AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 412948 10.1155/2012/412948 412948 Research Article Application of Sumudu Decomposition Method to Solve Nonlinear System of Partial Differential Equations Eltayeb Hassan 1 Kılıçman Adem 2 Mursaleen M. 1 Mathematics Department College of Science King Saud University P.O. Box 2455 Riyadh 11451 Saudi Arabia ksu.edu.sa 2 Department of Mathematics and Institute for Mathematical Research Universiti Putra Malaysia, Serdang 43400 Selangor Malaysia upm.edu.my 2012 31 10 2012 2012 26 08 2012 22 10 2012 2012 Copyright © 2012 Hassan Eltayeb and Adem Kılıçman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We develop a method to obtain approximate solutions of nonlinear system of partial differential equations with the help of Sumudu decomposition method (SDM). The technique is based on the application of Sumudu transform to nonlinear coupled partial differential equations. The nonlinear term can easily be handled with the help of Adomian polynomials. We illustrate this technique with the help of three examples, and results of the present technique have close agreement with approximate solutions obtained with the help of Adomian decomposition method (ADM).

1. Introduction

Most of phenomena in nature are described by nonlinear differential equations. So scientists in different branches of science try to solve them. But because of nonlinear part of these groups of equations, finding an exact solution is not easy. Different analytical methods have been applied to find a solution to them. For example, Adomian has presented and developed a so-called decomposition method for solving algebraic, differential, integrodifferential, differential-delay and partial differential equations. In the nonlinear case for ordinary differential equations and partial differential equations, the method has the advantage of dealing directly with the problem [1, 2]. These equations are solved without transforming them to more simple ones. The method avoids linearization, perturbation, discretization, or any unrealistic assumptions [3, 4]. It was suggested in  that the noise terms appears always for inhomogeneous equations. Most recently, Wazwaz  established a necessary condition that is essentially needed to ensure the appearance of “noise terms” in the inhomogeneous equations. In the present paper, the intimate connection between the Sumudu transform theory and decomposition method arises in the solution of nonlinear partial differential equations is demonstrated.

The Sumudu transform is defined over the set of the functions (1.1)A={f(t):M,τ1,τ2>0,|f(t)|<Met/τj,if  t(-1)j×[0,)} by the following formula: (1.2)G(u)=S[f(t);u]=0f(ut)e-tdt,u(-τ1,τ2).

The existence and the uniqueness were discussed in , for further details and properties of the Sumudu transform and its derivatives we refer to . In , some fundamental properties of the Sumudu transform were established.

In , this new transform was applied to the one-dimensional neutron transport equation. In fact one can easily show that there is a strong relationship between double Sumudu and double Laplace transforms, see .

Further in , the Sumudu transform was extended to the distributions and some of their properties were also studied in . Recently Kılıçman et al. applied this transform to solve the system of differential equations, see .

A very interesting fact about Sumudu transform is that the original function and its Sumudu transform have the same Taylor coefficients except a factor n!. Thus if (1.3)f(t)=n=0antn then (1.4)F(u)=n=0n!antn, see .

Similarly, the Sumudu transform sends combinations, C(m,n), into permutations, P(m,n) and hence it will be useful in the discrete systems. Further (1.5)S(H(t))=£(δ(t))=1,£(H(t))=S(δ(t))=1u.

Thus we further note that since many practical engineering problems involve mechanical or electrical systems where action is defined by discontinuous or impulsive forcing terms. Then the Sumudu transform can be effectively used to solve ordinary differential equations as well as partial differential equations and engineering problems. Recently, the Sumudu transform was introduced as a new integral transform on a time scale 𝕋 to solve a system of dynamic equations, see . Then the results were applied on ordinary differential equations when 𝕋=, difference equations when 𝕋=0, but also, for q-difference equations when 𝕋=q0, where q0:={qt:t0  for  q>1} or 𝕋=q¯:=q{0} for q>1 which has important applications in quantum theory and on different types of time scales like 𝕋=h0, 𝕋=02, and 𝕋=𝕋n the space of the harmonic numbers. During this study we use the following Sumudu transform of derivatives.

Theorem 1.1.

Let f(t) be in A, and let Gn(u) denote the Sumudu transform of the nth derivative, fn(t) of f(t), then for n1(1.6)Gn(u)=G(u)un-k=0n-1f(k)(0)un-k. For more details, see .

