# Study of Convergence of Reduced Differential Transform Method for Different Classes of Differential Equations

Authors: Seyyedeh Roodabeh Moosavi Noori; Nasir Taghizadeh
Journal: International Journal of Differential Equations
 (2021)

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## Abstract

In this work, we study the sufficient condition for convergence of the reduced differential transform method for nonlinear differential equations. The main power of this method is its ability and flexibility in solving linear and nonlinear problems properly and easily and obtain solutions both numerically and analytically. Simple approaches of reduced differential transform method and the convergence results for different classes of differential equations such as linear and nonlinear ordinary, partial, fractional, and system of differential equations are briefly discussed. Eight examples are checked to confirm convergence results as well as the strength and efficiency of the method.


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## Body

## 1. Introduction

The reduced differential transform method (RDTM) is an analytical-numerical technique introduced for the first time by Keskin [1, 2] to study the analytical solutions of linear and nonlinear wave equations [3, 4]. This suggested technique is highly efficient and powerful in obtaining the exact solutions as well as approximate solutions of mathematical modeling of many problems in technology, finance, engineering disciplines, natural sciences such as biology, physics, chemistry, and Earth science, gives the solution in the form of rapidly convergent successive approximations, and is capable of handling linear and nonlinear equations in a similar manner. In recent years, the reduced differential transform method has been widely adopted by many researchers such as in [5–9] and by the references therein. Also, it was shown by many authors [10–14] that the solution procedure of the RDTM is simpler and more straightforward than the homotopy perturbation method (HPM), differential transform method (DTM), variational iteration method (VIM), Adomian decomposition method (ADM), etc. On the contrary, the size of computational work has been reduced while still maintaining high precision from the numerical solution and rapid convergence has been guaranteed. The advantage of this method is its simplicity in using, and it solves the equations straightforward and directly without using Adomian’s polynomial, perturbation, linearization or any other transformation, and restrictive conditions and gives the solution as convergent power series with simply determinable components. Therefore, the RDTM can overcome the foregoing limitations and restrictions of perturbation techniques and complicated computational so that it provides us with a possibility to analyse accurately nonlinear equations. The RDTM was successfully applied to ordinary differential equations [15], partial differential equations [16–18], fractional differential equations [19–23], Volterra integral equation [24–26], and integro-differential equations [27–29].In the present work, in light of the abovementioned method, we will study linear and nonlinear problems. In addition, our main objective is to study the sufficient condition for convergence of the method for nonlinear equations. The main ideas explained in this paper are expected to be used for more linear and nonlinear models.The rest of this study is presented as follows. In Section2, we simply introduce the reduced differential transform method (RDTM). In Section 3, we prove the convergences of the considered method. In Section 4, RDTM approaches and convergence results are addressed. In Section 5, we apply this method to obtain the exact solutions for linear and nonlinear ordinary, partial, fractional, and system of differential equations. Finally, we offer some summaries and conclusions in Section 6.


