基于扩展G′/G-展开法的两个非线性拟抛物型方程的精确解
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摘要
对两个重要的非线性拟抛物物理模型,Benjamin-Bona-Mahony-Peregrine-Burgers(BBMPB)和Oskolkov-Benjamin-Bona-Mahony-Burgers(OBBMB)方程进行了研究.使用扩展的G′/G-函数展开法,借助符号计算软件Maple,获得了它们新的精确行波解,并验证了它们的正确性.这些解包括双曲函数精解,三角周期解和有理函数解.精确解的获得,有助于解释以这两个方程为模型的一些实际物理现象、为数值解的进一步研究提供一定的参考.获得的结果证实该方法也可以用于求解一些其他的非线性拟抛物模型方程.
Two important nonlinear pseudoparabolic physical models,the Benjamin-Bona-Mahony-Peregrine-Burgers(BBMPB)equation,and the Oskolkov-Benjamin-Bona-Mahony-Burgers(OBBMB)equation were studied.The extended(G′/G)-expansion method and symbolic computation software Maple were used to obtain the new exact traveling wave solutions of these equations,and verified the correctness.These solutions include function fine solutions,triangular periodic solutions and rational function solutions.These exact solutions can help to explain some practical physical phenomena which are modelled on these equations,and provide a certain reference for the further study of the numerical solution.The obtained results confirm that the proposed methods can also be used to solve some of the other nonlinear pseudoparabolic model equations.
Two important nonlinear pseudoparabolic physical models,the Benjamin-Bona-Mahony-Peregrine-Burgers(BBMPB)equation,and the Oskolkov-Benjamin-Bona-Mahony-Burgers(OBBMB)equation were studied.The extended(G′/G)-expansion method and symbolic computation software Maple were used to obtain the new exact traveling wave solutions of these equations,and verified the correctness.These solutions include function fine solutions,triangular periodic solutions and rational function solutions.These exact solutions can help to explain some practical physical phenomena which are modelled on these equations,and provide a certain reference for the further study of the numerical solution.The obtained results confirm that the proposed methods can also be used to solve some of the other nonlinear pseudoparabolic model equations.
引文
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[15]Omer F G,Samil A.The tanh-coth method for some nonlinear pseudoparabolic equations with exact solutions[J].Advances in Difference Equations,2013,143(1):2-18.
[16]Jahani M,Manafian J.Improvement of the Exp-function method for solving the BBM equation with timedependent coefficients[J].Eur.Phys.J.Plus.,2016,131(3):1-12.
[17]Elsayed M E,Abdul-Ghani A N.Solitons and other exact solutions for variant nonlinear Boussinesq equations[J].Optik-International Journal for Light and Electron Optics,2017,139:166-177.
[18]Mirzazadeh M,Eslami M.Dark optical solitons of Biswas-Milovic equation with dual-power law nonlinearity[J].Eur.Phys.J.Plus.,2015,130(1):1-7.
[19]Jalil M,Mehdi F A.Application of the generalized G′/G-expansion method for nonlinear PDEs to obtaining soliton wave solution[J].Optik-International Journal for Light and Electron Optics,2017,135:395-406.
[20]Pang Jing,Bian Chunquan.A new auxiliary equation method for finding travelling wave solutions to KdVequation[J].Applied Mathematics and Mechanics,2010,31(7):929-936.
[2]Korpusov M O,Sveshnikov A G.Blow-up of solutions of strongly nonlinear equations of pseudoparabolic type[J].J.Math.Sci.,2008,148(1):1-142.
[3]Kaikina E I,Naumkin P I,Shishmarev I A.The Cauchy problem for an equation of Sobolev type with power non-linearity[J].Izv.Math.,2005,69(1):59-111.
[4]Karch G.Asymptotic behaviour of solutions to some pseudoparabolic equations[J].Applied Mathematics Letters,2007,25(2):111-114.
[5]Abdel A S,Osman E S.The homogeneous balance method and its application to the Benjamin-Bona-Mahoney(BBM)equation[J].Applied Mathematics and Computation,2010,217(4):1385-1390.
[6]Lu B,Yuan G X,Yang J K.A class of exact solutions of the BBM equations[J].Advances in Computer Science,Intelligent System and Environment,2011,105(1):557-580.
[7]Abbasbandy S.Shirzadi A.The first integral method for modified Benjamin-Bona-Mahony equation[J],Communications in Nonlinear Science and Numerical Simulation,2010,15(7):1759-1764.
[8]Oskolkov A P.Nonlocal problems for one class of nonlinear operator equations that arise in the theory of Sobolev-type equations[J].Journal of Soviet Mathematics,1993,64(1):724-735.
[9]Oskolkov A P.On stability theory for solutions of semilinear dissipative equations of the Sobolev type[J].Journal of Mathematical Sciences,1995,77(3):3225-3231.
[10]Dehghan M,Manafian J.Solving nonlinear fractional partial differential equations using the homotopy analysis method[J].Numer.Methods Partial Differ.Equ.,2010,26(2):448-479.
[11]Yusufoglu E,Bekir A.The tanh and the sine-cosine methods for exact solutions of the MBBM and the Vakhnenko equations[J].Chaos Solitons Fractals,2008,38(4):1126-1133.
[12]Wazwaz A M.The tanh method for travelling wave solutions of nonlinear equations[J].Applied Mathematics&Computation,2004,154(3):713-723.
[13]Abdel-All N H,Abdel-Razek M A.Expanding the tanh-function method for solving nonlinear equations[J].Applied Mathematics and Computation,2011,2(9):1096-1104.
[14]Akbar M A,Norhashidah H M,Syed T D.Assessment of the further improved(G′/G)-expansion method and the extended tanh-method in probing exact solutions of nonlinear PDEs[J].Springer Plus,2013,2(1):2-9.
[15]Omer F G,Samil A.The tanh-coth method for some nonlinear pseudoparabolic equations with exact solutions[J].Advances in Difference Equations,2013,143(1):2-18.
[16]Jahani M,Manafian J.Improvement of the Exp-function method for solving the BBM equation with timedependent coefficients[J].Eur.Phys.J.Plus.,2016,131(3):1-12.
[17]Elsayed M E,Abdul-Ghani A N.Solitons and other exact solutions for variant nonlinear Boussinesq equations[J].Optik-International Journal for Light and Electron Optics,2017,139:166-177.
[18]Mirzazadeh M,Eslami M.Dark optical solitons of Biswas-Milovic equation with dual-power law nonlinearity[J].Eur.Phys.J.Plus.,2015,130(1):1-7.
[19]Jalil M,Mehdi F A.Application of the generalized G′/G-expansion method for nonlinear PDEs to obtaining soliton wave solution[J].Optik-International Journal for Light and Electron Optics,2017,135:395-406.
[20]Pang Jing,Bian Chunquan.A new auxiliary equation method for finding travelling wave solutions to KdVequation[J].Applied Mathematics and Mechanics,2010,31(7):929-936.