Non-homogeneous time fractional heat equation

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1 33 JACM 3, No., 33-4 (8) Journal of Abtract and Computational Mathematic Non-homogeneou time fractional heat equation A. Aghili Department of Applied Mathematic, Faculty of Mathematical Science, Univerity of Guilan, Raht - Iran Received: 3 Aug 7, Accepted: Sep 7 Publihed online: 3 Jan 8 Abtract: In thi article, the author conidered certain non-homogeneou time fractional heat equation which i a generalization of the problem of a vicou ring damper for a freely proceing atellite. Tranform method i a powerful tool for olving partial fractional differential equation. The reult reveal that the tranform method i very convenient and effective. Keyword: Caputo fractional derivative, Non-homogeneou time fractional heat equation, Laplace tranform, Fourier tranform. Introduction Fractional differential equation arie in the unification of diffuion and wave propagation phenomenon. The time fractional heat equation, which i a mathematical model of a wide range of important phyical phenomena, i a partial differential equation obtained from the claical heat equation by replacing the firt time derivative of a fractional derivative of order. In recent year, it ha turned out that many phenomena in fluid mechanic, phyic, biology, engineering and other area of the cience can be uccefully modeled by the ue of fractional derivative. That i becaue of the fact that, a realitic modeling of a phyical phenomenon having dependence not only at the time intant, but alo the previou time hitory can be uccefully achieved by uing fractional calculu. In thi work, we conider method and reult for the partial fractional diffuion equation which arie in application. Several method have been introduced to olve fractional differential equation, the popular Laplace tranform method, [,, 3]. Atanackovic and Stankovic [4,5] and Stankovic [3] ued the Laplace tranform in a certain pace of ditribution to olve a ytem of partial differential equation with fractional derivative, and indicated that uch a ytem may erve a a certain model for a vico-elatic rod. Wy [5] and Schneider [] conidered the time fractional diffuion and wave equation and obtained the olution in term of Fox function. In recent year, the implementation of extended G/G- method for the olution of nonlinear evolution equation, nonlinear Klein - Gordon equation, Bouineq equation have been well-etablihed by reearcher [4].. Definition And Notation Definition. The left Caputo fractional derivative of order α ( < α < ) of φ(t) i a follow [8] D c,α a φ(t) = t Γ ( α) a (t ξ ) α φ (ξ )dξ. () Correponding author armanaghili@yahoo.com, arman.aghili@gmail.com c 8 BISKA Biliim Technology

2 JACM 3, No., 33-4 (8) / 34 Definition. The Laplace tranform of function f (t) i defined a follow If L { f (t)} = F(), then L {F()} i given by L { f (t)} = e t f (t)dt = F() () f (t) = c+i e t F()d, (3) πi c i where F() i analytic in the region Re() > c. The above integral i known a Bromwich complex inverion formula Lemma. Let L { f (t)} = F() then, the following identitie hold true..l (e k ) = k ( dξ, π).e ωβ = π e rβ (ωcoβπ) in(ωr β inβπ)( e τ rτ dτ)dr, 3.L (F( α )) = π f (u) e tr urα coαπ in(ur α inαπ)drdu, 4.L (F( ) = t u πt ue 4t f (u)du. Proof. [,] e tξ k Example. By uing an appropriate integral repreentation for the modified Beel function of the econd kind of order ν, K ν (), how that L { K η(a µ) ( µ) η K ν (b + β) t ( + β) ν } = e (µ β)t τ η µτ a e 4τ (t τ) ν e (a) +η β(t τ) b 4(t τ) (b) +ν dτ. (4) Solution. It i well known that K ν (a ) ha the following integral repreentation [6] K ν (a ) = (a ) ν ν+ e ξ a dξ. (5) ξ ν+ At thi tage, uing complex inverion formula for the Laplace tranform and the above integral repreentation we get L { K η(a ) η } = c+i iπ c i Changing the order of integration and implifying to obtain L { K η(a ) η } = a η ).5η η ((a η+ The value of the inner integral i δ(t a ), we have the following e ξ ξ η+ ( c+i iπ c i e ξ a dξ )d. (6) ξ η+ e t a d)dξ. (7) η+ L { K η(a ) η } = a η e ξ a δ(t )dξ, (8) ξ η+ making a change of variable (t a ) = u and uing elementary propertie of Dirac - delta function, we arrive at L { K η(a ) η } = tη e a 4t. (9) (a) η+ c 8 BISKA Biliim Technology

