ON SOLVABILITY OF THE INTEGRODIFFERENTIAL HYPERBOLIC EQUATION WITH PURELY NONLOCAL CONDITIONS

doi:10.1016/S0252-9602(15)30006-0

数学物理学报(英文版) ›› 2015, Vol. 35 ›› Issue (3): 601-609.doi: 10.1016/S0252-9602(15)30006-0

ON SOLVABILITY OF THE INTEGRODIFFERENTIAL HYPERBOLIC EQUATION WITH PURELY NONLOCAL CONDITIONS

Ahcene MERAD¹, Abdelfatah BOUZIANI², Cenap OZEL³, Adem KILIÇMAN⁴

1. Department of Mathematics, Larbi Ben M'hidi University, Oum El Bouaghi, Algeria;
2. Department of Mathematics, Larbi Ben M'hidi University, Oum El Bouaghi, Algeria;
3. Department of Mathematics, Faculty of Science, Dokuz Eylul University 35160 Buca/I.zmir, Turkey;
4. Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, UPM, Serdang, Selangor, 43400, Malaysia

收稿日期:2013-12-30 出版日期:2015-05-01 发布日期:2015-05-01
基金资助:
The fourth author gratefully acknowledges the partially support by the University Putra Malaysia under the Grant Scheme having project number GP-IBT/2013/9420100.

ON SOLVABILITY OF THE INTEGRODIFFERENTIAL HYPERBOLIC EQUATION WITH PURELY NONLOCAL CONDITIONS

Ahcene MERAD¹, Abdelfatah BOUZIANI², Cenap OZEL³, Adem KILIÇMAN⁴

1. Department of Mathematics, Larbi Ben M'hidi University, Oum El Bouaghi, Algeria;
2. Department of Mathematics, Larbi Ben M'hidi University, Oum El Bouaghi, Algeria;
3. Department of Mathematics, Faculty of Science, Dokuz Eylul University 35160 Buca/I.zmir, Turkey;
4. Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, UPM, Serdang, Selangor, 43400, Malaysia

Received:2013-12-30 Online:2015-05-01 Published:2015-05-01
Supported by:
The fourth author gratefully acknowledges the partially support by the University Putra Malaysia under the Grant Scheme having project number GP-IBT/2013/9420100.

摘要/Abstract

摘要：

In this study, we prove the existence, uniqueness, and continuous dependence upon the data of solution to integro-differential hyperbolic equation with purely nonlocal (integral) conditions. The proofs are based on a priori estimates and Laplace transform method. Finally, we obtain the solution using a numerical technique (Stehfest algorithm) by inverting the Laplace transform.

关键词: Integro-differential hyperbolic equation, approximate solution, nonlocal purely conditions

Abstract:

Key words: Integro-differential hyperbolic equation, approximate solution, nonlocal purely conditions

中图分类号:

44A10

Ahcene MERAD, Abdelfatah BOUZIANI, Cenap OZEL, Adem KILIÇMAN. ON SOLVABILITY OF THE INTEGRODIFFERENTIAL HYPERBOLIC EQUATION WITH PURELY NONLOCAL CONDITIONS[J]. 数学物理学报(英文版), 2015, 35(3): 601-609.

Ahcene MERAD, Abdelfatah BOUZIANI, Cenap OZEL, Adem KILIÇMAN. ON SOLVABILITY OF THE INTEGRODIFFERENTIAL HYPERBOLIC EQUATION WITH PURELY NONLOCAL CONDITIONS[J]. Acta mathematica scientia,Series B, 2015, 35(3): 601-609.

