In this paper, we used the New Iterative Method (NIM) developed by Daftardar-Gejji and Jafari for the solution of linear and nonlinear systems of partial differential equations. This method is very simple as it reduces the size of computation and readily converges to the exact solution. To demonstrate the efficiency of the method, some illustrative examples were provided. The results obtained confirmed that the method is an efficient method for a wide variety of systems of linear and nonlinear PDEs.&nbsp

Dehinsilu, O. A.

Hammed, F. A.

Shittu, M. T.

Solanke, O. O.

Usman, M. A.

English

Faculty of Science, Federal University of Lafia

FULafia Journal of Science & Technology Vol.6 No.3 September, 202042THE NEW ITERATIVE METHOD FOR SOLVING LINEAR AND NONLINEAR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONSShittu, M.T., *Usman, M.A., Solanke, O.O., Hammed, F.A and Dehinsilu, O.A.Department of Mathematical Science, Olabisi Onabanjo University, Ago –Iwoye.Nigeria*Corresponding Email:usman.mustapha@oouagoiwoye.edu.ngManuscript Received:18/05/2020      Accepted:11/08/2020        Published: September, 2020ABSTRACT In this paper, we used the New Iterative Method (NIM) developed by Daftardar-Gejji and Jafari for the solution of linear and nonlinear systems of partial differential equations. This method is very simple as it reduces the size of computation and readily converges to the exact solution. To demonstrate the efficiency of the method, some illustrative examples were provided. The results obtained confirmed that the method is an efficient method for a wide variety of systems of linear and nonlinear PDEs.  Keywords: New Iterative Method, Systems , Partial Differential Equations,Linear,NonlinearISSN (Print): 24490954ISSN (Online): 26364972PHYSICAL  SCIENCESFULafia Journal of Science & Technology Vol.6 No.3 September, 202043INTRODUCTIONSystems of partial differential equations, linear or nonlinear, have attracted much concern in studying evolution equations that describe wave propagation, in investigating shallow water waves, and in examining the chemical reaction-diffusion model of Brusselator Abdul-Majid Wazwaz(2009).  Recently, many powerful methods have been presented in solving the partial differential equation systems, such as Variational Iterative Method (VIM) Abdul-Majid Wazwaz(2007) M. Akbarzade(2011), Adomian Decomposition Method (ADM) Abdul-Majid Wazwaz(2000) Mohammed E. A. Rabie and Tarig M. Elzali(2014), Aboodh Adomian Decomposition Method (AADM) Mohand M. Abdelrahim Mahgoub and Abdelilah K. Hassan Sedeeg(2017), Laplace Adomian Decomposition Method (LADM) Jasem Fadaei (2011), Homotopy Perturbation Method Jafar Biazar and Mostafa Eslami(2011), Differential Transformation Method (DTM) Raslan K.R and Zain F.Abu Sheer(2013). In this paper, we applied the New Iterative Method (NIM) to obtain the solution of partial differential equation systems. This method was developed by Daftardar-Gejji and Jafari in 2006 Daftargar-Geji V. and Jafari H(2006) and has been extensively used by many researchers for the treatment of linear and nonlinear ordinary and partial differential equations of integer and fractional order such as fractional physical differential equations Hemeda A.A.(2013), higher order KDV Manoj Kumar and Anuj shanker Saxena (2016), Fractional Gas Dynamics and Coupled Burger’s Equations Al-luhaibi Mohammed S.(2015), Nonlinear Abel type integral equation Gupta Praveen Kumar(2012), linear and nonlinear Klein-Gordon Equation Yaseem M. and Samraiz M. (2012)  and many more. This method is highly accurate and requires reduced amount of calculations compared with the existing iterative methods.Considering a nonlinear system of partial differential equation in an operator form asLt u+R1 (u,v,w)+N1 (u,v,w)=g1       Lt v+R2 (u,v,w)+N2 (u,v,w)=g2 ...........(1)Lt w+R3 (u,v,w)+N3 (u,v,w)=g3       With initial datau(x,y,o)=f1 (x) v(x,y,o)=f2 (x)       w(x,y,o)=f3 (x)        Where Lt is considered a first order partial differential operator, Rj , 1 ≤  j  ≤  3 and  Nj , 1 ≤  j  ≤  3 , are linear and nonlinear operators respectively, and g1,g2  and  g3 are source terms [Wazwaz]MATERIALS AND METHODSNEW ITERATIVE METHOD (NIM)To illustrate the idea of the NIM, we consider the following general functional equation:u=f+N(u) ..........................................