Application of Sumudu Decomposition Method to Solve Nonlinear System of Partial Differential Equations

Abstract

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 [5] that the noise terms appears always for inhomogeneous equations. Most recently, Wazwaz [6] 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 𝐴=𝑓(𝑡)∶∃𝑀,𝜏₁,𝜏₂>0,𝑓(𝑡)<𝑀𝑒^{𝑡/𝜏𝑗},if𝑡∈(−1)^𝑗×[0,∞) (1.1) by the following formula: 𝐺(𝑢)𝑓(𝑡)==𝑆;𝑢_∞0𝑓(𝑢𝑡)𝑒^−𝑡𝑑𝑡,𝑢∈−𝜏₁,𝜏₂. (1.2)

The existence and the uniqueness were discussed in [7], for further details and properties of the Sumudu transform and its derivatives we refer to [8]. In [9], some fundamental properties of the Sumudu transform were established.

In [10], 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 [7].

Further in [11], the Sumudu transform was extended to the distributions and some of their properties were also studied in [12]. Recently Kılıçman et al. applied this transform to solve the system of differential equations, see [13].

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 𝑓(𝑡)=^∞_𝑛=0𝑎_𝑛𝑡^𝑛 (1.3) then 𝐹(𝑢)=^∞_𝑛=0𝑛!𝑎_𝑛𝑡^𝑛, (1.4) see [14].

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 𝑆(𝐻(𝑡(𝛿(𝑡£(𝐻(𝑡(𝛿(𝑡=1))=£))=1,))=𝑆))𝑢. (1.5)

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 [15]. Then the results were applied on ordinary differential equations when 𝕋=ℝ, difference equations when 𝕋=ℕ₀, but also, for 𝑞-difference equations when 𝕋=𝑞^ℕ0, where 𝑞^ℕ0∶={𝑞^𝑡∶𝑡∈ℕ₀for𝑞>1} or 𝕋=𝑞^ℤ∶=𝑞^ℤ∪{0} for 𝑞>1 which has important applications in quantum theory and on different types of time scales like 𝕋=ℎℕ₀, 𝕋=ℕ²₀, and 𝕋=𝕋_𝑛 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 𝐺^𝑛(𝑢) denote the Sumudu transform of the 𝑛th derivative, 𝑓^𝑛(𝑡) of 𝑓(𝑡), then for 𝑛≥1𝐺^𝑛(𝑢)=𝐺(𝑢)𝑢^𝑛−^𝑛−1_𝑘=0𝑓^(𝑘)(0)𝑢^𝑛−𝑘. (1.6) For more details, see [16].

We consider the general inhomogeneous nonlinear equation with initial conditions given below: 𝐿𝑈+𝑅𝑈+𝑁𝑈=ℎ(𝑥,𝑡), (1.7) 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 𝑆[𝐿𝑈]+𝑆[𝑅𝑈]+𝑆[𝑁𝑈]=𝑆[ℎ(𝑥,𝑡)]. (1.8)

Using the differential property of Laplace transform and initial conditions we get 1𝑢^𝑛𝑆[𝑈(−1𝑥,𝑡)]𝑢^𝑛𝑈()−1𝑥,0𝑢^𝑛−1𝑈^′()𝑈𝑥,0−⋯−^𝑛−1(𝑥,0)𝑢+𝑆[𝑅𝑈]+𝑆[𝑁𝑈]=𝑆[ℎ(𝑥,𝑡)]. (1.9)

By arrangement we have 𝑆[𝑈(𝑥,𝑡)]=𝑈(𝑥,0)+𝑢𝑈^′(𝑥,0)+⋯+𝑢^𝑛−1𝑈^𝑛−1(𝑥,0)−𝑢^𝑛𝑆[𝑅𝑈]−𝑢^𝑛𝑆[𝑁𝑈]+𝑢^𝑛𝑆[ℎ(𝑥,𝑡)]. (1.10)

The second step in Sumudu decomposition method is that we represent solution as an infinite series: 𝑈(𝑥,𝑡)=^∞_𝑖=0𝑈_𝑖(𝑥,𝑡) (1.11) and the nonlinear term can be decomposed as 𝑁𝑈(𝑥,𝑡)=^∞_𝑖=0𝐴_𝑖, (1.12) where 𝐴_𝑖 are Adomian polynomials [6] of  𝑈₀,  𝑈₁,  𝑈₂,  …,  𝑈_𝑛 and it can be calculated by formula 𝐴_𝑖=1𝑑𝑖!𝑑𝜆^𝑖𝑖𝑁^∞_𝑖=0𝜆^𝑖𝑈_𝑖𝜆=0,𝑖=0,1,2,…. (1.13)

