A Numerical Approach Based on Exponential Polynomials for solving of Fredholm Integro-Differential-Difference Equations
Yıl 2015,
Cilt: 3 Sayı: 2, 44 - 54, 19.01.2015
Mehmet Balci
Mehmet Sezer
Öz
In this study, a matrix method based on exponential polynomials by means of collocation points is proposed to solvethe higher-order linear Fredholm integro-differential-difference equations under the initial-boundary conditions. In addition, an erroranalysis technique based on residual function is developed for our method. Illustrative examples are included to demostrate the validityand applicability of the presented technique
Kaynakça
- S.Yalc¸ınbas¸, M.Sezer, The approximate solution of high-order linear Voltera-fredholm Integro-Differential equations in term of Taylor Polynomials, Apply. Math. Comput., 112, 291-308, 2000.
- W.Wang, An Algorithm for solving the high-order nonlinear Voltera-fredholm Integro-Differential equations with mechanization, Apply. Math. Comput., 172, 1-23, 2006.
- Y.Ben, B. Zhang, H. Qiao, A simple Taylor series expansion method for a class of second kind integral equations, J. Comp Appl. Math., 110, 15-24, 1999.
- K. Maleknejad, Y. Mahmoud, Numerical solution of linear Fredholm Integral Equations by using hybrid Taylor and block-pulse functions, Apply. Math. Comput., 149, 799-806, 2004.
- M.T. Rashed, Numerical solution of functional differential, integral and integro-differential equations, Appl. Numer. Math., 156, 485-492, 2004.
- W.Wang, C. Lin, A new algorithm for integral of trigonometric functions with mechanization, Apply. Math. Comput., 164(1), 71-82, 2005.
- M.Sezer, M.G¨ulsu, A new polynomial approach for solving difference and Fredholm integro-differential equations with mixed argument, Apply. Math. Comput., 171, 332-344, 2005.
- S.Yalc¸ınbas¸, M.Sezer, H.H. Sorkun, Legendre polynomial solutions of high-order linear Fredholm integro-differential equations, Apply. Math. Comput., 210, 334-349, 2009.
- M.G¨ulsu, M.Sezer, A Taylor polynomial approach for solving differential-difference equations, J. Comp Appl. Math., 186, 349- 364, 2006.
- L.M. Delves, J.L. Mohamed, Computational Methods for integral equations, Cambridge University Press, Cambridge, 1985.
- M. Razzagi, S. Yousefi, Legendre wavelets method for the nonlinear Voltera-Fredholm integral equations, Math. Comput. Simul., 70, 1-8, 2005.
- S.Shahmorad, Numerical solution of general form linear Fredholm-Voltera integro-differential equations by the Tau Method with an error estimation, Appl. Math. Comput., 167, 1418-1424, 2005.
- S.M. Hosseini, S. Shahmorad, A matrix formulation of the Tau method and Voltera linear integro diferential equations, Korean J. Comput., 216, 2183-2198, 2002.
- M.G¨ulsu, Y. ¨Ozt¨urk, M.Sezer, A new colacation method for solution of mixed linear integro-differential-difference equations, Appl. Math. Comput., 216, 2183-2198, 2010.
- M.Sezer, M.G¨ulsu, Polynomial solution of the most general linear Fredholm-integro diferential difference equation by means of Taylor matrix method, Int.J.Complex Variables,50,5367 - 382,2005.
- M.G¨ulsu, M.Sezer, A method for the approximate solution of the high-order linear difference equations in terms of Taylor polynomials, Int.J.Comput. Math. 82, 5, 629 - 642, 2005.
- S¸.Y¨uzbas¸ı, N. S¸ahin, M.Sezer, Bessel polynomial solutions of the high-order linear Voltera integro-diferential equations, Comput. Math. Appl. 62, 4, 1940 - 1956, 2011.
- N.Kurt, M.Sezer, Polynomial solution of high-order Linear Fredholm integro-diferential equa-tions with constant coeficients, Journal of Franklin Institute, 345, 839 - 850, 2008.
- M.Sezer, A.A. Das¸cıo˘glu, Taylor polynomial solutions of general linear diferential-difference equations with variable coeficients, Apply. Math. Comput. 174, 1526 - 1538, 2006.
- S.Yalc¸ınbas¸, N. ¨Ozsoy, M.Sezer, Approximate solution of higher order linear diferential equations by means of a new rational Chebyshev collocation method, mathematical and computational Applications, 5, 1, 45 - 56, 2010.
