BOUNDARY VALUE PROBLEM WITH INTEGRAL CONDITIONS FOR A LINEAR THIRD-ORDER EQUATION

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1 BOUNDARY VALUE PROBLEM WITH INTEGRAL CONDITIONS FOR A LINEAR THIRD-ORDER EQUATION M. DENCHE AND A. MEMOU Received 6 March 3 and in revised form 9 July 3 We prove the existence and uniqueness of a strong solution for a linear third-order equation with integral boundary conditions. The proof uses energy inequalities and the density of the range of the generated operator.. Introduction In the rectangle =[,] [,T], we consider the equation with the initial conditions u = 3 u + 3 x ax,t u x = fx,t,.a ux,=, u x,=, x,,.b the final condition u x,t=, x,,.c the Dirichlet condition u,t=, t,t,.d Copyright c 3 Journal of Applied Mathematics 3: Mathematics Subject Classification: 35B45, 35G URL:

2 554 Boundary value problem with integral conditions and the integral condition ux,tdx =, t,t..e In addition, we assume that the function ax,t is bounded with <a ax,t a,. and has bounded partial derivatives such that c k k a x,t c k, x,, t,t, k=,3, with c k, a x x,t b, for x,t..3 Various problems arising in heat conduction [4, 6, 4, 5], chemical engineering [9], underground water flow [3], thermoelasticity [], and plasmaphysics [9] can be reduced to the nonlocal problems with integral boundary conditions. This type of boundary value problems has been investigated in [,, 3, 5, 6, 7, 9, 4, 5, 6,, 3] for parabolic equations, in [8, ] for hyperbolic equations, and in [,, ] for mixed-type equations. The basic tool in [4,,,, 6, 3] is the energy inequality method which, of course, requires appropriate multipliers and functional spaces. In this paper, we extend this method to the study of a linear third-order partial differential equation. This type of problems is encountered in the study of thermal conductivity [7] and microscale heat transfer [8].. Preliminaries In this paper, we prove the existence and uniqueness of a strong solution of problem.. For this, we consider the solution of problem. as a solution of the operator equation Lu = F, where L is the operator with domain of definition DL consisting of functions u E such that x k+ u/ k xx,t L, k =,3 and u satisfies conditions.d and.e. The operator L is considered from E to F, wheree is the Banach space of the functions u, u L, with the finite norm { u E = x 3 u 3 + u } dxdt x x + u. x + u dxdt,

3 M. Denche and A. Memou 555 and F is the Hilbert space of the functions F =f,,,, f L, with the finite norm F F = x f dxdt.. Then we establish an energy inequality u E k Lu F, u DL,.3 and we show that the operator L has the closure L. Definition.. A solution of the operator equation Lu = F is called a strong solution of problem.. Inequality.3 can be extended to u DL, that is, u E k Lu F, u D L..4 From this inequality, we obtain the uniqueness of a strong solution, if it exists, and the equality of the sets RL and RL. Thus, to prove the existence of a strong solution of problem. for any F F, it remains to prove that the set RL is dense in F. 3. An energy inequality and its applications Theorem 3.. For any function u DL, there exists the a priori estimate u E k Lu F, 3. where k = 7expct[ / b c 3 3cc + 3c c c3 a b] + min,a,c 3 3cc + 3c c c3 a b, 3. with the constant c satisfying a sup c< x,t a inf x,t a a +, c 3 3cc + 3c c c3 a b >, 3.3 c cc + c a c + ca <.

4 556 Boundary value problem with integral conditions Proof. Let where Mu = x 3 u + xj 3 u 3 x, J x u = x uζ,tdζ. 3.5 We consider the quadratic form Φu,u=Re exp ct umudxdt, 3.6 with the constant c satisfying 3.3, obtained by multiplying.a by exp ctmu, integrating over, and taking the real part. Substituting the expression of Mu in 3.6,weobtain Re exp ct umudxdt = Re exp ct x 3 u 3 dxdt Re exp ct x 3 u J 3 u 3 x dxdt 3 + Re exp ct ax,t u Mudxdt. x x Integrating the last two terms on the right-hand side by parts with respect to x in 3.7 and using the Dirichlet condition.d, weobtain Re xexp ct 3 u J 3 u 3 x dx = 3 Re exp ct a u x x Mudxdt exp ct J x 3 u 3 = Re exp ct x a u 4 u x 3 x dxdt Re exp ct a x uj 3 u x dxdt 3 Re exp ctau 3 u 3 dxdt. dx,