Substitution of (1.11) and (1.12) into (1.10) yields 𝑆^∞_𝑖=0𝑈_𝑖(𝑥,𝑡)=𝑈(𝑥,0)+𝑢𝑈^′(𝑥,0)+⋯+𝑢^𝑛−1𝑈^𝑛−1(𝑥,0)−𝑢^𝑛𝑆[𝑅𝑈(𝑥,𝑡)]−𝑢^𝑛𝑆^∞_𝑖=0𝐴_𝑖+𝑢^𝑛𝑆[ℎ(𝑥,𝑡)]. (1.14)

On comparing both sides of (1.14) and by using standard ADM we have: 𝑆𝑈₀(𝑥,𝑡)=𝑈(𝑥,0)+𝑢𝑈^′(𝑥,0)+⋯+𝑢^𝑛−1𝑈^𝑛−1(𝑥,0)+𝑢^𝑛𝑆[ℎ(𝑥,𝑡)]=𝑌(𝑥,𝑢) (1.15) then it follows that 𝑆𝑈₁(𝑥,𝑡)=−𝑢^𝑛𝑆𝑅𝑈₀(𝑥,𝑡)−𝑢^𝑛𝑆𝐴₀,𝑆𝑈₂(𝑥,𝑡)=−𝑢^𝑛𝑆𝑅𝑈₁(𝑥,𝑡)−𝑢^𝑛𝑆𝐴₁. (1.16)

In more general, we have 𝑆𝑈_𝑖+1(𝑥,𝑡)=−𝑢^𝑛𝑆𝑅𝑈_𝑖(𝑥,𝑡)−𝑢^𝑛𝑆𝐴_𝑖,𝑖≥0. (1.17)

On applying the inverse Sumudu transform to (1.15) and (1.17), we get 𝑈₀()(),𝑈𝑥,𝑡=𝐾𝑥,𝑡_𝑖+1(𝑥,𝑡)=−𝑆⁻¹𝑢^𝑛𝑆𝑅𝑈_𝑖(𝑥,𝑡)+𝑢^𝑛𝑆𝐴_𝑖,𝑖≥0, (1.18) 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 Ψ=Φ+𝑆⁻¹[ℎ(𝑥,𝑡)], (1.19) where the function Ψ, obtained from a term by using the initial condition is given by Ψ=Ψ₀+Ψ₁+Ψ₂+Ψ₃+⋯+Ψ_𝑛, (1.20) the terms Ψ₀,Ψ₁,Ψ₂,Ψ₃,…,Ψ_𝑛 appears while applying the inverse Sumudu transform on the source term ℎ(𝑥,𝑡) and using the given conditions. We define 𝑈₀=Ψ_𝑘+⋯+Ψ_𝑘+𝑟, (1.21) 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: 𝐿𝑈+𝑅𝑈+𝑁𝑈=ℎ(𝑥,𝑡) (1.22) with the initial condition 𝑈(𝑥,0)=𝑓(𝑥),𝑈_𝑡(𝑥,0)=𝑔(𝑥), (1.23) 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), 𝑆[𝑈(𝑥,𝑡)]=𝑓(𝑥)+𝑢𝑔(𝑥)−𝑢²𝑆[𝑅𝑈]−𝑢²𝑆[𝑁𝑈]+𝑢²𝑆[ℎ(𝑥,𝑡)]. (1.24)

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

2. Applications

Now in order to illustrate STDM we consider some examples. Consider a nonlinear partial differential equation 𝑈_𝑡𝑡+𝑈²−𝑈²_𝑥=0,𝑡>0 (2.1) with initial conditions 𝑈()𝑈𝑥,0=0,_𝑡(𝑥,0)=𝑒^𝑥. (2.2)

By taking Sumudu transform for (2.1) and (2.2) we obtain 𝑆[𝑈(𝑥,𝑡)]=𝑢𝑒^𝑥+𝑢²𝑆𝑈²_𝑥−𝑈². (2.3)

By applying the inverse Sumudu transform for (2.3), we get [𝑈(𝑥,𝑡)]=𝑡𝑒^𝑥+𝑆⁻¹𝑢²𝑆𝑈²_𝑥−𝑈² (2.4) which assumes a series solution of the function 𝑈(𝑥,𝑡) and is given by 𝑈(𝑥,𝑡)=^∞_𝑖=0𝑈_𝑖(𝑥,𝑡). (2.5)