(2)where  N is a nonlinear operator and f is a given function. We can find the solution of equation (2) having the series form                               ..............................................3 .....4Substituting equations (3) and (4) into equation (2) gives  .......5We define the recurrence relation of equation in the following way:u0=fu1=N(u_0)u2=N(u0+u1 )-N(u0)              .................................(6)u3=N(u0+u1+u2 )-N(u0+u1), n =1, 2,then ; m = 1,2, 3and     .....................................7The approximate solution of (2) is given by u(x,t)=u0 +u1+u2+u3+...      ...........................8APPLICATIONSIn this section, we used the New Iterative Method (NIM) to solve homogeneous and inhomogeneous linear nonlinear system of partial differential equations.RESULTS AND DISCUSSIONThe homogeneous linear system of PDEConsider the homogeneous linear systemut+vx=0    vt+ux=0           ..................................................9IC:  u(x,0)=ex   ,v(x,0)=e-x                    THE NEW ITERATIVE METHOD FOR SOLVING LINEAR AND NONLINEAR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONSFULafia Journal of Science & Technology Vol.6 No.3 September, 202044The exact solutions are:u(x,t)=ex Cosh t+e-x Sinh tv(x,t)=e-x Cosh t-ex Sinh taccording to the New Iterative Method (NIM), we have      ..............................10   .................................11         ........................................12from the initial condition, we havef1=u0=ex   and  f2=v0= e-xusing (12), When k = 0u1=te-xu1=te-xv1=-texu2=N(u0+u1 )-N(u0)v2=N(v0+v1 )-N(v0)Accordingly, we obtain the successive approximations:   u0=ex v0= e-xu1=te-x v1= -texu3=v3=Therefore the solution is as followsWe know that the Taylor series of:Therefore, the exact solutions are:u(x,t)= ex Cosh t+e-x Sinh t  v(x,t)= e-x Cosh t-ex Sinh t  ...........................(14)The inhomogeneous linear systemConsider the inhomogeneous linear systemut-vx-(u-v)=-2    vt+ux-(u-v)=-2      ............................(15)IC:  u(x,0)=1+ex   ,v(x,0)=-1+e-x                   The exact solutions are:u(x,t)=1+ex+tv(x,t)=-1+ex-taccording to the New Iterative Method (NIM), we have  ...............16       ............17    ...............18from the initial condition, we havef1=u0=1+ex   and  f2=v0= -1+e-xusing (18), When k = 0,    THE NEW ITERATIVE METHOD FOR SOLVING LINEAR AND NONLINEAR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONSFULafia Journal of Science & Technology Vol.6 No.3 September, 202045u1=texv1=-texu2=N(u0+u1 )-N(u0)v2=N(v0+v1 )-N(v0)Accordingly, we obtain the successive approximations:   u0=1+ex v0= -1+exu1=tex v1= -tex                                    ...........................19u3=v3=Therefore the solution is as followsWe know that the Taylor series of:Therefore, the exact solutions are:u(x,t)= 1+ex.et=1+ex+t   v(x,t)= -1+ex.e-t=1+ex-t  ......................................20The inhomogeneous nonlinear systemConsider the inhomogeneous nonlinear systemut+vux+u=1    vt-uvx-v=1       .....................................................21IC:  u(x,0)=ex   ,v(x,0)=e-x                    The exact solutions are:u(x,t)=ex-tv(x,t)=e-x+taccording to the New Iterative Method (NIM), we have   and   ..............22from the initial condition, we havef1=u0=ex   and  f2=v0= e-xusing (22), When k = 0,  u1=-texN(v0)=v1=∫0t(1+u0 v0x+v0)dt=∫0te-x dt=te-x v1=te-xu2=N(u0+u1 )-N(u0)v2=N(v0+v1 )-N(v0)Accordingly, we obtain the successive approximations:u0=exv0= e-xu1=-tex v1= te-x                   ..................................23u3=THE NEW ITERATIVE METHOD FOR SOLVING LINEAR AND NONLINEAR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONSFULafia Journal of Science & Technology Vol.6 No.3 September, 202046Therefore the solution is as followsWe know that the Taylor series of:Therefore, the exact solutions are:u(x,t)= ex.e-t=ex-t   v(x,t)= e-x.et=e-x+t  .......................24The homogeneous nonlinear systemConsider the homogeneous nonlinear systemut+ux vx+uy vy+u=0    vt+vx wx-vy wy-v=0 wt+wx ux+wy uy-w=0                          ..........25IC:  u(x,y,0)=ex+y  ,v(x,y,0)=ex-y    ,w(x,y,0)=e-x+y                 The exact solutions are:u(x,y,t)=ex+y-tv(x,y,t)=ex-y+tw(x,y,t)=e-x+y+taccording to the New Iterative Method (NIM), we have    .........26        .........27       ........28Therefore,      .........29from the initial condition, we havef1=u0=ex+y    ,f2=v0= ex-y    and   f3=w0=e-x+y using (29), When k = 0,   u1=-tex+yv1=tex-yw1=te-x+yu2=N(u0+u1 )-N(u0 )=accordingly, we obtain the successive approximations:  u0=ex+yv0= ex-y  w0=e-x+yu1=-tex+y v1= tex-y  w1= te-x+yu3=-Therefore the solution is as followsThe Taylor series of:THE NEW ITERATIVE METHOD FOR SOLVING LINEAR AND NONLINEAR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONSFULafia Journal of Science & Technology Vol.6 No.3 September, 202047therefore, the exact solutions are:u(x,y,t)= ex+y.e-t=ex+y-t   v(x,y,t)= ex-y.et=ex-y+t  ............................