## 2. Summary of the Method

We present some important definitions and mathematical preliminaries operations of the reduced differential transform method in which can help to more understand of the stated method in this section. Now, consider the function of two variableswx,t and assume that it can be expressed as the product of two different variable functions, i.e., wx,t=ϕxψt. The function wx,t can be displayed due to the properties of differential transform as follows:(1)wx,t=∑i=0∞Φixi∑j=0∞Ψjtj=∑k=0∞Wkxtk,where Wkx is the t-dimensional spectrum function of the original function wx,t.Definition 1.
The reduced differential transform function ofwx,t can be yield in the following form:(2)Wkx=1k!∂k∂tkwx,tt=t0,where wx,t is analytic and differentiated continuously function with regard to space x and time t, in the domain of interest.Definition 2.
The reduced differential inverse transform ofWkx is determined as(3)wx,t=∑k=0∞Wkxt−t0k.
Afterward, consolidating equations (2) and (3) yields(4)wx,t=∑k=0∞1k!∂k∂tkwx,tt=t0t−t0k.
By the help of the upper definitions, to illustrate the basic idea of the RDTM, consider the following operator form of nonlinear partial differential equations written as(5)ℒwx,t+ℛwx,t+Nwx,t=hx,t,with considering the following initial condition:(6)wx,0=gx,where ℒ=∂/∂t, ℛ and N indicate linear and nonlinear operators which have partial derivatives, and h is a nonhomogeneous term.
After applying the RDTM definition on both sides of equation (5), we can write the following iteration formula:(7)k+1Wk+1x=ℋkx−ℛWkx−NWkx,where ℒWkx, ℛWkx, NWkx, and ℋkx are the reduced differential transform functions of ℒwx,t, ℛwx,t, Nwx,t, and hx,t, respectively.
Implementing the aforesaid method to the initial condition (6), we have(8)W0x=gx.
To discover the remaining iteration, we plug equation (8) into (7) by simple reiterative calculation, and we tend to get the subsequent Wkx values. Afterwards, the inverse transformation of the set of values Wkxk=0n admits the n-terms approximation solution as follows:(9)w˜nx,t=∑k=0nWkxt−t0k.
Thus, the exact solution of the considered problem can be gained by(10)wx,t=limn⟶∞w˜nx,t.
Table1 contains the basic mathematical operations carried out by RDTM.Table 1
The fundamental operations of RDTM.
Original formTransformed formwx,tWkx=1/k!∂k/∂tkwx,tt=0wx,t=λux,t±γvx,tWkx=λUkx±γVkx (λ,γ are constants)wx,t=xmtnWkx=xmδk−n,δx=1,k=00,k≠0.wx,t=xmtnux,tWkx=xmUk−nxwx,t=ux,tvx,tWkx=∑r=0kUrxVk−rx=∑r=0kVrxUk−rxwx,t=∂r/∂trux,tWkx=k+1…k+rUk+rx=k+r!/k!Uk+rxwx,t=∂/∂xux,tWkx=∂/∂xUkxwx,t=eλtWkx=λk/k!wx,t=sinλt+αxWkx=λk/k!sinkπ/2+αxwx,t=cosλt+αxWkx=λk/k!coskπ/2+αx


