ON TWO EXACT SOLUTIONS OF TIME FRACTIONAL HEAT EQUATIONS USING DIFFERENT TRANSFORMS

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1 Ibrahim, R. W., et al.: O Two Exact Solutios of Time Fractioal Heat Equatios THERMA SCIENCE, Year 05, Vol. 9, Suppl., pp. S43-S49 S43 ON TWO EXACT SOUTIONS OF TIME FRACTIONA HEAT EQUATIONS USING DIFFERENT TRANSFORMS by Rabha W. IBRAHIM a b ad Hamid A. JAAB a Istitute of Mathematical Scieces, Uiversity Malaya, Kuala umpur, Malaysia b Faculty of Computer Sciece ad Iformatio Techology, Uiversity Malaya, Kuala umpur, Malaysia Origial scietific paper DOI: 0.98/TSCI5SS43I Two solutios of time fractioal differetial equatios are illustrated. The first oe coverges to fuctioal space i term of Weyl trasform i (R, while the secod solutio approaches to the Fox fuctio with respect to time, by usig the Fourier ad aplace-melli trasforms. The fractioal calculus is take i the sese of the Riema-iouville fractioal differetial operator. Key words: fractioal calculus, fractioal diffusio equatio, fractioal heat equatio Itroductio I the last few decades, the class of fractioal differetial equatios has bee cotemplated to be successful patters of real life pheomeo. Oe of the essetial applicatios of the fractioal calculus is formed by the itermediate physical process. A very sigificat type is the fractioal diffusio ad wave equatios. It has bee foud that the diversity of the uiversal electromagetic, acoustic, mechaical, ad resposes ca be formulated precisely applyig fractioal diffusio-wave equatios [-6]. Various cosideratios of the fractioal diffusio equatios have bee imposed i differet fractioal operators such as the Riema-iouville, Caputo, ad Rize fractioal differetial operators [7, 8]. Furthermore, the authors studied a maximal solutio of the timespace fractioal heat equatio i a complex domai. The fractioal time is cosidered i the sese of the Riema-iouville operator, while the fractioal space is itroduced i the Srivastava-Owa operator for complex variables [9]. Trasform is a cosiderable mechaism to deal with mathematical problems by fidig exact ad approximate solutios. umerous beeficial trasforms for solvig ulike problems adapted i wide literatures, such as the aplace trasform, the Fourier trasform, wavelet trasformatio, the Bucklud trasformatio, the local fractioal itegral trasforms, the itegral trasform, the fractioal complex trasforms, the Hakel itegral trasforms, ad matrix trasform Miura type [0-5]. Recetly, the fractioal heat equatio is studied by may authors. Awar et al. [6] imposed the double aplace trasform of the partial fractioal itegrals ad derivatives to get a solutio of partial differetial equatios i the sese of the Caputo operator. Yag ad Baleau [7] processed the heat equatio of fractioal order by usig the variatios iterative Correspodig author; rabhaibrahim@yahoo.com

2 S44 Ibrahim, R. W., et al.: O Two Exact Solutios of Time Fractioal Heat Equatios THERMA SCIENCE, Year 05, Vol. 9, Suppl., pp. S43-S49 method. Che et al. [8] established the existece ad uiqueess of weak solutio for a class of fractioal heat equatio. I this study, the two solutios of time fractioal differetial equatios are illustrated. The first oe coverges to fuctioal space i term of Weyl trasform i (R while the secod solutio approaches to the Fox fuctio with respect to time, by usig the Fourier ad aplace-melli trasforms. The fractioal calculus based o the the Riema-iouville fractioal differetial operator. Mathematical settig et σ ( C. The Weyl trasform þ : ( R ( R is defied by [9]: ( þσφ, y = σ ( x, y µϕy (, ( x, ydxd y, ϕy, ( R π ( R where µ is the Wiger trasform of ϕ ad ψ. Cosider the set of all fuctios ϕ by FC ( such that: C σ ϕ( z ψ( zd z < where dz refers to the ebesgue measure o the complex pla C, ad: ad ψ ( z=π exp( z, z C et θ be a measurable fuctio o C. The Bargma trasform Bθ is defied by: /4 π z ( z ( B ( z = π ( exp χ χ + θ θ χ d χ <, z C emma. et ϕ F( C. The: ( ( BþσηB ϕ( z = g( ηϕη ( z, z C et be a o egative differetial fuctio o (0, satisfyig the coditio: 0 < ( t <, t (0, lim ( t= t 0 et f ( R. Our aim is to determie the solutio of the fractioal heat equatio of the form: T k ( χ, t= w ( χ T( χ, t ( = for each k N, χ R, t > 0, (0, such that: where T(., t f ( R, as t 0 ( deoted the Riema-iouville fractioal differetial operators []: a t d ( t t υ( t = ( d d t υt t Γ( a