3 35 A. Aghili.: Non-homogeneou timefractional heat equation Finally, uing the hift and convolution theorem we obtain L { K η(a µ) ( µ) η K ν (b + β) t ( + β) ν } = e (µ β)t τ η µτ a e 4τ (t τ) ν e (a) +η β(t τ) b 4(t τ) (b) +ν dτ. () Definition 3. The The Laplace tranform of Caputo fractional derivative of order non integer. For n < α n, we have the following identity [5] L{ C Dα t f (t)} = α n F() α k f (k) (). () k= Definition 4. The The two-parameter function of the Mittag-Leffler type i defined by the erie expanion E α,β (z) = when α,β,z C. We have the following relationhip n= z n Γ (αn + β), () Definition 5. The implet Wright function i given by the erie L{t β E α,β (±at α )} = α β α a (Re() > a α. (3) W(α,β;z) = when α,β,z C. We have the following relationhip n= z n n!γ (αn + β), (4) Lemma. The following identitie hold true for < ν <. g(t) =L ( =e kt [L{t β E α,β (±at α )} = α β α a (Re() > a α ). (5) e (+ k ν +k )η ν + k dη) =.. e u J ( ktu)( e tr ηrν coπν in(ηr ν inπν)dr)du)dη. Proof. Let u aume that F() = k e (+ +k )η +k by complex inverion formula for the Laplace tranform, we have dη f (t) = c+i e t ( πi c i e (+ +k k )η + k dη)d, changing the order of integration which i permiible by Fubini theorem, lead to f (t) = e η ( c+i πi c i kη t e +k + k d)dη, c 8 BISKA Biliim Technology

4 JACM 3, No., 33-4 (8) / 36 in the inner integral by making change of variable + k = ξ, we obtain f (t) = e kt the value of the inner integral i J ( ktη) [7,8] o that e η ( c +i e tξ kη ξ πi c i ξ f (t) = e kt e η J ( ktη)dη, uing part three of () and theorem of Titchmarh [8] lead to g(t) = e kt dξ )dη, e u J ( ktu)( e tr ηrν coπν in(ηr ν inπ.ν)dr)du)dη. Main Reult In thi ection, the author conidered certain non-homogeneou time fractional heat equation which i a generalization of the problem of a vicou ring damper for a freely proceing atellite tudied by P.G.Bahuta [8]. In thi tudy, only the Laplace tranformation i conidered a it i eaily undertood and being popular among engineer and cientit. The baic goal of thi work ha been to implement the Laplace tranform method for tudying the above mentioned problem. The goal ha been achieved by formally deriving the exact analytical olution.. Non homogeneou time fractional heat equation Problem. Let u olve the following partial fractional differential equation D C,α t u = r r u (r ) λu + µ, (6) r λ,µ >, r a,u(r,) = β,u(a,t) = exp ω t, lim r > u(r,t) M. Solution. Solution: In order to obtain the olution of the fractional heat equation, the Laplace tranform i applied to PDE and boundary condition to obtain α U(r,) β α = r (U r(r,) + ru rr (r,)) λur,) + µ, (7) after implifying, we get the following Hence, the homogeneou equation i U rr (r,) + r U r(r,) ( α + λ)u(r,) = (β α + µ ). (8) [U rr (r,) + r U r(r,) ( α + λ)u(r,) =. (9) c 8 BISKA Biliim Technology

