt, (.)where the Wright function is presented by the following relation []:W (α,β ; t) =∞∑k=tkk!(αk + β) , α > –,β ∈C, t ∈C. (.)In this paper, in a general case we obtain an integral representation for eλα sα , α > , withorder one for s, and then we show how this operator can be applied to ﬁnd the formalsolutions of partial fractional diﬀerential equations (PFDEs).This problem for integral representation is referred to as the inverse of theMellin trans-form of eλαsα , α > , and in Section , we state main theorems and corollaries related to it.In Section , as an application of this technique, we ﬁnd formal solutions of the space-fractional Moshinskii’s equation and the time-fractional telegraph equation. Finally, inSection  the main conclusions are drawn.2 Main theorems and corollariesIn this section, we establish some theorems on the fractional exponential operator whichcan be useful for solving PFDEs. First, we derive an integral representation for the oper-ator eλnsn , which can be considered as a generalized representation of the relations (.)and (.).Theorem . The following identity holds true for c <s < c:eλnsn = π∫ ∞–∞esξAn(ξ ,λ)dξ , n = , , . . . , (.)where the function An(ξ ,λ) is presented byAn(ξ ,λ) =⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩∫∞ cos(rξ + (–)n+ λnrn))dr, n = k + ,nλ∫∞ e–rcos( λrn ξ )r–ndr, n = k + ,nλ∫∞ ercos( λrn ξ )r–ndr, n = k.(.)Proof By the deﬁnition of the inverse of the Mellin transform for a function eλnsn , we haveπ i∫ c+i∞c–i∞eλnsn t–s ds = π∫ ∞e(–iλr)n tir dr + π∫ ∞e(iλr)n t–ir dr= π∫ ∞eλnrne– nπ i tir dr + π∫ ∞eλnrnenπ i t–ir drAnsari Boundary Value Problems 2012, 2012:125 Page 3 of 6http://www.boundaryvalueproblems.com/content/2012/1/125= π∫ ∞eλnrn cos( nπ )e–iλnrn sin( nπ )+ir ln(t) dr+ π∫ ∞eλnrn cos( nπ )eiλnrn sin( nπ )–ir ln(t) dr= π∫ ∞eλnrn cos( nπ ) cos(r ln(t) – λnrn sin(nπ))dr. (.)The above relation implies that the Mellin transform of the last integral is equal to thefunction eλnsn , that is,eλnsn = πM{∫ ∞eλnrn cos( nπ ) cos(r ln(t) – λnrn sin(nπ))dr; s}= π∫ ∞ts–∫ ∞eλnrn cos( nπ ) cos(r ln(t) – λnrn sin(nπ))dr dt.By setting ln(t) = ξ , we get the relation (.). Theorem . (The Schouten-Van der Pol theorem for the Laplace transform []) Let cbe a suitable real constant such that F(s) and(s) are analytic functions in the half-planes > c and F(s) is the Laplace transform of f (t). Then the inverse of the Laplace transformF((s)) is given byg(t) =L–{F((s)); t} =∫ ∞f (τ )[ π i∫ c+i∞c–i∞e–(s)τ+ts ds]dτ . (.)Proof Using the deﬁnition of the Laplace transform for F((s))F((s))=∫ ∞e–(s)τ f (τ )dτ ,replacing in the inverse of the Laplace transform F((s))g(t) =L–{F((s)); t} = π i∫ c+i∞c–i∞F((s))ets dsand changing the order of integration, we get the relation (.). Corollary . It is obvious that by setting(s) = sα ,  < α < , in the relations (.) and us-ing the relation (.) for the inverse of the Laplace transform e–sα , the inverse of the Laplacetransform F(sα) can be presented byL–{F(sα); t} = t∫ ∞f (τ )W(–α, ; –τ t–α)dτ . (.)Corollary . By setting F(s) = eλnsn and combining the relations (.) and (.), we get anew integral representation for the fractional exponential equation eλnsγeλnsγ = π∫ ∞–∞esξA(α)n (ξ ,λ)dξ , γ = nα,  < α ≤ , (.)Ansari Boundary Value Problems 2012, 2012:125 Page 4 of 6http://www.boundaryvalueproblems.com/content/2012/1/125where the function A(α)n (ξ ) is given byA(α)n (ξ ,λ) =ξ∫ ∞An(τ ,λ)W(–α, ; –τξ–α)dτ . (.)In view of the theorems of a fractional exponential operator expressed in this section,we may apply this operator to PFDEs in the next section.3 Application to partial fractional differential equationsExample . In connection with initial-value diﬀusions, we consider the space-fractionalMoshinskii’s equation of order γ in the Riemann-Liouville sense []∂∂t u(x, t) = + x(( + x) ∂∂x)γu(x, t), γ > ,x > , t >  (.)with the Cauchy-type initial condition as u(x, ) = f (x).In order to obtain the solution of (.), by solving the ﬁrst-order partial diﬀerential withrespect to t and applying the initial condition, the formal solution in the form of fractionalexponential operator gives rise tou(x, t) = et+x ((+x) ∂∂x )γf (x). (.)Now, by setting ( t+x )n = λ, s = ( + x) ∂∂x and applying Corollary . for the integral rep-resentation of eλnsγ , we can write the solution in terms of the integral transform asu(x, t) = π∫ ∞–∞e(+x) ∂∂x ξA(α)n (ξ ,λ)f (x)dξ , γ = nα,  < α ≤ ,where the functionA(α)n (ξ ,λ) is given by the relation (.). The above relation can be sim-pliﬁed in the following form:u(x, t) = π∫ ∞–∞A(α)n(ξ ,( t + (tan(ξ + tan–(x)))) n)f(tan(ξ + tan–(x)))dξ , (.)where we used the relations (.) and (.) by choosing