3 Ibrahim, R. W., et al.: O Two Exact Solutios of Time Fractioal Heat Equatios THERMA SCIENCE, Year 05, Vol. 9, Suppl., pp. S43-S49 S45 ad for a =0 we cosider the form / t. We eed the followig results: emma. [0] et oe of the followig assumptios be achieved: γ β µ ([0, K] ad ([0, K], < β ([0, K] ad µ ([0, K], < β d µ ([0, K] ad ([0, K], < β β + d, β, d (0, where ([0, K] = { :[0, K] R/ ( s ( s j = O( j uiformly for 0 < τ j < s K}. The where poit-wise. Space fuctio solutio s s s ( b(= s ( s bs ( + bs ( ( s + R ( s s [( τ (][( s bτ bs (] (( sbs R ( s: = dτ + Γ( 0 ( s τ Γ( s This sectio deals with the existece of solutios of the iitial value problem (-(. Note that the uiqueess follow by the geeral theory for the heat equatios. Theorem. The solutio T( χ, t of the iitial value problem (-( is preseted by: T( χ, t = ( þ f( χ, χ R, t > 0 σ ( Proof. Defie a fuctio Τ o the set: S: = {( ζ, τ : ζ C, τ > 0} β β γ formulated by: T( z, t = ( BT ( z, t, ( z, t S ad let ϕ be the fuctio o C itroduced by: ϕζ ( = ( Bf ( ζ, ζ C By emma, we receive that for each ζ C ad τ > 0: σ ( Τ( ζτ, =( Bþ B ϕ( ζ= g[ ( τ] ϕ[ ( τζ ] Cosequetly, by emma, we obtai: Τ ϕ g R ( ζt, = g[ ( t] [ ( tζ ] + [ ( t] ϕ[ ( tζ ] + ( t By the defiitio of yields that R ( τ = 0. Thus, we obtai:

4 S46 Ibrahim, R. W., et al.: O Two Exact Solutios of Time Fractioal Heat Equatios THERMA SCIENCE, Year 05, Vol. 9, Suppl., pp. S43-S49 Τ ϕ g ( ζt, = g[ ( t] [ ( tζ ] + [ ( t] ϕ[ ( tζ ] = ϕ g [ ( tζ ] [ ( t] = g[ ( t] ϕ[ ( tζ ] + g[ ( t] ϕ[ ( tζ ] = ϕ[ ( tζ ] g[ ( t] := w ( ζ Τ ( ζ + w ( ζ, t Τ( ζ, t where lim ( τ. Thus, agai i view of emma, we have: ( BT ( ζ, τ = w ( ( (, ( ( (, ζ BT ζ τ + w ζ BT ζ τ τ By takig the iverse Bargma trasform ad applyig emma, we attai to: (3 (4 where ( T k ( χτ, = w ( χ T ( χτ,, k =3 (5 τ = ( BT T B B ( χt, = ( χt, BT ( χ, t R ad from the coditio limt =0 ( t =, we coclude that R ( χτ, = 0. Thus: We proceed to prove that: 3 ( : = B w χ B τ T(., τ f ( R, τ 0 I virtue of the ebesgue ' s domiated covergece theorem ad spectral theorem, we coclude that: This completes the proof. Theorem. For each t > 0: þ s ( R ( f f 0, as τ 0 T(., t C[ ( t] f where C[ ( t] is a positive fuctio o (0,. Proof. It is clear that: ( R ( R T(., t = þ ( f ( R σ ( R Therefore, by the Cauchy-Schwartz iequality yields: þ σ ( f σ ( f ( R Assume that C[ ( t]:= σ ( ( R ( R ( R cosequetly, we obtai the desired assertio.