参考文献

[1] Abramowitz M, Stegun I A. Hand Book of Mathematical Functions. New York: Dover, 1972
[2] Ang W T. A Method of Solution for the One-Dimentional Heat Equation Subject to Nonlocal Conditions. Southeast Asian Bulletin of Mathematics, 2002, 26: 185-191
[3] Be¨?lin S A. Existence of solutions for one-dimentional wave nonlocal conditions. Electron J Differential Equations, 2001, 200176): 1-8
[4] Bouziani A. Mixed problem with boundary integral conditions for a certain parabolic equation. J Appl Math Stochastic Anal, 1996, 9(3): 323-330
[5] Bouziani A. Solution forte d'un problème mixte avec une condition non locale pour une classe d'équations hyperboliques [Strong solution of a mixed problem with a nonlocal condition for a class of hyperbolic equations]. Acad Roy Belg Bull Cl Sci, 1997, 8: 53-70
[6] Sur un problème mixte avec uniquement des conditions aux limites intégrales pour une classe d'équations paraboliques. Maghreb Mathematical Review, 2000, 9(1/2): 55-70
[7] Bouziani A, Mechri R. The Rothe Method to a Parabolic Integro-differential Equation with a Nonclassical Boundary Conditions. Int Jour of Stochastic Analysis, Article ID 519684/16 page, doi: 10.1155/519684/(2010)
[8] Byszewski L. Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J Math Anal Appl, 1991, 162: 494-505
[9] Byszewski L. Theorems about the existence and uniqueness of continuous solutions of nonlocal problem for nonlinear hyperbolic equation. Appl Anal, 1991, 40: 173-180
[10] Byszewski L, Acka H. Existence of solutions of semilinear functional differential evolution nonlocal problems. Nonlinear Analysis, 1998, 34: 65-72
[11] Balachandran K, Park J Y. Existence of mild solution of a functional integrodifferential equation with nonlocal condition. Bulletin of the Korean Mathematical Society, 2001, 38: 175-182
[12] Balachandran K, Uchiyama K. Existence of solutions of quasilinear integrodifferential equations with nonlocal condition. Tokyo Journal of Mathematics, 2000, 23: 203-210
[13] Balachandran K, Park D G. Existence of solutions of quasilinear integrodifferential evolution equations in Banach spaces. Bulletin of the Korean Mathematical Society, 2009, 46: 691-700
[14] Balachandran K, Samuel F P. Existence of solutions for quasilinear delay integrodifferential equations with nonlocal condition. Electronic Journal of Differential Equations, 2009, 2009(6): 1-7
[15] N'Guérékata G M. A Cauchy problem for some fractional abstract differential equation with non local conditions. Nonlinear Analysis: Theory Methods and Applications, 2009, 70: 1873-6
[16] Balachandran K, Park J Y. Nonlocal Cauchy problem for abstract fractional semilinear evolution equations. Nonlinear Analysis: Theory Methods and Applications, 2009, 71: 4471-5
[17] Ahmad B. Some existence results for boundary value problems of fractional semilinear evolution equations. Electronic Journal of Qualitative Theory of Differential Equations, 2009, (28): 1-7
[18] Balachandran K, Trujillo J J. The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces. Nonlinear Analysis: Theory Methods and Applications, 2010, 72: 4587-93
[19] Elbeleze A A, K?l?çman A, Taib B M. Application of homotopy perturbation and variational iteration method for Fredholm integro-differential equation of fractional order. Abstract & Applied Analysis, 2012, 2012: 14 pages
[20] Elbeleze A A, K?l?çman A, Taib B M. Homotopy Perturbation Method for Fractional Black-Scholes European Option Pricing Equations Using Sumudu Transform. Mathematical Problems in Engineering, 2013, 2013: 7 pages
[21] Gordeziani D G, Avalishvili G A. Solution of nonlocal problems for one-dimensional oscillations of a medium. Mat Model, 2000, 12(1): 94-103
[22] Graver D P. Observing stochastic processes and aproximate transform inversion. Oper Res, 1966, 14: 4440-459
[23] Hassanzadeh H, Pooladi-Darvish M. Comparision of different numerical Laplace inversion methods for engineering applications. Appl Math Comp, 2007, 189: 1966-1981
[24] Kythe P K, Puri P. Computational methods for linear integral equation. New Orlans: Universty of New Orlans, 2002
[25] Kac?r J. Method of Rothe in Evolution Equations//Teubner-Texte zur Mathematik, Vol 80. Leipzig: BSB B. G. Teubner Verlagsgesellschaft, 1985
[26] Merad A. Adomian Decomposition Method for Solution of Parabolic Equation to Nonlocal Conditions. Int Journal of Contemp Math Sciences, 2011, 6(29/32): 1491-1496
[27] Merad A, Marhoune A L. Strong Solution for a High Order Boundary Value Problem with Integral Condition. Turk Jour Math, 2013, 37(3): 1-9
[28] Merazga N, Bouziani A. Rothe time-discretization method for a nonlocal problem arising in thermoelasticity. Journal of Applied Mathematics and Stochastic Analysis, 2005, 2005(1): 13-28
[29] Mesloub S, Bouziani A. Mixed problem with integral conditions for a certain class of hyperbolic equations. Journal of Applied Mathematics, 2001, 1(3): 107-116
[30] Pul'kina L S. A non-local problem with integral conditions for hyperbolic equations. Electron J Differential Equations, 1999, 1999(45): 1-6
[31] Pul'kina L S. On the solvability in L2 of a nonlocal problem with integral conditions for a hyperbolic equation. Differ Equ, 2000, 36(2): 316-318
[32] Pul'kina L S. A mixed problem with integral condition for the hyperbolic equation. Matematicheskie Zametki, 2003, 74(3): 435-445
[33] Stehfest H. Numerical Inversion of the Laplace Transform. Comm ACM, 1970, 13: 47-49
[34] Shruti A D. Numerical Solution for Nonlocal Sobolev-type Differential Equations. Electronic Journal of Differential Equations, 2010, 19: 75-83
[35] Wazwaz AM. A comparison study between the modified decomposition method and the traditional methods for solving nonlinear integral equation. Appl Math and Comp, 2006, 181(2): 1703-1712