5 M. Denche and A. Memou 557 Integrating each term by parts in 3.9 with respect to t and using the initial and final conditions.b and.c,weget Re exp ct x = Re a u Mudxdt x exp ct a x uj 3 u x dxdt 3 3 a 3c a a + 3 3c [ x u x + u ]dxdt + exp ct c3 a [ a 3 exp ct ca x u x + u ] dxdt [ x + exp cta u x + u ] dx T=t a a [ exp ct c + x c a u x + u ]dx t=t { } a + Re exp ct ca x u u x x + u u dx. T=t 3. Substituting 3.8 and 3. in 3.7 and using conditions.,.3, and 3.3,weobtain exp ct x 3 u 3 dxdt + exp ct { c 3 3cc + 3c c c3 a b } [ x u x + u ]dxdt Re exp ct um udxdt. 3. Again, substituting the expression of Mu in 3. and using elementary inequality, we get

6 558 Boundary value problem with integral conditions x exp ct 3 u 3 dxdt + exp ct { c 3 3cc + 3c c c3 a b } [ x u x + u ]dxdt 7 exp ct x f dxdt. By virtue of.a, we have u a x x dxdt x f dxdt + x 3 u 3 dxdt { x + 4 b u x + u }dxdt This last inequality combined with 3. yields x 3 u 3 dxdt + c 3 3cc + 3c c c3 a b { x u x + u }dxdt + a x u x dxdt { [ ] } 4b 7 expct 5 + c 3 3cc + 3c c + c3 a b x f dxdt. 3.4 Thus, this inequality implies { x 3 u 3 + u } dxdt + x k x f dxdt, x u x + u dxdt 3.5

7 M. Denche and A. Memou 559 where Then, k = 7expcT[ 5 + 4b / c 3 3cc + 3c c c3 a b ] + min,a,c 3 3cc + 3c c c3 a b. 3.6 u E k Lu F, u DL. 3.7 Thus, we obtain the desired inequality. Lemma 3.. The operator L from E to F admits a closure. Proof. Suppose that u n DL is a sequence such that u n ine, Lu n F in F. 3.8 We need to show that F =. We introduce the operator v = x 3 v + { ax,t [ x v ]}, x x with domain D consisting of functions v W,3 satisfying v t= =, v =, t= v =, v x= =, t= v x =. x= 3. We note that D is dense in the Hilbert space obtained by completing L with respect to the norm x v dxdt = v. 3. Since x fvdxdt= lim n + = lim n + x u n vdxdt u n vdxdt=, 3. for any function v D, it follows that f =.

8 56 Boundary value problem with integral conditions Theorem 3. is valid for a strong solution, then we have the inequality Hence we obtain the following corollary. u E k Lu F, u D L. 3.3 Corollary 3.3. A strong solution of problem. is unique if it exists, and depends continuously on F. Corollary 3.4. The range RL of the operator L is closed in F, andrl= RL. 4. Solvability of problem. To prove the solvability of problem.,it is sufficient to show that RL is dense in F. The proof is based on the following lemma. Lemma 4.. Suppose that ax,t and its derivatives 4 a/ 3 x and a/ x are bounded. Let D L ={u DL : ux, =, u/x, =, u/ x,t=}. If, for u D L and for some functions w L, x uwdxdt=, 4. then w =. Proof. Equality 4. can be written as follows: xw 3 u dxdt = a x u 3 x x { x w For a given wx,t, we introduce the function vx,t such that x vx,t=wx,t } w ζ dζ dxdt. 4. wζ,t dζ. 4.3 ζ From 4.3, we conclude that vx,tdx =, and thus, we have 3 u Nvdxdt = Atuvdxdt, where Atu = / xa x u/ x and Nv = xv + Jv.

9 M. Denche and A. Memou 56 Following [3], we introduce the smoothing operators J 3 = I ɛ, 3 J 3 = I + ɛ, with respect to t, which provide the solutions of the respective problems g ɛ ɛ 3 g 3 = g, g ɛ=, g ɛ + ɛ 3 g ɛ 3 = g, g ɛ=, g ɛ =, g ɛ T=, g ɛ T=, g ɛ T=. 4.6 We also have the following properties: for any g L,T, the functions Jɛ g, Jɛ g W 3,T.Ifg DL,thenJ ɛ g DL and we have lim Jɛ g g L = forɛ, [,T] lim Jɛ g g L [,T] = forɛ. 4.7 Substituting the function u in 4.4 by the smoothing function u and using the relation Atu = J Au ɛj β ɛ tu, 4.8 where we obtain β ɛ tu = 3 At u + 3 At u t + 3 At 3 u, 4.9 un 3 v 3 t dxdt = Atuvdxdt ɛ β ɛ tu vdxdt. 4. Passing to the limit, the equality in the relation 4. remains true for all functions u L such that x u/ x, / x x u/ x L, and satisfying condition.d.