5 37 A. Aghili.: Non-homogeneou timefractional heat equation The equation (9) i modified Beel differential equation of order zero. The general olution i U p (r,) = AI (r α + λ) + BK (r α + λ). () The function K for ome r, i unbounded. However, U p (r,) i a bounded function. Therefore B = and () read U p (r,) = AI (r α + λ). () Now, in order to obtain the olution of nonhomogeneou equation (6), we uppoe that U c (r,) = γ i the olution to nonhomogeneou equation. Then we get the following From relation () and (), we obtain U c (r,) = α β + µ α + λ = γ. () U(r,) = U p (r,) +U c (r,) = AI (r α + λ) + α β + µ α + λ. (3) In order to obtain contant A, in relation (3), we ue boundary condition to get from the above relation, we get the value of contant A a below U(a,) = + ω = AI (a α + λ) + α β + µ α + λ, (4) A = ( + ω α β + µ α + λ ) I (a α + λ). (5) By ubtitution of the value of A in relation (3), we obtain the general olution to non - homogeneou equation in the following form In cae α =, we have U(r,) = ( + ω α β + µ α + λ ) I (r α + λ) I (a α + λ) + α µ β + α + λ. (6) U(r,) = ( + ω β + µ + λ ) I (r + λ) I (a + λ) + β + µ + λ. (7) At thi tage, the Bromwich integral i utilized to invert U(r,) a follow. u(r,t) = c+i (( πi c i... = (Re[(( k + ω β + µ + λ ) I (r + λ) + λ) µ I (a + λ) + β + + λ )et d =.. (8) + ω β + µ + λ ) I (r I (a + λ) + β + + λ )et ], = k,). The ingularitie of the integrand are =, = λ, = ω, = k n a,n =,,3,..., then the reidue at the ingularitie of (8) are a follow. At =, we have b = lim > (( + ω β + µ + λ ) I (r + λ) I (a + λ) + β + µ + λ )et = µ λ ( I (r λ) I (a λ ). µ c 8 BISKA Biliim Technology

6 JACM 3, No., 33-4 (8) / At = λ, we have 3. At = ω, we have b = lim > λ (( + ω β + µ + λ ) I (r + λ) I (a + λ) + β + µ + λ )( + λ)et =. b 3 = lim > ω (( + ω β + µ + λ ) I (r + λ) I (a + λ) + β + µ + λ )( + ω )e t = I (r λ ω ) I (a λ ω ) e ωt. 4. If k,k,k 3,... are the root of the function J (ξ ), then J (k n ) = for n =,,3,... Uing the fact that I (ξ ) = J (iξ ), one get ξ = ik n, the root of I (a + λ), are n = (λ + k n a ). Finally, the reidue at n = (λ + k n a ), are At = (λ + k n a ), we have b 4 = lim k > (λ+ n )(( + ω β + µ + λ ) I (r + λ) a I (a + λ) + β + + λ )( + λ)et = = e ( k n a λ) J ( a r k n)( k n )( k a n λ) + µ(λ + k a n ω ) β(λ + k a n ω )( k a n J (k n )k n (λ + k n )(λ + k a n ω a ) Let u uppoe that λ = µ = and β =. Hence, u(r,t) i in the following form In cae α =.5, (emi - derivative) we have u(r,t) = e ω t J (ωr) J (ωa) + a U(r,) = ( + ω the above relation can be re-writen a below k e ( k n a ) J (( r a )k n) µ k n (a k n ω )J (k n ) β + µ + λ ) I (r + λ) I (a + λ) + U(r,) = β + µ ( + λ) + I (r + λ) + ω I (a β + µ + λ) ( + λ) At thi point, we find inverion of the above relation term wie, o that a ).. (9) β + µ + λ, (3) I (r + λ) I (a. (3) + λ) u(r,t) = L β ( ) + L µ ( ( + λ) ( + λ) ) + L ( + ω ) L ( I (r + λ) I (a )... (3) + λ)... L ( β + µ ( + λ). I (r + λ) I (a ). + λ) Let u introduce the following then G(r, ) = ( β + µ ( + λ). I (r + λ) I (a ), (33) + λ) G(r,) = ( β + µ ( + λ). I (r + λ) I (a ), (34) + λ) c 8 BISKA Biliim Technology