We consider the general inhomogeneous nonlinear equation with initial conditions given below: (1.7)LU+RU+NU=h(x,t), where L is the highest order derivative which is assumed to be easily invertible, R is a linear differential operator of order less than L, NU represents the nonlinear terms and h(x,t) is the source term. First we explain the main idea of  SDM: the method consists of applying Sumudu transform (1.8)S[LU]+S[RU]+S[NU]=S[h(x,t)].

Using the differential property of Laplace transform and initial conditions we get (1.9)1unS[U(x,t)]-1unU(x,0)-1un-1U(x,0)--Un-1(x,0)u+S[RU]+S[NU]=S[h(x,t)].

By arrangement we have (1.10)S[U(x,t)]=U(x,0)+uU(x,0)++un-1Un-1(x,0)-unS[RU]-unS[NU]+unS[h(x,t)].

The second step in Sumudu decomposition method is that we represent solution as an infinite series: (1.11)U(x,t)=i=0Ui(x,t) and the nonlinear term can be decomposed as (1.12)NU(x,t)=i=0Ai, where Ai are Adomian polynomials  of  U0,  U1,  U2,  ,  Un and it can be calculated by formula (1.13)Ai=1i!didλi[Ni=0λiUi]λ=0,i=0,1,2,.

Substitution of (1.11) and (1.12) into (1.10) yields (1.14)S[i=0Ui(x,t)]=U(x,0)+uU(x,0)++un-1Un-1(x,0)-unS[RU(x,t)]-unS[i=0Ai]+unS[h(x,t)].

On comparing both sides of (1.14) and by using standard ADM we have: (1.15)S[U0(x,t)]=U(x,0)+uU(x,0)++un-1Un-1(x,0)+unS[h(x,t)]=Y(x,u) then it follows that (1.16)S[U1(x,t)]=-unS[RU0(x,t)]-unS[A0],S[U2(x,t)]=-unS[RU1(x,t)]-unS[A1].

In more general, we have (1.17)S[Ui+1(x,t)]=-unS[RUi(x,t)]-unS[Ai],i0.

On applying the inverse Sumudu transform to (1.15) and (1.17), we get (1.18)U0(x,t)=K(x,t),Ui+1(x,t)=-S-1[unS[RUi(x,t)]+unS[Ai]],i0, where K(x,t) represents the term that is arising from source term and prescribed initial conditions. On using the inverse Sumudu transform to h(x,t) and using the given condition we get (1.19)Ψ=Φ+S-1[h(x,t)], where the function Ψ, obtained from a term by using the initial condition is given by (1.20)Ψ=Ψ0+Ψ1+Ψ2+Ψ3++Ψn, the terms Ψ0,Ψ1,Ψ2,Ψ3,,Ψn appears while applying the inverse Sumudu transform on the source term h(x,t) and using the given conditions. We define (1.21)U0=Ψk++Ψk+r, where k=0,1,,n, r=0,1,,n-k. Then we verify that U0 satisfies the original equation (1.7). We now consider the particular form of inhomogeneous nonlinear partial differential equations: (1.22)LU+RU+NU=h(x,t) with the initial condition (1.23)U(x,0)=f(x),Ut(x,0)=g(x), where L=2/t2 is second-order differential operator, NU represents a general non-linear differential operator where as h(x,t) is source term. The methodology consists of applying Sumudu transform first on both sides of (1.10) and (1.23), (1.24)S[U(x,t)]=f(x)+ug(x)-u2S[RU]-u2S[NU]+u2S[h(x,t)].

Then by the second step in Sumudu decomposition method and inverse transform as in the previous we have (1.25)U(x,t)=f(x)+tg(t)-S-1[u2S[RU]-u2S[NU]]+S-1[u2S[h(x,t)]].

2. Applications

Now in order to illustrate STDM we consider some examples. Consider a nonlinear partial differential equation (2.1)Utt+U2-Ux  2=0,t>0 with initial conditions (2.2)U(x,0)=0,Ut(x,0)=ex.

By taking Sumudu transform for (2.1) and (2.2) we obtain (2.3)S[U(x,t)]=uex+u2S[Ux2-U2].

By applying the inverse Sumudu transform for (2.3), we get (2.4)[U(x,t)]=tex+S-1[u2S[Ux2-U2]] which assumes a series solution of the function U(x,t) and is given by (2.5)U(x,t)=i=0Ui(x,t).