Using (2.4) into (2.5) we get ^∞_𝑖=0𝑈_𝑖(𝑥,𝑡)=𝑡𝑒^𝑥+𝑆⁻¹𝑢²𝑆^∞_𝑖=0𝐴_𝑖(𝑈)−^∞_𝑖=0𝐵_𝑖(𝑈). (2.6)

In (2.6) 𝐴_𝑖(𝑢) and 𝐵_𝑖(𝑢) are Adomian polynomials that represents nonlinear terms. So Adomian polynomials are given as follows: ^∞_𝑖=0𝐴_𝑖(𝑈)=𝑈²_𝑥,_∞𝑖=0𝐴_𝑖(𝑈)=𝑈². (2.7)

The few components of the Adomian polynomials are given as follows: 𝐴₀(𝑈)=𝑈²_0𝑥,𝐴₁(𝑈)=2𝑈_0𝑥𝑈_1𝑥,𝐴_𝑖(𝑈)=^𝑖_𝑟=0𝑈_𝑟𝑥𝑈_{𝑖−𝑟𝑥},𝐵₀(𝑈)=𝑈²₀,𝐵₁(𝑈)=2𝑈₀𝑈₁,𝐵_𝑖(𝑈)^𝑖_𝑟=0𝑈_𝑟𝑈_𝑖−𝑟. (2.8)

From the above equations we obtain 𝑈₀(𝑥,𝑡)=𝑡𝑒^𝑥,𝑈_𝑖+1(𝑥,𝑡)=𝑆⁻¹𝑆^∞_𝑖=0𝐴_𝑖(𝑈)−^∞_𝑖=0𝐵_𝑖(𝑈),𝑛≥0. (2.9)

Then the first few terms of 𝑈_𝑖(𝑥,𝑡) follow immediately upon setting 𝑈₁(𝑥,𝑡)=𝑆⁻¹𝑢²𝑆^∞_𝑖=0𝐴₀(𝑈)−^∞_𝑖=0𝐵₀(𝑈)=𝑆⁻¹𝑢²𝑆𝑈²_0𝑥−𝑈²₀=𝑆⁻¹𝑢²𝑆𝑡²𝑒^2𝑥−𝑡²𝑒^2𝑥=𝑆⁻¹𝑢²𝑆[0]=0. (2.10)

Therefore the solution obtained by LDM is given as follows: 𝑈(𝑥,𝑡)=^∞_𝑖=0𝑈_𝑖(𝑥,𝑡)=𝑡𝑒^𝑥. (2.11)