## 3. Convergence of Method

The principal aim of this section is to survey the sufficient conditions for convergence of the reduced differential transform method, according to the approach described by this method for solving the nonlinear equation (5) in Section 2. For this purpose, some theorems for convergence of the method and the error computation are addressed.The fundamental point views of RDTM for the solutions of nonlinear models include ascertaining power series expansion with the initial timet0:(11)wx,t=∑k=0∞akxt−t0k,t∈l,where l=t0,t0+r, r>0. The important results are proposed in the following theorems.Theorem 1.
Ifφkx,t=akxt−t0k, then the series solution ∑k=0∞φkx,t, stated in equation (11), ∀k∈N∪0.(i)
It is convergent if∃0<λ<1 such that φk+1≤λφk(ii)
It is divergent if∃λ>1 such that φk+1≥λφkTheorem1 is a specific case of Banach’s fixed point theorem. Using this theorem and a brief description of its proof, we investigate the truncation error of the series solution equation (11) as follows.Proof.
LetCl,. be the Banach space of all continuous functions on l with the norm φkx,t=akxt−t0k. Also, assume that a0x<N0, where N0 is a positive number. Define the sequence of partial sums Snn=0∞ as(12)Sn=φ0+φ1+⋯+φn.
We want to show thatSnn=0∞ is a Cauchy sequence in this Banach space. To reach this goal, we take(13)Sn+1−Sn=φn+1≤λφn≤⋯≤λn+1φ0≤λn+1N0.
Therefore, for anyn,m∈ℕ, n≥m, making use of (13) and the triangle inequality successively, we have(14)Sn−Sm=Sn−Sn−1+Sn−1−Sn−2+⋯+Sm+1−Sm≤Sn−Sn−1+Sn−1−Sn−2+⋯+Sm+1−Sm≤1−λn−m1−λλm+1φ0,and because 0<λ<1, we obtain(15)limn,m⟶∞Sn−Sm=0.
Hence,Snn=0∞ is a Cauchy sequence in the Banach space Cl,.. Thus, the series solution ∑k=0∞φkx,t, defined in equation (11), is convergent and it completes the proof.Remark 1.
According to the assumptions in (ii) and by using the ratio test, we have(16)φk+1φk≥λ>1.
As a result, the series is divergent.Remark 2.
If the series∑k=0∞akxt−t0k of the nonlinear equation (5) converges, then it is an exact solution.Theorem 2.
Suppose that the series solution∑k=0∞φkx,t, where φkx,t=akxt−t0k, converges to the solution wx,t. If the truncated series ∑k=0mφkx,t is used as an approximation to the solution ux,t, then the maximum absolute truncated error is estimated as(17)wx,t−∑k=0mφkx,t≤11−λλm+1φ0.Proof.
According to Theorem1, we have the inequality equation (14) as follows:(18)Sn−Sm≤1−λn−m1−λλm+1φ0,for n≥m. Also, since 0<λ<1, in the numerator, we have 1−λn−m<1; therefore, the inequality equation (18) can be reduced to(19)Sn−Sm≤11−λλm+1φ0.
It is clear whenn⟶∞, Sn⟶wx,t. Thus, inequality equation (17) is obtained and the theorem is proved.In summary, Theorems1 and 2 state that the reduced differential transform solution of nonlinear equation (5), obtained using the iteration formula (7) and (8), converges to an exact solution under the condition that ∃0<λ<1 such that φk+1≤λφk, ∀k∈ℕ∪0. In other words, if we define, for every i∈ℕ∪0, the parameters,(20)γi=φi+1φi,φi≠0,0,φi=0,then the series solution ∑k=0∞φkx,t of equation (5) converges to an exact solution wx,t, when 0≤γi<1,∀i∈ℕ∪0. In addition, the maximum absolute truncation error, as discussed in Theorem 2, is estimated to be(21)wx,t−∑k=0mφkx,t≤1γ−1γj+1φ0,where γ=maxγi,i=0,1,…,j.Remark 3.
The first finite terms have no effect the convergence of the series solution. In other words, if the first finiteγis, i=0,1,…,l, are not less than one and γi≤1 for i>l, then the series solution ∑k=0∞φkx,t of equation (5) converges to an exact solution. Because, according to Theorem 1, we have(22)Sn−Sj≤1−λn−j1−λλj−lφl+1and since 0<λ<1, for n≥j and fixed l, we get limn,j⟶∞Sn−Sj=0. In this case, the convergence of the RDTM approach depends on γi, for i>l.


## 4. RDTM Approaches and Convergence Results

The principal aim of this section is to summarize RDTM inclusive convergence results of the method for solving different classes of differential equations.


### 4.1. Ordinary Differential Equations (ODE)

Let us write the nonlinear ordinary differential equation as the following:(23)drdtrwt+ℛwt+Nwt=ht,t>0,where r∈ℕ, ℛ is a linear operator, N is a nonlinear operator, and h is a nonhomogeneous term, with the initial conditions wk0=ck, k=0,1,…,r−1. According to the operations of differential transformation, the series solution wt=∑k=0∞φkt is obtained using the following iteration formula:(24)k+r!k!Wk+r+ℛWk+NWk=ℋk,which converges to a solution of equation (23) if ∀k∈ℕ∪0, ∃0<λ<1 such that φk+1≤λφk.


### 4.2. Partial Differential Equations (PDE)

Let us write the nonlinear partial differential equation as the following:(25)∂r∂trwx,t+ℛwx,t+Nwx,t=hx,t,t>0,where r∈ℕ, ℛ is a linear operator, N is a nonlinear operator, and h is a nonhomogeneous term, with the initial conditions wkx,0=gkx, k=0,1,…,r−1. According to the operations of differential transformation, the series solution wx,t=∑k=0∞φkx,t is obtained using the following iteration formula:(26)k+r!k!Wk+rx+ℛWkx+NWkx=ℋkx,which converges to a solution of equation (25) if ∀k∈ℕ∪0, ∃0<λ<1 such that φk+1≤λφk.