the functions q(x) =  + x andv(x) = .Example . As another application, we consider the time-fractional telegraph equation[, ]∂∂x u(x, t) =[a ∂α∂tα + b∂α∂tα + c]u(x, t),  < α ≤ ,x, t > ,a,b, c ∈R (.)with initial and asymptotic conditions u(, t) = f (t), limx→∞ u(x, t) = .Similar to the previous problem by solving the equation with respect to x and applyingthe initial and asymptotic conditions, the formal solution takes the form:u(x, t) = e–x√a ∂α∂tα +b∂α∂tα +cf (t). (.)Ansari Boundary Value Problems 2012, 2012:125 Page 5 of 6http://www.boundaryvalueproblems.com/content/2012/1/125Now, by setting ∂∂t = s and writing an integral representation for e–x√as+bs+c in terms ofthe Bessel function of order one, we get []e–x√as+bs+c= e–x(b√a+√as) – x√a∫ ∞x√ae–su– baJ(√a√u – ax)√u – axdu,  = ac – b = . (.)We can rewrite the relation (.) in the following form:u(x, t) = e–xb√a∫ ∞τW(–α, ; –x√aτ–α)f (t – τ )dτ– x√a e– ba∫ ∞τf (t – τ )∫ ∞x√aJ(√a√u – ax)√u – ax×W(–α, ; –uτ–α)dudτ , (.)where we used the relation (.) for the linearization of a fractional exponential operatore–xsα , and then we applied the relations (.) and (.) by substituting q(x) =  and v(x) = .4 ConclusionsThis paper provides some new results in the theory of fractional derivative. These resultsshow the ﬂexible operational technique can be used in a fairly wide context beside theintegral transforms for obtaining the formal solutions of PFDEs.Also, this technique can be considered as a promising approach for many applicationsin applied sciences.Competing interestsThe author declares that he has no competing interests.AcknowledgementsThe author was partially supported by the Center of Excellence for Mathematics, University of Shahrekord.Received: 27 July 2012 Accepted: 12 October 2012 Published: 29 October 2012References1. Dattoli, G, Ottaviani, PL, Torte, A, Vazquez, L: Evolution operators equations: integration with algebraic and ﬁnitediﬀerence methods. Applications to physical problems in classical and quantum mechanics and quantum ﬁeldtheory. Riv. Nuovo Cimento 20, 1-133 (1997)2. Dattoli, G, Srivastava, HM, Zhukovsky, Z: Operational methods and diﬀerential equations with applications toinitial-value problems. Appl. Math. Comput. 184, 979-1001 (2007)3. Dattoli, G, Ricci, PE, Khomasuridze, I: Operational methods, special polynomial and functions and solution of partialdiﬀerential equations. Integral Transforms Spec. Funct. 15(4), 309-321 (2004)4. Babusci, D, Dattoli, G, Sacchetti, D: The Airy transform and the associated polynomials. Cent. Eur. J. Phys. 9(6), 1381(2011)5. Dattoli, G: Operational methods, fractional operators and special polynomials. Appl. Math. Comput. 141, 151-159(2003)6. Dattoli, G, Ricci, PE, Sacchetti, D: Generalized shift operators and pseudo-polynomials of fractional order. Appl. Math.Comput. 141, 215-224 (2003)7. Vallee, O, Soares, M: Airy Functions and Applications to Physics. Imperial College Press, London (2004)8. Aghili, A, Ansari, A: Solving partial fractional diﬀerential equations using the LA-transform. Asian-Eur. J math. 3(2),209-220 (2010)9. Aghili, A, Ansari, A: Solution to system of partial fractional diﬀerential equation using the L2-transform. Anal. Appl.9(1), 1-9 (2011)10. Ansari, A, Refahi Sheikhani, A, Saberi Najaﬁ, H: Solution to system of partial fractional diﬀerential equation using thefractional exponential operators. Math. Methods Appl. Sci. 35, 119-123 (2012)11. Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Diﬀerential Equations. North-HollandMathematics Studies, vol. 204. Elsevier, Amsterdam (2006)Ansari Boundary Value Problems 2012, 2012:125 Page 6 of 6http://www.boundaryvalueproblems.com/content/2012/1/12512. Duﬀy, DG: Transform Methods for Solving Partial Diﬀerential Equations. CRC Press, New York (2004)13. Chen, J, Liu, F, Anh, V: Analytical solution for the time-fractional telegraph equation by the method of separatingvariables. J. Math. Anal. Appl. 338, 1364-1377 (2008)14. Orsingher, E, Beghin, L: Time-fractional telegraph equations and telegraph processes with Brownian time. Probab.Theory Relat. Fields 128, 141-160 (2004)15. Ditkin, VA, Prudnikov, AP: Integral Transforms and Operational Calculus. Pergamon, Elansford (1965). (Englishtranslation)doi:10.1186/1687-2770-2012-125Cite this article as: Ansari: Fractional exponential operators and time-fractional telegraph equation. Boundary ValueProblems 2012 2012:125.

Fractional exponential operators and time-fractional telegraph equation

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Fractional exponential operators and time-fractional telegraph equation

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