Example 2.1. Consider the system of nonlinear coupled partial differential equation 𝑈_𝑡𝑥,𝑦,𝑡−𝑉_𝑥𝑊_𝑦𝑉=1,_𝑡𝑥,𝑦,𝑡−𝑊_𝑥𝑈_𝑦𝑊=5,_𝑡𝑥,𝑦,𝑡−𝑈_𝑥𝑉_𝑦=5 (2.12) with initial conditions 𝑈𝑉𝑊𝑥,𝑦,0=𝑥+2𝑦,𝑥,𝑦,0=𝑥−2𝑦,𝑥,𝑦,0=−𝑥+2𝑦. (2.13)
Applying the Sumudu transform (denoted by 𝑆) we have 𝑈𝑥,𝑦,𝑢=𝑥+2𝑦+𝑢+𝑢𝑆𝑉_𝑥𝑊_𝑦,𝑉𝑥,𝑦,𝑢=𝑥−2𝑦+5𝑢+𝑢𝑆𝑊_𝑥𝑈_𝑦,𝑊𝑥,𝑦,𝑢=−𝑥+2𝑦+5𝑢+𝑢𝑆𝑈_𝑥𝑉_𝑦. (2.14)
On using inverse Sumudu transform in (2.14), our required recursive relation is given by 𝑈𝑥,𝑦,𝑡=𝑥+2𝑦+𝑡+𝑆⁻¹𝑢𝑆𝑉_𝑥𝑊_𝑦,𝑉𝑥,𝑦,𝑡=𝑥−2𝑦+5𝑡+𝑆⁻¹𝑢𝑆𝑊_𝑥𝑈_𝑦,𝑈𝑥,𝑦,𝑡=−𝑥+2𝑦+5𝑡+𝑆⁻¹𝑢𝑆𝑈_𝑥𝑉_𝑦. (2.15)
The recursive relations are 𝑈₀𝑈𝑥,𝑦,𝑡=𝑡+𝑥+2𝑦,_𝑖+1𝑥,𝑦,𝑡=𝑆⁻¹𝑢𝑆^∞_𝑖=0𝐶_𝑖(𝑉𝑉,𝑊),𝑖≥0,₀𝑉𝑥,𝑦,𝑡=5𝑡+𝑥−2𝑦,_𝑖+1𝑥,𝑦,𝑡=𝑆⁻¹𝑢𝑆^∞_𝑖=0𝐷_𝑖(𝑊𝑈,𝑊),𝑖≥0,₀𝑊𝑥,𝑦,𝑡=5𝑡−𝑥+2𝑦,_𝑖+1𝑥,𝑦,𝑡=𝑆⁻¹𝑢𝑆^∞_𝑖=0𝐸_𝑖(𝑈,𝑉),𝑖≥0, (2.16) where 𝐶_𝑖(𝑉,𝑊),  𝐷_𝑖(𝑈,𝑊), and 𝐸_𝑖(𝑈,𝑉) are Adomian polynomials representing the nonlinear terms [1] in above equations. The few components of Adomian polynomials are given as follows 𝐶₀(𝑉,𝑊)=𝑉_0𝑥𝑊_0𝑦,𝐶₁(𝑉,𝑊)=𝑉_1𝑥𝑊_0𝑦+𝑉_0𝑥𝑊_1𝑦,⋮𝐶_𝑖(𝑉,𝑊)=^𝑖_𝑟=0𝑉_𝑟𝑥𝑊_{𝑖−𝑟𝑦},𝐷₀(𝑈,𝑊)=𝑈_0𝑦𝑊_0𝑥,𝐷₁(𝑈,𝑊)=𝑈_1𝑦𝑊_0𝑥+𝑊_1𝑥𝑈_0𝑦,⋮𝐷_𝑖(𝑈,𝑊)=^𝑖_𝑟=0𝑊_𝑟𝑥𝑈_{𝑖−𝑟𝑦},𝐸₀(𝑈,𝑉)=𝑈_0𝑥𝑉_0𝑦,𝐸₁(𝑈,𝑉)=𝑈_1𝑥𝑉_0𝑦+𝑈_0𝑥𝑉_1𝑦,⋮𝐸_𝑖(𝑉,𝑊)=^𝑖_𝑟=0𝑈_𝑟𝑥𝑉_{𝑖−𝑟𝑦}. (2.17)
By this recursive relation we can find other components of the solution 𝑈₁𝑥,𝑦,𝑡=𝑆⁻¹𝑢𝑆𝐶₀(𝑉,𝑊)=𝑆⁻¹𝑢𝑆𝑉_0𝑥𝑊_0𝑦=𝑆⁻¹[12𝑉𝑢𝑆[()()]]=2𝑡,₁𝑥,𝑦,𝑡=𝑆⁻¹𝑢𝑆𝐷₀(𝑈,𝑊)=𝑆⁻¹𝑢𝑆𝑊_0𝑥𝑈_0𝑦=𝑆⁻¹[2𝑊𝑢𝑆[(−1)()]]=−2𝑡,₁𝑥,𝑦,𝑡=𝑆⁻¹𝑢𝑆𝐸₀(𝑈,𝑉)=𝑆⁻¹𝑢𝑆𝑈_0𝑥𝑉_0𝑦=𝑆⁻¹[1𝑈𝑢𝑆[()(−2)]]=−2𝑡,₂𝑥,𝑦,𝑡=𝑆⁻¹𝑢𝑆𝐶₁(𝑉,𝑊)=𝑆⁻¹𝑢𝑆𝑉_1𝑥𝑊_0𝑦+𝑉_0𝑥𝑊_1𝑦𝑉=0,₂𝑥,𝑦,𝑡=𝑆⁻¹𝑢𝑆𝐷₁(𝑈,𝑊)=𝑆⁻¹𝑢𝑆𝑈_0𝑦𝑊_1𝑥+𝑈_1𝑦𝑊_0𝑥𝑊=0,₂𝑥,𝑦,𝑡=𝑆⁻¹𝑢𝑆𝐷₁(𝑈,𝑉)=𝑆⁻¹𝑢𝑆𝑈_1𝑥𝑉_0𝑦+𝑈_0𝑥𝑉_1𝑦=0. (2.18)
The solution of above system is given by 𝑈𝑥,𝑦,𝑡=^∞_𝑖=0𝑈_𝑖𝑉=𝑥,𝑦,𝑡=𝑥+2𝑦+3𝑡,𝑥,𝑦,𝑡^∞_𝑖=0𝑉_𝑖𝑊=𝑥,𝑦,𝑡=𝑥−2𝑦+3𝑡,𝑥,𝑦,𝑡^∞_𝑖=0𝑊_𝑖𝑥,𝑦,𝑡=−𝑥+2𝑦+3𝑡. (2.19)