10 56 Boundary value problem with integral conditions The operator At has a continuous inverse in L, defined by x A tg = + Ct ζ x aζ,t ζ gη,tdηdζ ζ aζ,t dζ, 4. where Ct= ζ dζ/aζ,t gη,tdη. 4. dζ/aζ,t Then, we have A tgdx=, hence the function u =J u can be represented in the form u = J A tatu. 4.3 Then B tg = [ 4 a 3 x J + 3 a 3 J + 3 a x + a J x ax,t [ g a a x x a x,t J g a ] gη,tdη Ct ] gη,tdη Ct x gη,tdη Ct ax,t x gη,tdη Ct. a x a x,t 4.4 The adjoint of B t has the form B t= a [ J 3 ] a h a + G h x x J /aη,t dη a h /ax,t dx G h, 4.5

11 where Gɛ h x= x + 3 a ζ aζ,t 3 aζ,t a ζ,t J ɛ J ɛ J 4 a 3 ζ h Consequently, equality 4. becomes M. Denche and A. Memou 563 h ζ a h + a ζ a ζ,t J 3 a h dζ un 3 v dxdt = Atuh 3 dxdt, 4.7 where h = v B v. The left-hand side of 4.7 is a continuous linear functional of u. Hence the function h has the derivatives x h / x, / x x h / x L and the following conditions are satisfied: h x= =, h x= =, and x h / x x= =. From the equality x h [ x = I J a 3 a 3 a J 3 ] x v x a x v x, 4.8 and since the operator J is bounded in L,forsufficiently small, we have /aj 3 a/ 3 <. Hence the operator I /aj 3 a/ 3 has a bounded inverse in L. We conclude that x v/ x L. Similarly, we conclude that / x x v/ x exists and belongs to L, and the following conditions are satisfied: v x= =, v x= =, x v x =. 4.9 x= Substituting u = t η T ζ expcτv τdτ dζdη in 4.4, where the constant c satisfies 3.3, weobtain expctvnvdxdt= Atuvdxdt. 4.

12 564 Boundary value problem with integral conditions Using the properties of smoothing operators, we have expctvnvdxdt= Atuvdxdt Atu 3 v dxdt, 3 4. and from Re Atu 3 v dxdt = xa u 3 v 3 x x dxdt 3 = Re x a u v x x dxdt + Re x a u v x x dxdt v + aexp ct x dxdt x + Re x a u v x x dxdt, we have Re Atu 3 v dxdt 3 v aexp+ct x dxdt x x a exp ct 4a 3 u x dxdt v aexp+ct x dxdt x x a exp ct u x dxdt exp+ct x 3 v dxdt x exp+ct v dxdt x x a exp ct u x dxdt

13 M. Denche and A. Memou 565 Integrating the first term on the right-hand side by parts in 4., we obtain Re Atuvdxdt 3 a xexp ct ca u x dxdt + xexp ct a a ca u x dx t=t { a a x exp ct c +c a+ a } ca u dx x + { 3 a xexp ct Combining 4.3 and 4.4,weget 3c a a + 3 3c c3 a Re expctvnvdxdt 3 xexp ct c ca u x dxdt xexp ct { a c ca } u x dx t=t t=t } u x dxdt xexp ct { c c c c a c + ca } u dx x t=t xexp ct { c 3 3c c + 3c c } c3 a u x dxdt + xexp ct c 4a 3 u x dxdt + xexp ct c u x dxdt x + expct 3 v x dxdt + xexp ct c u x dxdt x + expct v x dxdt. 4.5

