On The Cauchy Problem For Some Parabolic Fractional Partial Differential Equations With Time Delays

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1 Journal of Mathematics and System Science 6 (216) doi: / / D DAVID PUBLISHING On The Cauchy Problem For Some Parabolic Fractional Partial Differential Equations With Time Delays Mahmoud M.El-Borai, Wagdy G.El-Sayed, Faez N. Ghaffoori Department of Mathematics, Facultyof Science, Alexandria University, Alexandria Egypt Received: February 17, 216 / Accepted: March 16, 216 / Published: May 25, 216. Abstract: The Cauchy problem for some parabolic fractional partial differential equation of higher orders and with time delays is considered. The existence and unique solution of this problem is studied. Some smoothness properties with respect to the parameters of these delay fractional differential equations are considered. Keywords: Cauchy problem- fractional partial differential equations with time delays- successive approximations. 1. Introduction We will adhere the following notations, RR nn is n-dimensional Euclidean space, xx, yy are elements of this space, qq = (qq 1, qq 2,, qq nn ) is a multi-index, DD xx =,,, xx 1 xx 2 xx nn qq DD qq xx = xx qq 1 1 xx2 qq 2 xxnn qq nn qq = qq 1 + qq qq nn DD =, xx 2 = xx xx xx nn Consider the parabolic fractional partial differential equations with time delay of the form DD αα uu(xx,, pp) + aa qq (xx, )DD qq xx uu(xx,, pp) = bb qqqq (xx, )DD xx qq xx, pp jj, pp + h(xx, ) jj =1 qq <2mm (1.1) < αα 1, < TT, pp = (pp 1,, pp ), < pp jj < νν < TT, jj = 1,2,,, where νν is a given positive number Corresponding author: Mahmoud M.El-Borai, Department of Mathematics, Facultyof Science, Alexandria University, Alexandria, Egypt. m_m_elborai@yahoo.com. Let us suppose that uu satisfies the Cauchy condition, uu(xx,, pp) = ff(xx) (1.2) Where ff is a given function Let CC rr (QQ) denote the set of all functions defined in a neighborhood of each point of a domain QQ RR ll, and having bounded continuous partial derivatives of all orders le-ss than or equal r in QQ (rr denotes a nonnegative integer and ll a positive integer). We assume that ff CC 2mm 1 (RR nn ). Let uu satisfy uu(xx,, pp) = FF(xx, ), (xx, ) QQ 1 (1.3) QQ 1 = {(xx, ): xx RR nn, νν < <, < pp jj < νν } Where F is a given bounded continuous function on QQ 1. Assume that F CC 2mm 1 (RR nn ) for every ( νν, ), and FF(xx, ) = ff(xx), xx εε RR nn (1.4) The linear partial differential operator qq qq =2mm aa qq (xx, )DD xx is supposed to be uniform elliptic on RR nn, i.e. there exists a positive number λλ such that for every (xx, ) εε QQ 2 = {(xx, ): xx RR nn, TT }, and for every yyyyrr nn, ( 1) mm aa qq (xx, )yy qq λλ yy 2mm qq =2mm (1.5)

2 On The Cauchy Problem For Some Parabolic Fractional Partial Differential Equations With Time Delays 195 Where the number λλ is independent of xx, and yy. It is suitable to rewrite the problem (1.1), (1.2) in the form 1 ( ηη)αα 1 ΓΓ(αα) 1 ( ηη)αα 1 ΓΓ(αα) uu(xx,, pp) = ff(xx) + aa qq (xx, )DD qq xx uu(xx, ηη, pp) dddd = bb qqqq (xx, ηη)dd xx qq uu(xx, ηη jj =1 qq <2mm pp jj, pp) dddd Where Γ is Gamma function (1.6) Following Friedman [11], we assume the following conditions. (CC 1 ) All the coefficients aa qq, qq 2mm, are continuous and bounded functions on QQ 2. (CC 2 ) All the coefficients aa qq, qq 2mm, satisfy a Holder condition with respect to uniformly in QQ 2, namely there exists a positive constant K and a constant γγγγ(,1) such that aa qq (xx, ) aa qq (yy, ) KK xx yy γγ, for all xx, yy εε RR nn, εε [, TT] (CC 3 ) The