### 4.3. Fractional Partial Differential Equations (FPDE)

Let us write the nonlinear fractional partial differential equation as follows:(27)∂αr∂tαrwx,t+ℛwx,t+Nwx,t=hx,t,t>0,m−1<α≤m,where m∈ℕ, ∂αr/∂tαr is the Caputo fractional derivative of order αr, ℛ is a linear operator, N is a nonlinear operator, and h is a nonhomogeneous term, with the initial conditions wkx,0=gkx, k=0,1,…,m−1.Definition 3.
The Caputo fractional derivative operator is defined as(28)Daαwt=1Γm−α∫att−ξm−α−1wmξdξ,where α>0 and a are the order of the derivative and the initial value of function w, respectively.Properties of Caputo fractional derivative operator can be found in [30–32].Also, to determine the result, we introduce the subsequent Riemann–Liouville fractional integral operator.Definition 4.
The Riemann–Liouville fractional integral operator is defined as(29)Jaαwt=1Γα∫att−ξα−1wξdξ,t>0,α>0.Properties of Riemann–Liouville fractional integral operator can be found in [33, 34].Definition 5.
Let functionwx,t be analytic and differentiated continuously with respect to space x and time t in the domain of interest, and the fractional reduced differential transform function (FRDTM) is(30)Wkαx=1Γαk+1∂αk∂tαkwx,tt=t0,where 0<α≤1, the t-dimensional spectrum function Wkαx is the transformed function, and Γ is gamma function is defined as(31)Γp=∫0∞xp−1e−xdx.Definition 6.
The fractional reduced differential inverse transform ofWkαx is determined as follows:(32)wx,t=∑k=0∞Wkαxt−t0αk.
Afterward, combining equations (30) and (32), we write(33)wx,t=∑k=0∞1Γαk+1∂αk∂tαkwx,tt=t0t−t0αk,which in practical application can be approximated by a finite series:(34)w˜nx,t=∑k=0nWkαxt−t0αk.
Thus, the exact solution to the problem can be obtained by(35)wx,t=limn⟶∞w˜nx,t.In case ofα=1, FRDTM is reduced to classical RDTM. The basic properties of mathematical operations performed by FRDTM can be found in [33]. According to RDTM and Caputo differential operator and then following the same analysis presented in the previous section, the series solution wx,t=∑k=0∞φkx,t is obtained using the following iteration formula:(36)Γαk+r+1Γαk+1Wk+rαx+ℛWkx+NWkx=ℋkx,which converges to a solution of equation (27) if ∀k∈ℕ∪0 and ∃0<λ<1 such that φk+1≤λφk.


### 4.4. Systems of Fractional Partial Differential Equations

Let us write the following system of nonlinear fractional partial differential equations:(37)∂α∂tαw1x,t+ℛ1w1,w2,…,wn+N1w1,w2,…,wn=h1x,t,∂α∂tαw2x,t+ℛ2w1,w2,…,wn+N2w1,w2,…,wn=h2x,t,⋮∂α∂tαwnx,t+ℛnw1,w2,…,wn+Nnw1,w2,…,wn=hnx,t,where m−1<α≤m, m,n∈ℕ, ℛ1,ℛ2,…,ℛn are linear operators, N1,N2,…,Nn are nonlinear operators, and h1,h2,…,hn are the nonhomogeneous terms, with the initial conditions(38)w1kx,0=g1,kx,w2kx,0=g2,kx,⋮wnkx,0=gn,kx,for k=0,1,…,m−1. Then, for i=1,2,…,n, the series solution wix,t=∑k=0∞φi,kx,t is obtained using the iteration formula:(39)Γαk+r+1Γαk+1Wi,k+1αx+ℛWi,kx+NWi,kx=ℋi,kx,which converges to a solution of equation (37) if ∀k∈ℕ∪0 and ∃0<λi<1 such that φi,k+1≤λiφi,k.?


## 4.1. Ordinary Differential Equations (ODE)

Let us write the nonlinear ordinary differential equation as the following:(23)drdtrwt+ℛwt+Nwt=ht,t>0,where r∈ℕ, ℛ is a linear operator, N is a nonlinear operator, and h is a nonhomogeneous term, with the initial conditions wk0=ck, k=0,1,…,r−1. According to the operations of differential transformation, the series solution wt=∑k=0∞φkt is obtained using the following iteration formula:(24)k+r!k!Wk+r+ℛWk+NWk=ℋk,which converges to a solution of equation (23) if ∀k∈ℕ∪0, ∃0<λ<1 such that φk+1≤λφk.


