求解m阶非线性Volterra-Fredholm型积分微分方程的一种算法
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摘要
针对m阶非线性Volterra-Fredholm型积分微分方程,利用勒让德-伽辽金方法进行求解.勒让德多项式被选作基函数,通过基函数与残差正交得到有限维方程组,求解有限维方程组得到待定系数,便能求出方程的近似解.一些数值算例的给出证明了方法的可行性和有效性.
For the mth-order nonlinear Volterra-Fredholm integro-differential equations,the Legendre-Galerkin method is proposed to solve them.The Legendre polynomials are chosen as basis functions,the finite dimensional equations are obtained by orthogonal functions of the basis functions and the residuals,and the approximate solutions of the equations can be obtained by solving the finite dimensional equations.Some numerical examples are provided to illustrate the accuracy and computational efficiency of the method.
For the mth-order nonlinear Volterra-Fredholm integro-differential equations,the Legendre-Galerkin method is proposed to solve them.The Legendre polynomials are chosen as basis functions,the finite dimensional equations are obtained by orthogonal functions of the basis functions and the residuals,and the approximate solutions of the equations can be obtained by solving the finite dimensional equations.Some numerical examples are provided to illustrate the accuracy and computational efficiency of the method.
引文
[1]Wazwaz A M.A comparison study between the modified decomposition method and traditional method for solving nonlinear litegral equations[J].Applied Mathematics and Computation,2006,181(2):1703-1712.
[2]Rashed M T.Numerical solution of functional differential,integral and integro-differential equations[J].Applied Mathematics and Computation,2004,156(2):485-492.
[3]Reihani M H,Abadi Z.Rationalized Haar functions method for solving Fredholm and Volterraintegral equations[J].Journal of Computational and Applied Mathematics,2007,200(1):12-20.
[4]Maleknejed K,Mirzaee F.Numerical solution of integro-differential equations by using ratioalized Haar functions method[J].Kybernetes,2006,35(10):1735-1744.
[5]Ebadi G,Rahimi-Ardabili M Y,and Shahm oad S. Numerical solution of the nonlinear Volterra integro-differential equations by the Tau method[J].Applied Mathematics and Computation,2007,188(2):1580-1586.
[6]Abbasbandy S,Taati A.Numerical solution of the system of nonlinear Volterra integro-differential equations with nonlinear differential part by the operational Tau method and error estimation[J].Journal of Computational and Applied Mathematics,2009,231(1):106-113.
[7]Wazwaz A M.The combined Laplace transform Adomian decomposition method for handling nonlinear Volterra integro-differential equations[J].Applied Mathematics and Computation,2010,216(4):1304-1309.'
[8]Araghi M A,Sh S.Behzadi.Solving nonlinear Volterra-Fredholm integro-differential equations using the modified Adomian decomposition method[J].Computational Methods in Applied Mathematics and Computation,2009,9(4):321-331.
[9]Phillips D L.A technique for the numerical solution of certain integral equations of the first kind[J].Journal of the Acm,1962,9(1):84-96.
[10]A.N.Tikhonov.On the solution of incorrectly posed problem and the method of regularization[J].Soviet Mathematics,1963:501-505.
[11]Maleknejad K,Mohmoudi Y.Taylor polynomial solution of high-order nonlineax Volterra-Fredholm integro-differential equations[J].Applied Mathematics and Computation,2003,145(2-3):641-653.
[12]Daxania P,Ivaz K.Numerical solution of nonlineax Volterrar-Fredholm integro-differential equar tions[J].Computers and Mathematics with Applications,2008,56(9):2197-2209.
[13]Yalcinbas S.Taylar polynomial solution of nonlinear Volterra-Fredholm integral equations[J].Applied Mathematics and Com putation,2002,127(2-3):195-206.
[14]Yuzbasi S,Sahin N,and Yildirim A.A collocation approach for solving high-order linear FredholmVolterra integro-differential equations[J].Mathematical and Computer Modelling,2012,55(3-4):547-563.
[15]Kanwal R P,Liu K C.A Taylor expansion approach for solving integral equations[J].International Journal of Mathematical Education in Science,1989,20(3):411-414.
[16]Sezer M.Taylor polynomial solution of Volterra integral equations[J].International Journal of Mathematical Education in Science and Technology,1994,25(5):625-633.
[17]Ramadan M A,Lashien I F,and Zahra W K.Quintic nonpolynomial spline functions for fourth order two-point boundary value problems[J].Communications in Nonlinear Science and Numerical,2009,14(4):1105-1114.
[18]Siraj-ul-Islam I A.Tirmizi and Ashraf S.A class of methods based on non-polynomial spline functions for the solution of a special fourth order boundary value problems with engineering applicar tions[J].Applied Mathematics and Computation,2006,174(2):1169-1180.
[19]EI-Gamel M,Behiry S H,and Hashish H.Numerical method for the solution of special nonlinear fourth-order boundary value problems[J].Applied Mathematics and Computation,2003,145(2-3):717-734.
[20]El-Mikkawy M E A,Cheon G s.Combinatorial and hypergeometric identities via the Legendre polynomials-A computational approach[J].Applied Mathematics and Computation,2005,166(1):181-195.