7 39 A. Aghili.: Non-homogeneou timefractional heat equation and g(r,t) = L [G(r,)]. (35) At thi point, in order to invert (33), we may ue the table of invere Laplace -tranform or part four of () to get v(r,t) = L [G(r, + ηe η 4t )] = t πt g(r,η)dη, in order to invert (34), we need to evaluate the reidue at all the ingulartie. The ingulartie of (34) are a follow. A imple pole at = λ.. If k,k,...k n,.. are the root of the function J (x), thu for J (k n ) =.and I (x) = J (ix) therefore, ik n are the root of I (x). Hence, the root of I (a + λ) are n = a λ+k n a. At thi point, we ue the Bromwich integral to invert (35) the reidue at imple pole = λ i (35) the reidue at imple pole = λ i (i) b = lim > λ ( + λ)( β + µ ( + λ). I (r + λ) I (a + λ) )et =, the reidue at pole = λa +kn a (ii) i, b = lim > λa +k n a the above limit can be written a follow ( + λa + k n a )( β + µ ( + λ). I (r + λ) I (a + λ) )et, b (β + µ)(i (r + λ) = lim λa +k e t > n ( + λ) a By uing the relation, I (x) = J (ix) and J = J, we obtain I (a +λ) I (ik n ) ( λa +k n a ) (I (a + λ)) = (J (ia + λ)) = ai + λ (J (ia + λ). At thi tage, let u take the limit a tend to n = a λ+k n, we arrive at a Finally, we may find the reidue b a below lim >n (I (a + λ)) = a k n J (k n ). b = e ( a λ+kn a )t βk (µ βλ n a )J ( rkn J (k n ), by following the ame procedure, we may find b 3 a below L ( I (r + λ) I (a + λ) ) = b3 = lim > λa +k n a a ) ( + λa + k n ). a )( I (r + λ) I (a + λ) )et, c 8 BISKA Biliim Technology

8 JACM 3, No., 33-4 (8) / 4 after implifying we get b 3 = a λ+kn e ( a )t βk (µ βλ n a ) J ( rkn a ) ( a k n )J (k n ). After ubtitution of the value for each term in relation (3), we get the formal olution a follow u(r,t) = βe λ t Er f c(λ t) + µ π t e λ η Er f c(λ η) dη+ t η n= n= e ωt e ( a λ+kn a )t J ( rk n a ) ( k a n )J (k n ) n=+ n= e ( a λ+kn a )t βk (µ βλ n a ) a )J ( rkn J (k n ) (H(t) t πt ξ e ξ 4t (k n+a λ)ξ a )dξ. In the above relation * i convolution for the Laplace tranform and H(.), tand for the Heaviide unit tep function. 3 Concluion The main purpoe of thi work i to develop a method for finding an exact analytic olution of the time fractional heat equation. In thi work, the author conidered the time fractional heat equation (Time fractional in the Caputo ene). Many linear boundary value and initial value problem in applied mathematic, mathematical phyic, and engineering cience can be effectively olved by the ue of the Fourier tranform, the Laplace tranform, the Fourier coine/ine tranform. The Fourier and Laplace type integral tranform are wonderful alternative method for olving different type of PDE of fractional order. There are a lot of application of PFDE in the field of Vico elaticity a well.the paper i devoted to tudy application of one dimenional Laplace tranform in detail. One dimenional Laplace tranform provide a powerful method for analyzing linear ytem. The tranform method introduce a ignificant improvement in thi field over exiting technique. We hope that it will alo benefit many reearcher in the dicipline of applied mathematic, mathematical phyic and engineering. 4 Acknowledgment The author would like to thank the referee/ and editor/ for careful and thoughtful reading of the manucript which helped to improve the preentation of the reult. Reference [] A. Aghili, B. Salkhordeh Moghaddam. Laplace tranform pair of n-dimenion and a wave equation. Intern. Math. Journal, Vol. 5, (4), no.4, [] A. Aghili, B. Salkhordeh Moghaddam. Multi-dimenional Laplace tranform and ytem of partial differential equation. Intern. Math.Journal, Vol., (6), no.6, -4 [3] A. Aghili, B. Salkhordeh Moghaddam. Laplace tranform pair of n-dimenion and econd order linear differential equation with contant coefficient. Annale Mathematicae et informaticae, 35 (8), 3-. c 8 BISKA Biliim Technology

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