ON SOLVABILITY OF THE INTEGRODIFFERENTIAL HYPERBOLIC EQUATION WITH PURELY NONLOCAL CONDITIONS

ON SOLVABILITY OF THE INTEGRODIFFERENTIAL HYPERBOLIC EQUATION WITH PURELY NONLOCAL CONDITIONS

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 5

Metrics

本文评价

推荐阅读 10

[1]	龚循华. EKELAND´S PRINCIPLE FOR SET-VALUED VECTOR EQUILIBRIUM PROBLEMS[J]. 数学物理学报(英文版), 2014, 34(4): 1179-1192.
[2]	Svetlana Jankovic, Dejan Ilic. ONE LINEAR ANALYTIC APPROXIMATION FOR STOCHASTIC INTEGRODIFFERENTIAL EQUATIONS[J]. 数学物理学报(英文版), 2010, 30(4): 1073-1085.
[3]	Helge Holden, Kenneth H. Karlsen, Darko Mitrovic, Evgueni Yu. Panov. STRONG COMPACTNESS OF APPROXIMATE SOLUTIONS TO DEGENERATE ELLIPTIC-HYPERBOLIC EQUATIONS WITH DISCONTINUOUS FLUX FUNCTION[J]. 数学物理学报(英文版), 2009, 29(6): 1573-1612.
[4]	张万雄; 徐冰. HYERS-ULAM-RASSIAS STABILITY FOR A MULTIVALUED ITERATIVE EQUATION[J]. 数学物理学报(英文版), 2008, 28(1): 54-62.
[5]	陈方年, 段志文. THE ASYMPTOTIC BEHAVIOR OF SOLUTION FOR THE NONLINEAR HEAT-CONDUCTION EQUATION AND ITS APPLICATION[J]. 数学物理学报(英文版), 2001, 21(4): 561-571.