5 Ibrahim, R. W., et al.: O Two Exact Solutios of Time Fractioal Heat Equatios THERMA SCIENCE, Year 05, Vol. 9, Suppl., pp. S43-S49 S47 Time solutio I this sectio, we process aother solutio for the iitial value problem (-( i term of time. Theorem 3. For each t > 0: Txt (, :, (0, t Proof. By employig the fractioal itegral operator: o eq. (, we have: ( t t It υ( t = υt ( dt Γ ( k T( χ, t= f( χ + w ( χ I T( χ, t (6 = The Fourier trasform of the fuctio T is computed by the itegral: t T ( x, t= T ( χ, texp( ixχdχ Thus eq. (6 i the Fourier domai yields: t Txt (, = + I W( xtxt (,, N (7 where W ( x is the Fourier trasform of the coefficiets. A aplace trasform of eq. (7 implies that: To reduce x i term of time, we utilize the mea square displacemet, which is give by: ˆ (, ˆ E T x E= E ˆ W ( x where c 0. This leads to: < c ( t> = T ( c, t c d c = ct ( x, t x=0 By applyig the Melli trasforms: ad utilizig the relatio: where Ł is the aplace trasform, we obtai: < χ ( t >: t t ς λς ( = M[ λt (, ς]= λt ( dt M[ λτ (, ς] = M{Ł[ λτ (, E ˆ ], ς} Γ( ς 0 (8

6 S48 Ibrahim, R. W., et al.: O Two Exact Solutios of Time Fractioal Heat Equatios THERMA SCIENCE, Year 05, Vol. 9, Suppl., pp. S43-S49 ˆ ς E T( x, ς = dˆ ˆ Γ E + E ( ς 0 W where the assertio is the iverse aplace trasform of eq. (8. Cosequetly, we receive that: / T( x, = W ς ς β[( ς/, ( ς/ ] Γ( ς where β deoted the beta fuctio. The iversio of eq. (9 ito the time domai implies: where H is the Fox-Wright fuctio: H Txt (, = W t H, / (0,/, (0,/, (0, ( b, b,...,( bq, bq η = H [ η;( b, b ; ( a, a ] ( a, a,...,( ap, ap Γ ( b + b... Γ ( b q + bq η := Γ ( a + a... Γ ( a + a! q p q p j j, q j j, p = =0 q p q j= p =0 j= Γ ( bj + b j η! Γ ( a + a with aj, bj R, β j >0 for all j =,..., q, α j >0 for all j =,..., p ad + j=α j q for η <. But H ca be expaded i power series: j=β j 0,, ( T( xt, = ( Wt Γ( =0 Hece, the solutio is mootoic decreasig fuctio with the asymptotic behavior at t : This completes the proof. T( xt, :, t>0 t Coclusios We coclude that the solutios of the fractioal differetial heat equatio, uder some special coditios, ca be coverted ito two formulas. The first oe depeds o the fuctio space, which is give by the iitial coditio. While, the secod oe traslated ito time domai by usig the cocept of the asymptotic behavior, whe t. We utilized differet trasoms i both cases. O the first case, we employed the Weyl trasform i term of the Wiger trasform ad the Bargma trasform (see Theorem 3. The secod case deled j j p (9