14 566 Boundary value problem with integral conditions Using conditions 3.3 and inequalities 4.3 and 4.4, weobtain Re expctvnvdxdt, as. 4.6 Since Re expctvj xvdxdt=, then v = a.e. Finally, from the equality xv + J x v = xw, we conclude w =. Theorem 4.. The range RL of L coincides with F. Proof. Since F is Hilbert space, then RL=F if and only if the relation x ufdxdt=, 4.7 for arbitrary u D L and F F, implies that f =. Taking u D L in 4.7 and using Lemma 4., we obtain that w = xf =, then f =. References [] G. W. Batten Jr., Second-order correct boundary conditions for the numerical solution of the mixed boundary problem for parabolic equations, Math. Comp , [] S. A. Beilin, Existence of solutions for one-dimensional wave equations with nonlocal conditions, Electron. J. Differential Equations, no. 76, 8. [3] N. E. Benouar and N. I. Yurchuk, Mixed problem with an integral condition for parabolic equations with the Bessel operator, Differ. Equ. 7 99, no., [4] A. Bouziani and N. E. Benouar, Mixed problem with integral conditions for a third order parabolic equation, KobeJ.Math.5 998, no., [5] B.Cahlon,D.M.Kulkarni,andP.Shi,Stepwise stability for the heat equation with a nonlocal constraint,siamj.numer.anal , no., [6] J. R. Cannon, Thesolutionoftheheatequationsubjecttothespecificationofenergy, Quart. Appl. Math. 963, [7], The One-Dimensional Heat Equation, Encyclopedia of Mathematics and Its Applications, vol. 3, Addison-Wesley Publishing, Massachusetts, 984. [8] A. U. Çoskun, Investigation of microscale heat transfer effects in nanoscale electronic devices by spectral methods, Abstracts of the Intenational Conference on Mathematical Modeling and Scientific Computing Konya, Turkey,. [9] Y. S. Choi and K. Y. Chan, A parabolic equation with nonlocal boundary conditions arising from electrochemistry, Nonlinear Anal. 8 99, no. 4,

15 M. Denche and A. Memou 567 [] M. Denche and A. L. Marhoune, High-order mixed-type differential equations with weighted integral boundary conditions, Electron. J. Differential Equations, no. 6,. [], A three-point boundary value problem with an integral condition for parabolic equations with the Bessel operator, Appl. Math. Lett. 3, no.6, [], Mixed problem with nonlocal boundary conditions for a third-order partial differential equation of mixed type, Int. J. Math. Math. Sci. 6, no. 7, [3] R. E. Ewing and T. Lin, A class of parameter estimation techniques for fluid flow in porous media, Adv.inWaterRes.4 99, no., [4] N. I. Ionkin, The solution of a certain boundary value problem of the theory of heat conduction with a nonclassical boundary condition, Differ. Uravn , no., Russian. [5] N. I. Kamynin, A boundary value problem in the theory of heat conduction with non-classical boundary conditions, U.S.S.R. Comput. Math. and Math. Phys , [6] A. V. Kartynnik, A three-point mixed problem with an integral condition with respect to the space variable for second-order parabolic equations, Differ. Equ. 6 99, no. 9, [7] V. I. Korzyuk and V. B. Kiselev, A boundary value problem for a third-order linear nonclassical equation generated by the heat equation, Vestsī Nats. Akad. Navuk Belarusī Ser. Fīz.-Mat. Navuk, no. 3, 5 Russian. [8] L. S. Pulkina, A non-local problem with integral conditions for hyperbolic equations, Electron. J. Differential Equations , no. 45, 6. [9] A. A. Samarski, Some problems in the modern theory of differential equations,differ. Uravn. 6 98, Russian. [] P. Shi, Weak solution to an evolution problem with a nonlocal constraint, SIAMJ. Math. Anal , no., [] P. Shi and M. Shillor, On design of contact patterns in one-dimensional thermoelasticity, Theoretical Aspects of Industrial Design Wright-Patterson Air Force Base, Ohio, 99, SIAM, Pennsylvania, 99, pp [] V. F. Volkodavov and V. E. Zhukov, Two problems for the string vibration equation with integral conditions and special matching conditions on the characteristic,differ. Equ , no. 4, [3] N. I. Yurchuk, Mixed problem with an integral condition for certain parabolic equations,differ. Equ. 986, M. Denche: Laboratoire Equations Différentielles, Département de Mathématiques, Faculté des Sciences, Université Mentouri, 5 Constantine, Algeria address: denech@wissal.dz A. Memou: Laboratoire Equations Différentielles, Département de Mathématiques, Faculté des Sciences, Université Mentouri, 5 Constantine, Algeria

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MIXED PROBLEM WITH INTEGRAL BOUNDARY CONDITION FOR A HIGH ORDER MIXED TYPE PARTIAL DIFFERENTIAL EQUATION

Applied Mathematics and Stochastic Analysis, 16:1 (200), 6-7. Printed in the U.S.A. 200 by North Atlantic Science Publishing Company MIXED PROBLEM WITH INTEGRAL BOUNDARY CONDITION FOR A HIGH ORDER MIXED