- N.Akg¨on¨ul, N.S¸ahin, M.Sezer, A Hermite collocation method for the approximate solutions of high-order linear Fredholm integro- diferential equations, 27, 6, 1707 - 1721, 2011.
- O.R.Is¸ık, M.Sezer, Z.G¨uney, Berstein series solution of a class of linear integro-diferential equations with weakly singular kernel, Appl. Math. Comput. 217, 16, 7009 - 7020, 2011.
- B.G¨urb¨uz, M.G¨ulsu, M.Sezer, Numerical approach of high-order linear delay difference equations with variable in terms of Laguerre polynomials, Mathematical and Computational Applications, 16, 1, 267 - 278, 2011.
- F.Alharbi, Predened exponential basis set for half-bounded multi-domain spectral method, Applied mathematics, Scientific Research, 1, 146 - 152, 2010.
- J.H.Laning, R.H.Battin, Random processes in Automatic Control, McGraw-Hill, New York, 9, 1956.
- V.Cizek, Methods of Time Domain Synthes Research Report z-44, Czechoslovak Academy of Sciences, Institute of Radioelektronics, Praha, 1960.
- A.A.Dimitriyev, Orthogonal Exponential Functions in Hydrometeorology, Gidro-meteoizdat, Leningrad, 1973.
- O.Jaroch, Approximation by Exponential Functions, Aplikace matematiky, 7, 4, 249 - 264,1962.
- V.S.Chelyshkov, Sequence of exponential polynomials which are orthogonal on the semi-exis, Reports of the Academy of Sciences of the Uk SSR, ( Dohlady AN Uk SSR ), ser.A, 14- 47, 1997.
- B.J.C.Baxter, A.Iserles, On approximation by exponentials, Annals of Num. Math., 4, 39 -54, 1997.
- V.S.Chelyshkov, A variant of spectral method in the theory of hydrodynamic stability, Hydromachanics ( Gidromekhanica ), 68, 105 - 109, 1994.
- V.T.Grinchenko, V.S.Chelyshkov, Direct Numerical simulation of boundary layer transition, in Near Wall Turbulent Flows R.M.C. So, C.G.Speziale and B.E. Launder ( Editors ), Elsevier Science Publishers B.V., 889 - 897, 1993.
Mehmet Ali Balci1and Mehmet Sezer2
Yıl 2015,
Cilt: 3 Sayı: 2, 44 - 54, 19.01.2015
Mehmet Balci
Mehmet Sezer
Kaynakça
- S.Yalc¸ınbas¸, M.Sezer, The approximate solution of high-order linear Voltera-fredholm Integro-Differential equations in term of Taylor Polynomials, Apply. Math. Comput., 112, 291-308, 2000.
- W.Wang, An Algorithm for solving the high-order nonlinear Voltera-fredholm Integro-Differential equations with mechanization, Apply. Math. Comput., 172, 1-23, 2006.
- Y.Ben, B. Zhang, H. Qiao, A simple Taylor series expansion method for a class of second kind integral equations, J. Comp Appl. Math., 110, 15-24, 1999.
- K. Maleknejad, Y. Mahmoud, Numerical solution of linear Fredholm Integral Equations by using hybrid Taylor and block-pulse functions, Apply. Math. Comput., 149, 799-806, 2004.
- M.T. Rashed, Numerical solution of functional differential, integral and integro-differential equations, Appl. Numer. Math., 156, 485-492, 2004.
- W.Wang, C. Lin, A new algorithm for integral of trigonometric functions with mechanization, Apply. Math. Comput., 164(1), 71-82, 2005.
- M.Sezer, M.G¨ulsu, A new polynomial approach for solving difference and Fredholm integro-differential equations with mixed argument, Apply. Math. Comput., 171, 332-344, 2005.
- S.Yalc¸ınbas¸, M.Sezer, H.H. Sorkun, Legendre polynomial solutions of high-order linear Fredholm integro-differential equations, Apply. Math. Comput., 210, 334-349, 2009.
- M.G¨ulsu, M.Sezer, A Taylor polynomial approach for solving differential-difference equations, J. Comp Appl. Math., 186, 349- 364, 2006.
- L.M. Delves, J.L. Mohamed, Computational Methods for integral equations, Cambridge University Press, Cambridge, 1985.