7 Ibrahim, R. W., et al.: O Two Exact Solutios of Time Fractioal Heat Equatios THERMA SCIENCE, Year 05, Vol. 9, Suppl., pp. S43-S49 S49 with the Fourier trasform together with the aplace trasform as well as the Melli trasform. This viewed that the theory of trasforms is very active to covert solutios. Ackowledgmets Malaya. Refereces This research is supported by Project No.: RG3-4AFR from the Uiversity of [] Podluby, I., Fractioal Differetial Equatios, Mathematics i Sciece ad Egieerig, Academic Press, Sa Diego, Cal., USA, 999 [] Hilfer, R., Applicatios of Fractioal Calculus i Physics, World Scietific, Sigapore, 000 [3] Kilbas, A. A., et al., Theory ad Applicatios of Fractioal Differetial Equatios, Elsevier, Amsterdam, The Netherlads, 006 [4] Sabatier, J., et al., Advaces i Fractioal Calculus: Theoretical Developmets ad Applicatios i Physics ad Egieerig, Spriger, The Netherlads, 007 [5] akshmikatham, V. et al., Theory of Fractioal Dyamic Systems, Cambridge Scietific Pub., Cambridge, UK, 009 [6] Jumarie, G., Fractioal Differetial Calculus for No-Differetiable Fuctios. Mechaics, Geometry, 00 Stochastics, Iformatio Theory, ambert Academic Publishig, Saarbrucke, Germay, 03 [7] He, J.-H., et al., Covertig Fractioal Differetial Equatios ito Partial Differetial Equatio, Themal Sciece, 6 (0,, pp [8] Guo, P., et al., Numerical Simulatio of the Fractioal agevi Equatio, Thermal Sciece, 6 (0,, pp [9] Ibrahim, R. W., Jalab, H. A., Time-Space Fractioal Heat Equatio i the Uit Disk, Abstract ad Applied Aalysis, 03 (03, ID [0] Gordoa, P. R. et al., Bucklud Trasformatios for a Matrix Secod Pailev Equatio, Physics etters A, 374 (00, 34, pp [] Molliq, R., Batiha, B., Approximate Aalytic Solutios of Fractioal Zakharov-Kuzetsov Equatios by Fractioal Complex Trasform, Iteratioal Joural of Egieerig ad Techology, (0,, pp. -3 [] Ibrahim, R. W., Fractioal Complex Trasforms for Fractioal Differetial Equatios, Advaces i Differece Equatios 0 (0, 9, pp. - [3] Ibrahim, R. W., Jahagiri, J. M., Existece of Fractioal Differetial Chais ad Factorizatios Based o Trasformatios, Mathematical Methods i the Applied Scieces, (04 DOI: 0.00/mma.35 [4] Ibrahim, R. W., Sokol, J., O a Class of Aalytic Fuctio Derived by Fractioal Differetial Operator, Acta Mathematica Scietia, 34B (04, 4, pp. -0 [5] Ibrahim, R. W., Jahagiri, J. M., Boudary Fractioal Differetial Equatio i a Complex Domai, Boudary Value Problems, 04 (04, 66, pp. - [6] Awar, A. M. O., et al., Fractioal Caputo Heat Equatio with the Double aplace Trasform, Rom. Jour. Phys., 58 (03, -, pp. 5- [7] Yag, X.-J., Baleau, D., Fractal Heat Coductio Problem Solved by ocal Fractioal Variatioal Iteratio Method, Thermal Sciece, 7 (03,, pp [8] Che, H., Wag, Y., Fractioal Heat Equatios Ivolvig Iitial Measure Data ad Subcritical Absorptio, Aalysis of PDE, (04,, pp. -38 [9] Wog, M. W., Weyl Trasforms, Spriger Verlag, New York, USA, 998 [0] Alsaedi, A. et al., Maximum Priciple for Certai Geeralized Time ad Space-Fractioal Diffusio Equatios, J. Quarterly Appl. Math., 73 (05,, pp Paper submitted: October 0, 04 Paper revised: Jauary, 05 Paper accepted: February, 05