Example 2.2. Consider the following homogeneous linear system of PDEs: 𝑈_𝑡(𝑥,𝑡)−𝑉_𝑥()−()𝑉𝑥,𝑡𝑈−𝑉=2,_𝑡(𝑥,𝑡)+𝑈_𝑥(𝑥,𝑡)−(𝑈−𝑉)=2, (2.20) with initial conditions 𝑈(𝑥,0)=1+𝑒^𝑥,𝑉(𝑥,0)=−1+𝑒^𝑥. (2.21)
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 𝑆[𝑈(𝑥,𝑡)]=1+𝑒^𝑥−2𝑢+𝑢𝑆𝑉_𝑥[],𝑆[𝑉(+𝑢𝑆𝑈−𝑉𝑥,𝑡)]=−1+𝑒^𝑥−2𝑢−𝑢𝑆𝑈_𝑥[],𝑈+𝑢𝑆𝑈−𝑉_𝑥(𝑥,𝑡)=^∞_𝑖=0𝑈_𝑥𝑖(𝑥,𝑡),𝑉_𝑥(𝑥,𝑡)=^∞_𝑖=0𝑉_𝑥𝑖(𝑥,𝑡). (2.22)(2.23)
Using the decomposition series (2.23) for the linear terms 𝑈(𝑥,𝑡), 𝑉(𝑥,𝑡) and 𝑈_𝑥, 𝑉_𝑥, we obtain 𝑆^∞_𝑖=0𝑈_𝑖(𝑥,𝑡)=1+𝑒^𝑥−2𝑢+𝑢𝑆^∞_𝑖=0𝑉_𝑖𝑥+𝑢𝑆^∞_𝑖=0𝑈_𝑖−^∞_𝑖=0𝑉_𝑖,𝑆^∞_𝑖=0𝑉_𝑖(𝑥,𝑡)=−1+𝑒^𝑥−2𝑢−𝑢𝑆^∞_𝑖=0𝑈_𝑖𝑥+𝑢𝑆^∞_𝑖=0𝑈_𝑖−^∞_𝑖=0𝑉_𝑖. (2.24)
The SADM presents the recursive relations 𝑆𝑈₀(𝑥,𝑡)=1+𝑒^𝑥𝑆𝑉−2𝑢,₀(𝑥,𝑡)=−1+𝑒^𝑥𝑆𝑈−2𝑢,_𝑖+1=𝑢𝑆𝑉_𝑖𝑥+𝑢𝑆𝑈_𝑖−𝑉_𝑖𝑆𝑉,𝑖≥0,_𝑖+1=−𝑢𝑆𝑈_𝑖𝑥+𝑢𝑆𝑈_𝑖−𝑉_𝑖,𝑖≥0. (2.25)
Taking the inverse Sumudu transform of both sides of (2.25) we have 𝑈₀(𝑥,𝑡)=1+𝑒^𝑥𝑉−2𝑡,₀(𝑥,𝑡)=−1+𝑒^𝑥𝑈−2𝑡,₁=𝑆⁻¹𝑢𝑆𝑉_0𝑥+𝑢𝑆𝑈₀−𝑉₀=𝑆⁻¹𝑢𝑒^𝑥+2𝑢=𝑡𝑒^𝑥𝑉+2𝑡,₁=𝑆⁻¹−𝑢𝑆𝑈_0𝑥+𝑢𝑆𝑈₀−𝑉₀=𝑆⁻¹−𝑢𝑒^𝑥+2𝑢=−𝑡𝑒^𝑥𝑈+2𝑡,₂=𝑆⁻¹𝑢²𝑒^𝑥=𝑡²𝑒2!^𝑥,𝑉₂=𝑆⁻¹𝑢²𝑒^𝑥=𝑡²𝑒2!^𝑥, (2.26) and so on for other components. Using (1.11), the series solutions are given by 𝑈(𝑥,𝑡)=1+𝑒^𝑥𝑡1+𝑡+²+𝑡2!³⋯,𝑉()3!𝑥,𝑡=−1+𝑒^𝑥𝑡1−𝑡+²−𝑡2!³⋯.3! (2.27)
Then the solutions follows 𝑈(𝑥,𝑡)=1+𝑒^𝑥+𝑡,𝑉(𝑥,𝑡)=−1+𝑒^𝑥−𝑡. (2.28)