(31)w(x,y,t)= e-x+y.et=e-x+y+tCONCLUSIONIn this work, the New Iterative Method (NIM) is employed for the solution of systems of linear and nonlinear partial differential equations. This method was applied to four examples successfully and the results show that NIM is a powerful and efficient Mathematical tool for solving systems of linear and nonlinear partial differential equations.  The method reduces the size of computational work thus the method is powerful and effective and can be utilized to tackle complex situations arising out of real world.REFERENCESAbdul-Majid Wazwaz (2009) ‘Partial Differential Equations and Solitary Waves Theory’ Higher Education  Press, Beijing and Springer-Verlag Berlin Heidelberg.Abdul-Majid Wazwaz (2007): ‘The variational iteration method for solving linear and nonlinear systems of  PDEs’, Computers and Mathematics with Applications 54,) 895–902M. Akbarzade(2011); ‘Application of Variational Iteration Method to Partial Differential Equation Systems’,  Int. Journal of Math. Analysis,  5(18) , 863 – 870Abdul-Majid Wazwaz,(2000): ‘The Decomposition Method Applied to Systems of Partial Differential Equations  and to the Reaction-Diffusion Brusselator’, Applied Mathematics and Computation,  110(2,3,15) ,  251-264.Mohammed E. A. Rabie and Tarig M. Elzali,(2014): ‘A Study of some Systems of Nonlinear Partial Differential  Equations by using Adomian and Modified Decomposition Methods’, African Journal of Mathematics  and Computer Science Research,  7(6),  61-67, October,  ISSN 2006-9731, Article Number: 98918D47930.Mohand M. Abdelrahim Mahgoub and Abdelilah K. Hassan Sedeeg(2017):. ‘An Efficient Method for Solving  Linear and Nonlinear System of Partial Differential Equations’ , British Journal of Mathematics &  Computer Science, 20(1): 1-10,  Article no.BJMCS.30259,ISSN: 2231-0851Jasem Fadaei(2011):, ‘Application of Laplace−Adomian Decomposition Method on Linear and Nonlinear  System of PDEs’ Applied Mathematical Sciences,  5(. 27), 1307 – 1315Jafar Biazar and Mostafa Eslami,(2011): ‘A new homotopy perturbation method for solving systems of partial  differential equations, Computers and Mathematics with Applications 62 , 225–234Raslan K. R. and Zain F. Abu Sheer(2013):, ‘Differential transform method for solving non-linear systems of  partial differential equations’ International Journal of Physical Sciences,  8(38),  1880-1884, , ISSN 1992 – 1950.Daftargar-Gejji V. and Jafari H.(2006) , ‘An iterative method for solving nonlinear functional equations’, J.  Math. Anal. Appl. 316., 753 – 763Hemeda A. A.(2013):, ‘New Iterative Method: An Application for solving Fractional Physical Differential  Equations’, Abstract and Applied Analysis, 13Manoj Kumar and Anuj Shanker Saxena(2016):, ‘New Iterative Method for solving higher order KDV  equations’, 4th International Conference on Science, Technology and Management (ICSTM-16): ,  ISBN 978-81-932074-8-2Al-luhaibi Mohamed S.(2015):  ‘New Iterative Method for Fractional Gas Dynamics and Coupled Burger’s  Equations’, The Scientific World Journal Gupta Praveen Kumar (2012), ‘Modified New Iterative Method for Solving Nonlinear Abel Type Integral  Equations’, International Journal of Nonlinear Science, .14(.3)  307-315Yaseem M. and Samraiz M.(2012):  ‘The Modified New Iterative Method for Solving Linear and Nonlinear  Klein-Gordon Equations’, Applied Mathematical Sciences,  6( 60), 2979-2987, 2012THE NEW ITERATIVE METHOD FOR SOLVING LINEAR AND NONLINEAR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS

THE NEW ITERATIVE METHOD FOR SOLVING LINEAR AND NONLINEAR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS

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THE NEW ITERATIVE METHOD FOR SOLVING LINEAR AND NONLINEAR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS

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Faculty of Science, Federal University of Lafia