### SOLVABILITY THE TELEGRAPH EQUATION WITH PURELY INTEGRAL CONDITIONS

Year 2013, Volume 3, Issue 1, 117 - 125, 01.06.2013

### Abstract

In this paper a numerical technique is developed for the one-dimensional telegraph equation, we prove the existence, uniqueness, and continuous dependence upon the data of solution to a telegraph equation with purely integral conditions. The proofs are based on a priori estimates and Laplace transform method. Finally, we obtain the solution by using a simple and efficient algorithm for numerical solution.

### References

• [1] Abramowitz, M., Stegun. I.A.,(1972), Hand book of Mathematical Functions, Dover, New York.
• [2] Ang, W.T., (2002),A Method of Solution for the One-Dimentional Heat Equation Subject to Nonlocal Conditions, Southeast Asian Bulletin of Mathematics 26 , 185-191.
• [3] Be¨ılin, S. A., (2001), Existence of solutions for one-dimentional wave nonlocal conditions, Electron. J. Differential Equations , no. 76, 1-8.
• [4] Bouziani, A., (1996), Problemes mixtes avec conditions int´egrales pour quelques ´equations aux d´eriv´ees partielles, Ph.D. thesis, Constantine University.
• [5] Bouziani, A.,(1996), Mixed problem with boundary integral conditions for a certain parabolic equation, J. Appl. Math. Stochastic Anal. 09 ,no. 3, 323-330.
• [6] Bouziani, A.,(1997), Solution forte d’un probleme mixte avec une condition non locale pour une classe d’´equations hyperboliques [Strong solution of a mixed problem with a nonlocal condition for a class of hyperbolic equations], Acad. Roy. Belg. Bull. Cl. Sci. 8 , 53-70.
• [7] Bouziani, A., (2000), Strong solution to an hyperbolic evolution problem with nonlocal boundary conditions, Maghreb Math. Rev., 9 , no. 1-2, 71–84.
• [8] Bouziani, A., (2002), Initial-boundary value problem with nonlocal condition for a viscosity equation, Int. J. Math. & Math. Sci. 30 , no. 6, 327-338.
• [9] Bouziani, A., (2002), On the solvabiliy of parabolic and hyperbolic problems with a boundary integral condition, Internat. J. Math. & Math. Sci., 31 , 435-447.
• [10] Bouziani, A., (2002), On a class of nonclassical hyperbolic equations with nonlocal conditions, J. Appl. Math. Stochastic Anal. 15 ,no. 2, 136-153.
• [11] Bouziani, A.,(2004), Mixed problem with only integral boundary conditions for an hyperbolic equation, Internat. J. Math. & Math. Sci., 26, 1279-1291.
• [12] Bouziani, A. and Benouar N.,(1996), Probleme mixte avec conditions int´egrales pour une classe d’´equations hyperboliques, Bull. Belg. Math. Soc. 3 , 137-145.
• [13] Bouziani, A. & Merazga N.,(2004), Rothe time-discretization method applied to a quasilinear wave equation subject to integral conditions, Advances in Difference Equations, Vol. 2004, N◦ 3, 211-235.
• [14] Graver D. P.,(1966), Observing stochastic processes and aproximate transform inversion, Oper. Res. 14, 444-459.
• [15] Gordeziani, D. G. & Avalishvili, G. A., (2000), Solution of nonlocal problems for one-dimensional oscillations of a medium, Mat. Model. 12 , no. 1, 94–103 (Russian).
• [16] Hassanzadeh Hassan; Pooladi-Darvish Mehran,(2007), Comparision of different numerical Laplace inversion methods for engineering applications, Appl. Math. Comp. 189 1966-1981.
• [17] Kac˘ur, J., (1985), Method of Rothe in Evolution Equations, Teubner-Texte zur Mathematik, vol. 80, BSB B. G. Teubner Verlagsgesellschaft, Leipzig.
• [18] Merad, A., (2011), Adomian Decomposition Method for Solution of Parabolic Equation to Nonlocal Conditions,Int. J. Contemp. Math. Sciences, Vol. 6, , no. 30, 1491 - 1496.
• [19] Merad, A. & Marhoune, A. L., (2012), Strong Solution for a High Order Boundary Value Problem with Integral condition, Turk. J. Math., doi: 10.3906/math-1105-34 .
• [20] Mesloub S. & Bouziani, A.,(1999), On a class of singular hyperbolic equation with a weighted integral condition, Int. J. Math. Math. Sci. 22 ), no. 3, 511–519.
• [21] Mesloub S. and Bouziani, A., (2001), Mixed problem with integral conditions for a certain class of hyperbolic equations, Journal of Applied Mathematics, Vol. 1 , no. 3, 107-116.
• [22] Pul’kina,L. S.,(1999), A non-local problem with integral conditions for hyperbolic equations, Electron. J. Differential Equations , no. 45, 1–6.
• [23] Pul’kina,L. S.,(2000), On the solvability in L2 of a nonlocal problem with integral conditions for a hyperbolic equation, Differ. Equ. 36 , no. 2, 316–318.
• [24] Pul’kina,L. S., (2003),A mixed problem with integral condition for the hyperbolic equation, Matematicheskie Zametki, vol. 74, no. 3, , pp. 435–445.
• [25] Stehfest,H., (1970), Numerical Inversion of the Laplace Transform, Comm. ACM 13, 47-49.
• [26] Shruti A.D.,(2010), Numerical Solution for Nonlocal Sobolev-type Differential Equations, Electronic Journal of Differential Equations, Conf. 19 , 75-83.
Year 2013, Volume 3, Issue 1, 117 - 125, 01.06.2013