Using (2.4) into (2.5) we get (2.6)i=0Ui(x,t)=tex+S-1[u2S[i=0Ai(U)-i=0Bi(U)]].

In (2.6) Ai(u) and Bi(u) are Adomian polynomials that represents nonlinear terms. So Adomian polynomials are given as follows: (2.7)i=0Ai(U)=Ux2,i=0Ai(U)=U2.

The few components of the Adomian polynomials are given as follows: (2.8)A0(U)=U0x2,A1(U)=2U0x  U1x  ,  Ai(U)=r=0iUrx  Ui-rx  ,B0(U)=U02,B1(U)=2U0U1,Bi(U)r=0iUr  Ui-r  .

From the above equations we obtain (2.9)U0(x,t)=tex,Ui+1(x,t)=S-1[S[i=0Ai(U)-i=0Bi(U)]],n0.

Then the first few terms of Ui(x,t) follow immediately upon setting (2.10)U1(x,t)=S-1[u2S[i=0A0(U)-i=0B0(U)]]=S-1[u2S[U0x2-U02]]=S-1[u2S[t2e2x-t2e2x]]=S-1[u2S]=0.

Therefore the solution obtained by LDM is given as follows: (2.11)U(x,t)=i=0Ui(x,t)=tex.

Example 2.1.

Consider the system of nonlinear coupled partial differential equation (2.12)Ut(x,y,t)-VxWy=1,Vt(x,y,t)-WxUy=5,Wt(x,y,t)-UxVy=5 with initial conditions (2.13)U(x,y,0)=x+2y,V(x,y,0)=x-2y,W(x,y,0)=-x+2y.

Applying the Sumudu transform (denoted by S) we have (2.14)U(x,y,u)=x+2y+u+uS[VxWy],V(x,y,u)=x-2y+5u+uS[WxUy],W(x,y,u)=-x+2y+5u+uS[UxVy].

On using inverse Sumudu transform in (2.14), our required recursive relation is given by (2.15)U(x,y,t)=x+2y+t+S-1[uS[VxWy]],V(x,y,t)=x-2y+5t+S-1[uS[WxUy]],U(x,y,t)=-x+2y+5t+S-1[uS[UxVy]].

The recursive relations are (2.16)U0(x,y,t)=t+x+2y,Ui+1(x,y,t)=S-1[uS[i=0Ci(V,W)]],i0,V0(x,y,t)=5t+x-2y,Vi+1(x,y,t)=S-1[uS[i=0Di(U,W)]],i0,W0(x,y,t)=5t-x+2y,Wi+1(x,y,t)=S-1[uS[i=0Ei(U,V)]],i0, where Ci(V,W),  Di(U,W), and Ei(U,V) are Adomian polynomials representing the nonlinear terms  in above equations. The few components of Adomian polynomials are given as follows (2.17)C0(V,W)=V0xW0y,C1(V,W)=V1xW0y+V0xW1y,Ci(V,W)=r=0iVrxWi-ry,D0(U,W)=U0yW0x,D1(U,W)=U1yW0x+W1xU0y,Di(U,W)=r=0iWrxUi-ry,E0(U,V)=U0xV0y,E1(U,V)=U1xV0y+U0xV1y,Ei(V,W)=r=0iUrxVi-ry.

By this recursive relation we can find other components of the solution (2.18)U1(x,y,t)=S-1[uS[C0(V,W)]]=S-1[uS[V0xW0y]]=S-1[uS[(1)(2)]]=2t,V1(x,y,t)=S-1[uS[D0(U,W)]]=S-1[uS[W0xU0y]]=S-1[uS[(-1)(2)]]=-2t,W1(x,y,t)=S-1[uS[E0(U,V)]]=S-1[uS[U0xV0y]]=S-1[uS[(1)(-2)]]=-2t,U2(x,y,t)=S-1[uS[C1(V,W)]]=S-1[uS[V1xW0y+V0xW1y]]=0,V2(x,y,t)=S-1[uS[D1(U,W)]]=S-1[uS[U0yW1x+U1yW0x]]=0,W2(x,y,t)=S-1[uS[D1(U,V)]]=S-1[uS[U1xV0y+U0xV1y]]=0.