[21]Marzban H R,Razzaghi M.Optimal control of linear delay systems via hybrid of block-pulse and Legendre polynomials[J]:Journal of the Franklin Institute,2004,341(3):279-293.
[22]Atkinson K E.An introduction to numerical analysis[M].John Wiley and Sons,NY,2~(nd)Edition,1989.
[23]Babolian E,Masouri Z,and Hatamzadeh-Varmazyar S.Numerical solution of nonlinear VolterraFredholm integro-differential equations Via direc.t method using Triangular functions[J].Computers Mathematics with Applications,2009,58(2):239-247.
[24]Pandey P K.Non-standard difference method for numerical solution of linear fredholm integrodifferential type two-point boundary value problems[J].Open Access Library Journal,2015,02(9):1-10.
[2]Rashed M T.Numerical solution of functional differential,integral and integro-differential equations[J].Applied Mathematics and Computation,2004,156(2):485-492.
[3]Reihani M H,Abadi Z.Rationalized Haar functions method for solving Fredholm and Volterraintegral equations[J].Journal of Computational and Applied Mathematics,2007,200(1):12-20.
[4]Maleknejed K,Mirzaee F.Numerical solution of integro-differential equations by using ratioalized Haar functions method[J].Kybernetes,2006,35(10):1735-1744.
[5]Ebadi G,Rahimi-Ardabili M Y,and Shahm oad S. Numerical solution of the nonlinear Volterra integro-differential equations by the Tau method[J].Applied Mathematics and Computation,2007,188(2):1580-1586.
[6]Abbasbandy S,Taati A.Numerical solution of the system of nonlinear Volterra integro-differential equations with nonlinear differential part by the operational Tau method and error estimation[J].Journal of Computational and Applied Mathematics,2009,231(1):106-113.
[7]Wazwaz A M.The combined Laplace transform Adomian decomposition method for handling nonlinear Volterra integro-differential equations[J].Applied Mathematics and Computation,2010,216(4):1304-1309.'
[8]Araghi M A,Sh S.Behzadi.Solving nonlinear Volterra-Fredholm integro-differential equations using the modified Adomian decomposition method[J].Computational Methods in Applied Mathematics and Computation,2009,9(4):321-331.
[9]Phillips D L.A technique for the numerical solution of certain integral equations of the first kind[J].Journal of the Acm,1962,9(1):84-96.
[10]A.N.Tikhonov.On the solution of incorrectly posed problem and the method of regularization[J].Soviet Mathematics,1963:501-505.
[11]Maleknejad K,Mohmoudi Y.Taylor polynomial solution of high-order nonlineax Volterra-Fredholm integro-differential equations[J].Applied Mathematics and Computation,2003,145(2-3):641-653.
[12]Daxania P,Ivaz K.Numerical solution of nonlineax Volterrar-Fredholm integro-differential equar tions[J].Computers and Mathematics with Applications,2008,56(9):2197-2209.
[13]Yalcinbas S.Taylar polynomial solution of nonlinear Volterra-Fredholm integral equations[J].Applied Mathematics and Com putation,2002,127(2-3):195-206.
[14]Yuzbasi S,Sahin N,and Yildirim A.A collocation approach for solving high-order linear FredholmVolterra integro-differential equations[J].Mathematical and Computer Modelling,2012,55(3-4):547-563.
[15]Kanwal R P,Liu K C.A Taylor expansion approach for solving integral equations[J].International Journal of Mathematical Education in Science,1989,20(3):411-414.
[16]Sezer M.Taylor polynomial solution of Volterra integral equations[J].International Journal of Mathematical Education in Science and Technology,1994,25(5):625-633.
[17]Ramadan M A,Lashien I F,and Zahra W K.Quintic nonpolynomial spline functions for fourth order two-point boundary value problems[J].Communications in Nonlinear Science and Numerical,2009,14(4):1105-1114.
[18]Siraj-ul-Islam I A.Tirmizi and Ashraf S.A class of methods based on non-polynomial spline functions for the solution of a special fourth order boundary value problems with engineering applicar tions[J].Applied Mathematics and Computation,2006,174(2):1169-1180.
[19]EI-Gamel M,Behiry S H,and Hashish H.Numerical method for the solution of special nonlinear fourth-order boundary value problems[J].Applied Mathematics and Computation,2003,145(2-3):717-734.
[20]El-Mikkawy M E A,Cheon G s.Combinatorial and hypergeometric identities via the Legendre polynomials-A computational approach[J].Applied Mathematics and Computation,2005,166(1):181-195.
[21]Marzban H R,Razzaghi M.Optimal control of linear delay systems via hybrid of block-pulse and Legendre polynomials[J]:Journal of the Franklin Institute,2004,341(3):279-293.
[22]Atkinson K E.An introduction to numerical analysis[M].John Wiley and Sons,NY,2~(nd)Edition,1989.
[23]Babolian E,Masouri Z,and Hatamzadeh-Varmazyar S.Numerical solution of nonlinear VolterraFredholm integro-differential equations Via direc.t method using Triangular functions[J].Computers Mathematics with Applications,2009,58(2):239-247.
[24]Pandey P K.Non-standard difference method for numerical solution of linear fredholm integrodifferential type two-point boundary value problems[J].Open Access Library Journal,2015,02(9):1-10.
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