coefficients aa qq, qq = 2mm, of the principle term are uniformly continuous as functions of relative to (xx, ) in QQ 2. (CC 4 ) DD xx ββ aa qq εε CC(QQ 2 ), ββ = (ββ 1,, ββ nn ) is a multi-index ββ 2mm, qq 2mm, and CC(QQ 2 ) is the set of all continuous bounded functions on QQ 2. We assume also that coefficients bb qqqq, qq 2mm, lative to iple term are uniformly continuous as functions of jj = 1,2,,, satisfy a Holder condition with respect to xx, (similar to cond. (CC 2 ) ) and DD xx ββ bb qqqq εε CC(QQ 2 ), ββ < 2mm. In section 2, we study the existence and uniqueness of the solution of the Cauchy problem (1.1), (1.2). In section 3, we consider the dependence of the parameters pp 1,, pp of the solution of the considerd Cauchy problem, compare [1], [1], [13], and [14]. 2. Uniqueness and Existence Theorems Consider the Cauchy problem DD αα ww(xx, ) + aa qq (xx, )DD qq xx ww(xx, ) = (2.1) ww(xx, ) = ff(xx) (2.2) It is well known under the conditions (CC 1 ) (CC 4 ) that the solution of the Cauchy problem (2.1), (2.2) represented in the form [7] ww(xx, ) = GG(xx yy, αα θθ) ff(yy)ζζ αα (θθ)dddddddd (2.3) ζζ αα (θθ) is a probability density function define on (,) G is the fundamental solution of equation (2.1), when αα = 1, which has the following conditions [11]. (CC 5 )DD qq xx GG, DD qq yy GG, DD GG εε (QQ 2 QQ 2 ), qq 2mm (CC 6 ) DD qq xx DD BB KKKK(xx yy, θθ) yy GG(xx,, yy, θθ) ( θθ) νν 1 Where zz(xx yy, θθ) = exp[ νν 2 ( xx yy 2mm / θθ) 1 2mm 1], (< θθ < ) νν 1 = 1/2mm[ qq + ββ + nn], KK, νν 2 are positive constants and qq < 2mm, ββ 2mm. (CC 7 ) DD qq xx [GG(xx,, yy, θθ) GG(zz,, yy, θθ)] KK xx zz νν 3 ( θθ) νν 4 Λ, Where Λ = exp[ νν 5 ( xx yy 2mm / θθ) 1 2mm] + exp[ νν 5 ( zz yy 2mm / θθ) 1 2mm ], νν 4 = 1/2mm( qq + νν 3 + nn), νν 3 is an arbitrary positive number 1, KK and νν 5 are positive constants and qq < 2mm, < θθ <. Let P = { pp εε RR : < pp ii < νν, ii = 1,, } QQ 3 = { (xx, ) xx εε RR nn, νν < < TT } We prove now the following uniqueness theorem. Theorem 2.1: Let uu be a function defined on QQ 3 P. Let DD uu, DD qq xx GGεε CC(QQ 3 ), qq 2mm, for every

3 196 On The Cauchy Problem For Some Parabolic Fractional Partial Differential Equations With Time Delays fixed pp εε PP. Let FF = on QQ 1. If uu is a solution of the Cauchy problem (1.1), (1.2) on QQ 2, which satisfies the condition (1.3) on QQ 1. Then the solution is unique Proof: We set (xx,, pp) =, h(xx, ) =, xx RR nn, pp εε PP, >. In this case we must prove uu(xx,, pp) =, for sufficiently small >, xx RR nn, pp εε PP Using (CC 5 ), (CC 6 ) of the fundamental solution G, we get GG(xx,, yy, θθ)dd yy qq uu(yy, θθ, pp)dddd ( 1) qq uu(yy, θθ, pp)dd yy qq GG(xx,, yy, θθ)dddd And so there exists a number cc εε (,1) such that supp xx GG(xx,, yy, θθ)dd qq yy uu(yy, θθ, pp)dddd = KK ( θθ) cc uu, > θθ (2.4) Where uu = ssssss xx uu(xx,, pp), qq < 2mm, KK is a positive constant. According to the conditions imposed on the considered problem, we can write uu(xx,, pp) = αα θθζζ αα (θθ)( ηη) αα 1 GG(xx jj=1 qq <2mm ηη jj () yy, ( ηη) αα θθ) bb qqqq (yy, ηη)dd qq yy uu(yy, ηη pp jj, pp)dddddddddddd Where pp jj ηη jj () = pp jj > ppjj Thus uu(xx,, pp) = αα ( 1) qq uu(yy, ηη jj=1 qq <2mm ηη jj () pp jj, pp)θθθθ(θθ)( ηη) αα 1 DD qq yy [bb qqqq (yy, ηη)gg(xx yy, ( ηη) αα θθ)] dddddddddddd Since DD qq xx bb qqqq εε CC(QQ 2 ), qq < 2mm, jj = 1,,. It follows by using (2.4) that gg(, pp) αααα αααα where for Thus jj =1 jj =1 ηη jj () ηη jj () θθ( ηη) cccc θθ cc ( ηη) αα 1 ζζ αα (θθ)gg ηη pp jj, pp dddddddd [ θθ 1 cc ζζ αα (θθ)dddd]( ηη) (1 cc)αα 1 gg