The solution of above system is given by (2.19)U(x,y,t)=i=0Ui(x,y,t)=x+2y+3t,V(x,y,t)=i=0Vi(x,y,t)=x-2y+3t,W(x,y,t)=i=0Wi(x,y,t)=-x+2y+3t.

Example 2.2.

Consider the following homogeneous linear system of PDEs: (2.20)Ut(x,t)-Vx(x,t)-(U-V)=2,Vt(x,t)+Ux(x,t)-(U-V)=2, with initial conditions (2.21)U(x,0)=1+ex,V(x,0)=-1+ex.

Taking the Sumudu transform on both sides of (2.20), then by using the differentiation property of Sumudu transform and initial conditions, (2.21) gives (2.22)S[U(x,t)]=1+ex-2u+uS[Vx]+uS[U-V],S[V(x,t)]=-1+ex-2u-uS[Ux]+uS[U-V],(2.23)Ux(x,t)=i=0Uxi(x,t),Vx(x,t)=i=0Vxi(x,t).

Using the decomposition series (2.23) for the linear terms U(x,t), V(x,t) and Ux, Vx, we obtain (2.24)S[i=0Ui(x,t)]=1+ex-2u+uS[i=0Vix]+uS[i=0Ui-i=0Vi],S[i=0Vi(x,t)]=-1+ex-2u-uS[i=0Uix]+uS[i=0Ui-i=0Vi].

The SADM presents the recursive relations (2.25)S[U0(x,t)]=1+ex-2u,S[V0(x,t)]=-1+ex-2u,S[Ui+1]=uS[Vix]+uS[Ui-Vi],i0,S[Vi+1]=-uS[Uix]+uS[Ui-Vi],i0.

Taking the inverse Sumudu transform of both sides of (2.25) we have (2.26)U0(x,t)=1+ex-2t,V0(x,t)=-1+ex-2t,U1=S-1[uS[V0x]+uS[U0-V0]]=S-1[uex+2u]=tex+2t,        V1=S-1[-uS[U0x]+uS[U0-V0]]=S-1[-uex+2u]=-tex+2t,U2=S-1[u2ex]=t22!ex,V2=S-1[u2ex]=t22!ex, and so on for other components. Using (1.11), the series solutions are given by (2.27)U(x,t)=1+ex(1+t+t22!+t33!),V(x,t)=-1+ex(1-t+t22!-t33!).

Then the solutions follows (2.28)U(x,t)=1+ex+t,V(x,t)=-1+ex-t.

Example 2.3.

Consider the system of nonlinear partial differential equations (2.29)Ut+VUx+U=1,Vt-UVx-V=1 with initial conditions (2.30)U(x,0)=ex,V(x,0)=e-x.

On using Sumudu transform on both sides of (2.29), and by taking Sumudu transform for the initial conditions of (2.30) we get (2.31)S[U(x,t)]=ex+u-uS[VUx]-uS[U],S[V(x,t)]=ex+u+uS[UVx]+uS[V].

Similar to the previous example, we rewrite U(x,t) and V(x,t) by the infinite series (1.11), then inserting these series into both sides of (2.31) yields (2.32)S[i=0Ui(x,t)]=ex+u-uS[i=0Ai]-uS[i=0Ui],S[i=0Vi(x,t)]=e-x+u+uS[i=0Bi]-uS[i=0Vi], where the terms Ai and Bi are handled with the help of Adomian polynomials by (1.12) that represent the nonlinear terms VUx and UVx, respectively. We have a few terms of the Adomian polynomials for VUx and UVx which are given by (2.33)A0=U0xV0,A1=U0xV1+U1xV0,  A2=U0xV2+U1xV1+U2xV0,B0=V0xU0,B1=V0xU1+V1xU0,  B2=V0xU2+V1xU1+V2xU0,

By taking the inverse Sumudu transform we have (2.34)U0=ex+t,V0=e-x+t,(2.35)Ui+1=S-1[u]-S-1[uS[Ai]]-S-1[uS[Ui]],Vi+1=S-1[u]+S-1[uS[Bi]]+S-1[uS[Vi]].

Using the inverse Sumudu transform on (2.35) we have (2.36)U1=-t-t22!-tex-t22!ex,V1=-t-t22!+te-x-t22!e-x,U2=t22!+t22!ex,V2=t22!+t22!e-x.