### References

• [1] Abramowitz, M., Stegun. I.A.,(1972), Hand book of Mathematical Functions, Dover, New York.
• [2] Ang, W.T., (2002),A Method of Solution for the One-Dimentional Heat Equation Subject to Nonlocal Conditions, Southeast Asian Bulletin of Mathematics 26 , 185-191.
• [3] Be¨ılin, S. A., (2001), Existence of solutions for one-dimentional wave nonlocal conditions, Electron. J. Differential Equations , no. 76, 1-8.
• [4] Bouziani, A., (1996), Problemes mixtes avec conditions int´egrales pour quelques ´equations aux d´eriv´ees partielles, Ph.D. thesis, Constantine University.
• [5] Bouziani, A.,(1996), Mixed problem with boundary integral conditions for a certain parabolic equation, J. Appl. Math. Stochastic Anal. 09 ,no. 3, 323-330.
• [6] Bouziani, A.,(1997), Solution forte d’un probleme mixte avec une condition non locale pour une classe d’´equations hyperboliques [Strong solution of a mixed problem with a nonlocal condition for a class of hyperbolic equations], Acad. Roy. Belg. Bull. Cl. Sci. 8 , 53-70.
• [7] Bouziani, A., (2000), Strong solution to an hyperbolic evolution problem with nonlocal boundary conditions, Maghreb Math. Rev., 9 , no. 1-2, 71–84.
• [8] Bouziani, A., (2002), Initial-boundary value problem with nonlocal condition for a viscosity equation, Int. J. Math. & Math. Sci. 30 , no. 6, 327-338.
• [9] Bouziani, A., (2002), On the solvabiliy of parabolic and hyperbolic problems with a boundary integral condition, Internat. J. Math. & Math. Sci., 31 , 435-447.
• [10] Bouziani, A., (2002), On a class of nonclassical hyperbolic equations with nonlocal conditions, J. Appl. Math. Stochastic Anal. 15 ,no. 2, 136-153.
• [11] Bouziani, A.,(2004), Mixed problem with only integral boundary conditions for an hyperbolic equation, Internat. J. Math. & Math. Sci., 26, 1279-1291.
• [12] Bouziani, A. and Benouar N.,(1996), Probleme mixte avec conditions int´egrales pour une classe d’´equations hyperboliques, Bull. Belg. Math. Soc. 3 , 137-145.
• [13] Bouziani, A. & Merazga N.,(2004), Rothe time-discretization method applied to a quasilinear wave equation subject to integral conditions, Advances in Difference Equations, Vol. 2004, N◦ 3, 211-235.
• [14] Graver D. P.,(1966), Observing stochastic processes and aproximate transform inversion, Oper. Res. 14, 444-459.
• [15] Gordeziani, D. G. & Avalishvili, G. A., (2000), Solution of nonlocal problems for one-dimensional oscillations of a medium, Mat. Model. 12 , no. 1, 94–103 (Russian).