ηη pp jj, pp)dddd gg(, pp) = uu = ssssss xx uu(xx,, pp) αααα pp jj ξξ() ηη Γ((1 cc)αα + 1) jj=1 Where ξξ() = ssssss pp gg(, pp) pp jj (1 cc)αα 1 ξξ(ηη)dddd > pp jj (2.5) If pp jj, jj = 1,2,, there is nothing to prove. Suppose > pp jj, then from equation (2.5) ξξ() αααα Γ (1 cc)αα + 1 ( ηη)(1 cc)αα 1 ξξ(ηη)dddd It is easy to get: for all ξξ() KK MM nn nnnn [Γ(δδ)] nn 1 Γ(nnnn + 1) nn = 1,2, Where MM = ssssss ξξ(), δδ = (1 cc)αα, KK is a positive integer, leingnn, we get ξξ(), for all t. We prove now the following existence theorem. Theorem 2.2: There exists a unique function uu, uu, DD uu, DD qq uuεε CC(QQ 3 ), qq 2mm Such thatuu represents the unique solution of the Cauchy problem (1.1), (1.2) on QQ 2, which satisfies the condition (1.3) on QQ 1. Proof: Let us try to find the solution of the integro-partial differential equation. uu = HH + EEEE (2.6) Where H is a given function defined on QQ 2 by,

4 On The Cauchy Problem For Some Parabolic Fractional Partial Differential Equations With Time Delays 197 HH(xx,, pp) = GG(xx, yy, αα θθ) ff(yy)ζζ αα (θθ)dddddddd + αα θθζζ αα (θθ)( ηη) αα 1 GG(xx yy, ( ηη) αα θθ)h(yy, ηη)dddddddddddd + ηη jj () αα ( 1) qq FF(yy, ηη jj =1 qq <2mm pp jj, pp)θθζζ αα (θθ)( ηη) αα 1 DD yy qq [bb qqqq (yy, ηη)gg(xx yy, ( ηη) αα θθ)] dddddddddddd And E is an integro-partial differential difference operator defined on CC(QQ 2 ) by EEEE = uu, uu (xx,, pp) = αα jj =1 qq <2mm ηη jj () ( 1) qq uu(yy, ηη pp jj, pp)θθθθ(θθ)( ηη) αα 1 DD yy qq [bb qqqq (yy, ηη)gg(xx yy, ( ηη) αα θθ)] dddddddddddd We apply the method of successive approximation to solve (2.7), to do this, set uu rr+1 = HH + EEuu rr, rr = 1,2, Where the zero approximation uu is taken to be zero. The linearity of the operator E leads to the following equation uu rr+1 uu rr = EE(uu rr uu rr 1 ), using the conditions imposed on the coefficients and the properties of the fundamental solution G of equation (2.2), we get by a similar method of the proof of theorem (2.1) u r+1 u r KKδδδδ [Γ(δδ)] rr 1 Γ(δδδδ + 1) where KK is a positive constant, δδ = (1 cc)αα. Thus the required solution of equation (2.6) is given by uu = (uu rr uu rr 1 ) rr=1 This series is absolutely and uniformly convergent on QQ 2 to the function uu εε CC(QQ 2 ). To prove the smoothness of the function u, we consider the following equation VV ii = DD xxii HH + EEuu ii + PP ii VV ii (2.7) Where EEuu ii = uu ii, PP ii VV ii = VV ii, uu ii (xx,, pp) = αα jj =1 ηη jj () θθζζ αα (θθ)( ηη) αα 1 bb,,,jj (yy, ηη) uu(yy, ηη pp jj, pp) DD xxii [GG(xx yy, ( ηη) αα θθ)] dddddddddddd VV ii (xx,, pp) = αα ( 1) qq 1 VV ii (yy, ηη jj =1 1 qq 2m ηη jj () pp jj, pp)θθθθ(θθ) ( ηη) αα 1 qq ee DD xxii DD ii yy [bb qqqq (yy,ηη)gg(xx DD xxii = yy, ( ηη) αα θθ)] dddddddddddd xx ii, ee 1 = (1,,,), ee 2 = (,1,,), ee nn are the unit vectors = (,,,1) If the solution VV ii of equation(2.7) exists in the space CC(QQ 2 ), then the partial derivative DD xxii uu exists for every ii = 1,, nn and VV ii = DD xxii uuuu CC(QQ 2 ) Using a similar technique to that used to prove the existence of solution of equation (2.6) in the space CC(QQ 2 ) and noting that u is a known function in the space CC(QQ 2 ) and using the conditions (CC 5 ) and (CC 6 ) of the function G, we deduce that equation (2.7) has a unique solution VV ii in the space CC(QQ 2 ), for every ii = 1,, nn. And so VV ii = DD xxii uu εε CC(QQ 2 ), compare [2] By induction we can prove that DD qq xxii uu εε CC(QQ 2 ) for qq < 2mm. Now to prove that DD qq xx εε CC(QQ 2 ) for qq = 2mm, we need to prove first all the partial derivatives DD qq xx uu satisfy for qq < 2mm, a Holder continuity condition with respect to x. To do this, we notice that DD ββ xx uu(xx,, pp) DD ββ zz uu(zz,, pp) = DD ββ xx HH(xx,, pp) DD ββ zz HH(zz,, pp) +