Example 2.3. Consider the system of nonlinear partial differential equations 𝑈_𝑡+𝑉𝑈_𝑥𝑉+𝑈=1,_𝑡−𝑈𝑉_𝑥−𝑉=1 (2.29) with initial conditions 𝑈(𝑥,0)=𝑒^𝑥,𝑉(𝑥,0)=𝑒^−𝑥. (2.30)
On using Sumudu transform on both sides of (2.29), and by taking Sumudu transform for the initial conditions of (2.30) we get 𝑆[𝑈(𝑥,𝑡)]=𝑒^𝑥+𝑢−𝑢𝑆𝑉𝑈_𝑥[𝑈],𝑆[𝑉(−𝑢𝑆𝑥,𝑡)]=𝑒^𝑥+𝑢+𝑢𝑆𝑈𝑉_𝑥+𝑢𝑆[𝑉]. (2.31)
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 𝑆^∞_𝑖=0𝑈_𝑖(𝑥,𝑡)=𝑒^𝑥+𝑢−𝑢𝑆^∞_𝑖=0𝐴_𝑖−𝑢𝑆^∞_𝑖=0𝑈_𝑖,𝑆^∞_𝑖=0𝑉_𝑖(𝑥,𝑡)=𝑒^−𝑥+𝑢+𝑢𝑆^∞_𝑖=0𝐵_𝑖−𝑢𝑆^∞_𝑖=0𝑉_𝑖, (2.32) where the terms 𝐴_𝑖 and 𝐵_𝑖 are handled with the help of Adomian polynomials by (1.12) that represent the nonlinear terms 𝑉𝑈_𝑥 and 𝑈𝑉_𝑥, respectively. We have a few terms of the Adomian polynomials for 𝑉𝑈_𝑥 and 𝑈𝑉_𝑥 which are given by 𝐴₀=𝑈_0𝑥𝑉₀,𝐴₁=𝑈_0𝑥𝑉₁+𝑈_1𝑥𝑉₀,𝐴₂=𝑈_0𝑥𝑉₂+𝑈_1𝑥𝑉₁+𝑈_2𝑥𝑉₀,⋮𝐵₀=𝑉_0𝑥𝑈₀,𝐵₁=𝑉_0𝑥𝑈₁+𝑉_1𝑥𝑈₀,𝐵₂=𝑉_0𝑥𝑈₂+𝑉_1𝑥𝑈₁+𝑉_2𝑥𝑈₀,⋮ (2.33)
By taking the inverse Sumudu transform we have 𝑈₀=𝑒^𝑥𝑉+𝑡,₀=𝑒^−𝑥𝑈+𝑡,_𝑖+1=𝑆⁻¹[𝑢]−𝑆⁻¹𝑢𝑆𝐴_𝑖−𝑆⁻¹𝑢𝑆𝑈_𝑖,𝑉_𝑖+1=𝑆⁻¹[𝑢]+𝑆⁻¹𝑢𝑆𝐵_𝑖+𝑆⁻¹𝑢𝑆𝑉_𝑖. (2.34)(2.35)
Using the inverse Sumudu transform on (2.35) we have 𝑈₁𝑡=−𝑡−²2!−𝑡𝑒^𝑥−𝑡²𝑒2!^𝑥,𝑉₁𝑡=−𝑡−²2!+𝑡𝑒^−𝑥−𝑡²𝑒2!^−𝑥,𝑈₂=𝑡²+𝑡2!²𝑒2!^𝑥𝑉⋯,₂=𝑡²+𝑡2!²𝑒2!^−𝑥⋯. (2.36)
The rest terms can be determined in the same way. Therefore, the series solutions are given by 𝑈(𝑥,𝑡)=𝑒^𝑥𝑡1−𝑡+²−𝑡2!³⋯,𝑉()3!𝑥,𝑡=𝑒^−𝑥𝑡1+𝑡+²+𝑡2!³⋯.3! (2.37)
Then the solution for the above system is as follows: 𝑈(𝑥,𝑡)=𝑒^𝑥−𝑡,𝑉(𝑥,𝑡)=𝑒^−𝑥+𝑡. (2.38)

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.