The rest terms can be determined in the same way. Therefore, the series solutions are given by (2.37)U(x,t)=ex(1-t+t22!-t33!),V(x,t)=e-x(1+t+t22!+t33!).

Then the solution for the above system is as follows: (2.38)U(x,t)=ex-t,V(x,t)=e-x+t.

3. Conclusion

The Sumudu transform-Adomian decomposition method has been applied to linear and nonlinear systems of partial differential equations. Three examples have been presented, this method shows that it is very useful and reliable for any nonlinear partial differential equation systems. Therefore, this method can be applied to many complicated linear and nonlinear PDEs.

Acknowledgment

The authors would like to express their sincere thanks and gratitude to the reviewers for their valuable comments and suggestions for the improvement of this paper. The first author acknowledges the support by the Research Center, College of Science, King Saud University. The second author also gratefully acknowledges the partial support by the University Putra Malaysia under the Research University Grant Scheme (RUGS) 05-01-09-0720RU and FRGS 01-11-09-723FR.

Alizadeh E. Farhadi M. Sedighi K. Ebrahimi-Kebria H. R. Ghafourian A. Solution of the Falkner-Skan equation for wedge by Adomian Decomposition method Communications in Nonlinear Science and Numerical Simulation 2009 14 3 724 733 Wazwaz A.-M. The modified decomposition method and Padé approximants for a boundary layer equation in unbounded domain Applied Mathematics and Computation 2006 177 2 737 744 10.1016/j.amc.2005.09.102 2291999 Bellomo N. Monaco R. A. A comparison between Adomian's decomposition methods and perturbation techniques for nonlinear random differential equations Journal of Mathematical Analysis and Applications 1985 110 2 495 502 10.1016/0022-247X(85)90311-7 805271 ZBL0575.60064 Race R. On the Adomian decomposition method and comparison with Picard’s method Journal of Mathematical Analysis and Applications 1987 128 480 483 Adomian G. Rach R. Noise terms in decomposition solution series Computers & Mathematics with Applications 1992 24 11 61 64 10.1016/0898-1221(92)90031-C 1186719 ZBL0777.35018 Wazwaz A. M. Necessary conditions for the appearance of noise terms in decomposition solution series Applied Mathematics and Computation 1987 81 1718 1740 Kılıçman A. Eltayeb H. A note on integral transforms and partial differential equations Applied Mathematical Sciences 2010 4 3 109 118 2576040 ZBL1194.35017 Asiru M. A. Sumudu transform and the solution of integral equations of convolution type International Journal of Mathematical Education in Science and Technology 2001 32 6 906 910 10.1080/002073901317147870 1872543 ZBL1008.45003 Aşiru M. A. Further properties of the Sumudu transform and its applications International Journal of Mathematical Education in Science and Technology 2002 33 3 441 449 10.1080/002073902760047940 1912446 ZBL1013.44001 Kadem A. Kılıçman A. Note on transport equation and fractional Sumudu transform Computers & Mathematics with Applications 2011 62 8 2995 3003 10.1016/j.camwa.2011.08.009 2837734 ZBL1232.44002 Eltayeb H. Kılıçman A. Fisher B. A new integral transform and associated distributions Integral Transforms and Special Functions 2010 21 5 367 379 10.1080/10652460903335061 2666504 ZBL1191.35017 Kılıçman A. Eltayeb H. On the applications of Laplace and Sumudu transforms Journal of the Franklin Institute 2010 347 5 848 862 10.1016/j.jfranklin.2010.03.008 Kılıçman A. Eltayeb H. Agarwal R. P. On Sumudu transform and system of differential equations Abstract and Applied Analysis 2010 2010 11 598702 10.1155/2010/598702 2609064 ZBL1197.34001 Kılıçman A. Eltayeb H. Atan K. A. M. A note on the comparison between Laplace and Sumudu transforms Bulletin of the Iranian Mathematical Society 2011 37 1 131 141 2850110 ZBL1242.44001 Agwa H. Ali F. Kılıçman A. A new integral transform on time scales and its applications Advances in Difference Equations 2012 2012, article 60 Belgacem F. B. M. Karaballi A. A. Kalla S. L. Analytical investigations of the Sumudu transform and applications to integral production equations Mathematical Problems in Engineering 2003 2003 3 103 118 10.1155/S1024123X03207018 2032184