5 198 On The Cauchy Problem For Some Parabolic Fractional Partial Differential Equations With Time Delays jj =1 qq <2mm ηη jj () θθθθ(θθ)( ηη) αα 1 bb qqqq (yy, ηη) DD yy qq uu(yy, ηη, pp)[dd xx ββ GG(xx yy, ( ηη) αα θθ) DD zz ββ GG(zz Where ββ < 2mm yy, ( ηη) αα θθ)] dddddddddddd Using condition (CC 7 ) of the function G and remembering the conditions imposed on the coefficients of equation (1.1) we can prove that there is a constant γγγγ (,1) such that DD xx ββ uu(xx,, pp) DD zz ββ uu(zz,, pp) K xx zz γγ for all xx, zz εε RR nn Where the positive constant K, is independent of xx, zz and. This completes the proof of the theorem. 3. Properties of Smoothness In this section we consider the dependence on the parameters pp 1,, pp. We shall deduce some smoothness properties of the solution uu (xx,, pp) with respect to p, compare [1]. Theorem 3.1: If FF has continuous bounded derivatives DD rr FF oooo QQ 11, then u has continuous bounded derivatives DD rr ppii uu oooo QQ 2 PP, DD ppii = pp ii, rr = 1,2,, ii = 1,, Proof: For a fixed εε (, TT] and a fixed xx εε RR nn, we consider the following equation DD αα VV rrii (xx,, pp) + aa qq (xx, )DD xx qq VV rrii (xx,, pp) = bb qqqq (xx, )DD qq jj =1 qq <2mm xx VV rrii xx, pp jj, pp (3.1) Where pp εε PP, VV rrii xx, pp jj, pp = DD pp rr ii FF(xx, pp ii ), pp ii, ii = jj, pp ii, ii jj ii = 1,, Assume that for some fixed VV rrii (xx,, pp) = (3.2) Equation (3.1) can be obtained from formally from equation (1.1) by applying formally the partial rr differential operator DD ppii. If the Cauchy problem (3.1), (3,2) has a solution, then we deduce immediately that DD rr ppii uu = VV rrii εε CC(QQ 2 PP). The Cauchy problem (3.1), (3.2) can be transformed to ηη jj () VV rrii (xx,, pp) = α ( 1) qq DD rr ppii FF(yy, ηη qq <2mm pp ii )θθζζ αα (θθ)( ηη) αα 1 DD yy qq [bb qqii (yy, ηη)gg(xx yy, ( ηη) αα θθ)] dddddddddddd jj =1 ηη jj () + αα ( 1) qq VV rrii (yy, ηη qq 2m pp jj, pp)θθζζ αα (θθ)( ηη) αα 1 DD yy qq [bb qqii (yy, ηη)gg(xx yy, ( ηη) αα θθ)] dddddddddddd (3.3) Similar to theorem (2.2) we see that equation (3.3) satisfies the required. It can be also proved that (xx,, pp) =, for ii jj pp ii pp jj References [1] H.T. Bank and P. I. Daniel, Estimation of delays and other parameter in nonlinear functional equations. Siam J. control. Optima. 21, (1993). [2] M. M. El-Borai, on the initial value problem for a partial differential equation with operator coefficients, int. J. math. And math. Sci.Vol. 3 No. 1, (198). [3] M. M. El-Borai, some probability densities and fundamental solutions of fractional evolution equation 14, (22). [4] M. M. El-Borai, the fundamental solutions for fractional evolution equation of pa- rabolic type, J. of Appl. Math. Stochastic Analysis (JAMSA) 24, [5] M. M. El-Borai, K. El-Said El-Nadi and I. G, El-Akabawi. On some integro-diffe- rential equations of fractional orders, the International J. of contemporary Mathem atics, Vol. 1, 26, No. 15, [6] M. M. El-Borai, Khairia El-Said El-Nadi, Iman G. El-Akabawy. Approximate sol- utions for nonlinear fractional heat equations. Int. j.contemp. Math.Sciences 27, 2 (27): [7] M. M. El-Borai, Khairia El-Said El-Nadi, Hoda A. Foad. On some fractional stoc- hastic delay differential equations. Computers and Mathematics with Applications 21, 59:

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