- M. Razzagi, S. Yousefi, Legendre wavelets method for the nonlinear Voltera-Fredholm integral equations, Math. Comput. Simul., 70, 1-8, 2005.
- S.Shahmorad, Numerical solution of general form linear Fredholm-Voltera integro-differential equations by the Tau Method with an error estimation, Appl. Math. Comput., 167, 1418-1424, 2005.
- S.M. Hosseini, S. Shahmorad, A matrix formulation of the Tau method and Voltera linear integro diferential equations, Korean J. Comput., 216, 2183-2198, 2002.
- M.G¨ulsu, Y. ¨Ozt¨urk, M.Sezer, A new colacation method for solution of mixed linear integro-differential-difference equations, Appl. Math. Comput., 216, 2183-2198, 2010.
- M.Sezer, M.G¨ulsu, Polynomial solution of the most general linear Fredholm-integro diferential difference equation by means of Taylor matrix method, Int.J.Complex Variables,50,5367 - 382,2005.
- M.G¨ulsu, M.Sezer, A method for the approximate solution of the high-order linear difference equations in terms of Taylor polynomials, Int.J.Comput. Math. 82, 5, 629 - 642, 2005.
- S¸.Y¨uzbas¸ı, N. S¸ahin, M.Sezer, Bessel polynomial solutions of the high-order linear Voltera integro-diferential equations, Comput. Math. Appl. 62, 4, 1940 - 1956, 2011.
- N.Kurt, M.Sezer, Polynomial solution of high-order Linear Fredholm integro-diferential equa-tions with constant coeficients, Journal of Franklin Institute, 345, 839 - 850, 2008.
- M.Sezer, A.A. Das¸cıo˘glu, Taylor polynomial solutions of general linear diferential-difference equations with variable coeficients, Apply. Math. Comput. 174, 1526 - 1538, 2006.
- S.Yalc¸ınbas¸, N. ¨Ozsoy, M.Sezer, Approximate solution of higher order linear diferential equations by means of a new rational Chebyshev collocation method, mathematical and computational Applications, 5, 1, 45 - 56, 2010.
- N.Akg¨on¨ul, N.S¸ahin, M.Sezer, A Hermite collocation method for the approximate solutions of high-order linear Fredholm integro- diferential equations, 27, 6, 1707 - 1721, 2011.
- O.R.Is¸ık, M.Sezer, Z.G¨uney, Berstein series solution of a class of linear integro-diferential equations with weakly singular kernel, Appl. Math. Comput. 217, 16, 7009 - 7020, 2011.
- B.G¨urb¨uz, M.G¨ulsu, M.Sezer, Numerical approach of high-order linear delay difference equations with variable in terms of Laguerre polynomials, Mathematical and Computational Applications, 16, 1, 267 - 278, 2011.
- F.Alharbi, Predened exponential basis set for half-bounded multi-domain spectral method, Applied mathematics, Scientific Research, 1, 146 - 152, 2010.
- J.H.Laning, R.H.Battin, Random processes in Automatic Control, McGraw-Hill, New York, 9, 1956.
- V.Cizek, Methods of Time Domain Synthes Research Report z-44, Czechoslovak Academy of Sciences, Institute of Radioelektronics, Praha, 1960.
- A.A.Dimitriyev, Orthogonal Exponential Functions in Hydrometeorology, Gidro-meteoizdat, Leningrad, 1973.
- O.Jaroch, Approximation by Exponential Functions, Aplikace matematiky, 7, 4, 249 - 264,1962.
- V.S.Chelyshkov, Sequence of exponential polynomials which are orthogonal on the semi-exis, Reports of the Academy of Sciences of the Uk SSR, ( Dohlady AN Uk SSR ), ser.A, 14- 47, 1997.
- B.J.C.Baxter, A.Iserles, On approximation by exponentials, Annals of Num. Math., 4, 39 -54, 1997.
- V.S.Chelyshkov, A variant of spectral method in the theory of hydrodynamic stability, Hydromachanics ( Gidromekhanica ), 68, 105 - 109, 1994.
- V.T.Grinchenko, V.S.Chelyshkov, Direct Numerical simulation of boundary layer transition, in Near Wall Turbulent Flows R.M.C. So, C.G.Speziale and B.E. Launder ( Editors ), Elsevier Science Publishers B.V., 889 - 897, 1993.