• [16] Hassanzadeh Hassan; Pooladi-Darvish Mehran,(2007), Comparision of different numerical Laplace inversion methods for engineering applications, Appl. Math. Comp. 189 1966-1981.
• [17] Kac˘ur, J., (1985), Method of Rothe in Evolution Equations, Teubner-Texte zur Mathematik, vol. 80, BSB B. G. Teubner Verlagsgesellschaft, Leipzig.
• [18] Merad, A., (2011), Adomian Decomposition Method for Solution of Parabolic Equation to Nonlocal Conditions,Int. J. Contemp. Math. Sciences, Vol. 6, , no. 30, 1491 - 1496.
• [19] Merad, A. & Marhoune, A. L., (2012), Strong Solution for a High Order Boundary Value Problem with Integral condition, Turk. J. Math., doi: 10.3906/math-1105-34 .
• [20] Mesloub S. & Bouziani, A.,(1999), On a class of singular hyperbolic equation with a weighted integral condition, Int. J. Math. Math. Sci. 22 ), no. 3, 511–519.
• [21] Mesloub S. and Bouziani, A., (2001), Mixed problem with integral conditions for a certain class of hyperbolic equations, Journal of Applied Mathematics, Vol. 1 , no. 3, 107-116.
• [22] Pul’kina,L. S.,(1999), A non-local problem with integral conditions for hyperbolic equations, Electron. J. Differential Equations , no. 45, 1–6.
• [23] Pul’kina,L. S.,(2000), On the solvability in L2 of a nonlocal problem with integral conditions for a hyperbolic equation, Differ. Equ. 36 , no. 2, 316–318.
• [24] Pul’kina,L. S., (2003),A mixed problem with integral condition for the hyperbolic equation, Matematicheskie Zametki, vol. 74, no. 3, , pp. 435–445.
• [25] Stehfest,H., (1970), Numerical Inversion of the Laplace Transform, Comm. ACM 13, 47-49.
• [26] Shruti A.D.,(2010), Numerical Solution for Nonlocal Sobolev-type Differential Equations, Electronic Journal of Differential Equations, Conf. 19 , 75-83.

### Details

Primary Language English Research Article A. MERAD This is me Department of Mathematics, Faculty of Sciences, Larbi Ben M’hidi University, Oum El Bouaghi, 04000, ALGERIA A. BOUZİANİ This is me Department of Mathematics, Faculty of Sciences, Larbi Ben M’hidi University, Oum El Bouaghi, 04000, ALGERIA June 1, 2013 Year 2013, Volume 3, Issue 1

### Cite

 Bibtex @ { twmsjaem761701, journal = {TWMS Journal of Applied and Engineering Mathematics}, issn = {2146-1147}, eissn = {2587-1013}, address = {Işık University ŞİLE KAMPÜSÜ Meşrutiyet Mahallesi, Üniversite Sokak No:2 Şile / İstanbul}, publisher = {Turkic World Mathematical Society}, year = {2013}, volume = {3}, number = {1}, pages = {117 - 125}, title = {SOLVABILITY THE TELEGRAPH EQUATION WITH PURELY INTEGRAL CONDITIONS}, key = {cite}, author = {Merad, A. and Bouziani, A.} }