## 4.2. Partial Differential Equations (PDE)

Let us write the nonlinear partial differential equation as the following:(25)∂r∂trwx,t+ℛwx,t+Nwx,t=hx,t,t>0,where r∈ℕ, ℛ is a linear operator, N is a nonlinear operator, and h is a nonhomogeneous term, with the initial conditions wkx,0=gkx, k=0,1,…,r−1. According to the operations of differential transformation, the series solution wx,t=∑k=0∞φkx,t is obtained using the following iteration formula:(26)k+r!k!Wk+rx+ℛWkx+NWkx=ℋkx,which converges to a solution of equation (25) if ∀k∈ℕ∪0, ∃0<λ<1 such that φk+1≤λφk.


## 4.3. Fractional Partial Differential Equations (FPDE)

Let us write the nonlinear fractional partial differential equation as follows:(27)∂αr∂tαrwx,t+ℛwx,t+Nwx,t=hx,t,t>0,m−1<α≤m,where m∈ℕ, ∂αr/∂tαr is the Caputo fractional derivative of order αr, ℛ is a linear operator, N is a nonlinear operator, and h is a nonhomogeneous term, with the initial conditions wkx,0=gkx, k=0,1,…,m−1.Definition 3.
The Caputo fractional derivative operator is defined as(28)Daαwt=1Γm−α∫att−ξm−α−1wmξdξ,where α>0 and a are the order of the derivative and the initial value of function w, respectively.Properties of Caputo fractional derivative operator can be found in [30–32].Also, to determine the result, we introduce the subsequent Riemann–Liouville fractional integral operator.Definition 4.
The Riemann–Liouville fractional integral operator is defined as(29)Jaαwt=1Γα∫att−ξα−1wξdξ,t>0,α>0.Properties of Riemann–Liouville fractional integral operator can be found in [33, 34].Definition 5.
Let functionwx,t be analytic and differentiated continuously with respect to space x and time t in the domain of interest, and the fractional reduced differential transform function (FRDTM) is(30)Wkαx=1Γαk+1∂αk∂tαkwx,tt=t0,where 0<α≤1, the t-dimensional spectrum function Wkαx is the transformed function, and Γ is gamma function is defined as(31)Γp=∫0∞xp−1e−xdx.Definition 6.
The fractional reduced differential inverse transform ofWkαx is determined as follows:(32)wx,t=∑k=0∞Wkαxt−t0αk.
Afterward, combining equations (30) and (32), we write(33)wx,t=∑k=0∞1Γαk+1∂αk∂tαkwx,tt=t0t−t0αk,which in practical application can be approximated by a finite series:(34)w˜nx,t=∑k=0nWkαxt−t0αk.
Thus, the exact solution to the problem can be obtained by(35)wx,t=limn⟶∞w˜nx,t.In case ofα=1, FRDTM is reduced to classical RDTM. The basic properties of mathematical operations performed by FRDTM can be found in [33]. According to RDTM and Caputo differential operator and then following the same analysis presented in the previous section, the series solution wx,t=∑k=0∞φkx,t is obtained using the following iteration formula:(36)Γαk+r+1Γαk+1Wk+rαx+ℛWkx+NWkx=ℋkx,which converges to a solution of equation (27) if ∀k∈ℕ∪0 and ∃0<λ<1 such that φk+1≤λφk.


## 4.4. Systems of Fractional Partial Differential Equations

Let us write the following system of nonlinear fractional partial differential equations:(37)∂α∂tαw1x,t+ℛ1w1,w2,…,wn+N1w1,w2,…,wn=h1x,t,∂α∂tαw2x,t+ℛ2w1,w2,…,wn+N2w1,w2,…,wn=h2x,t,⋮∂α∂tαwnx,t+ℛnw1,w2,…,wn+Nnw1,w2,…,wn=hnx,t,where m−1<α≤m, m,n∈ℕ, ℛ1,ℛ2,…,ℛn are linear operators, N1,N2,…,Nn are nonlinear operators, and h1,h2,…,hn are the nonhomogeneous terms, with the initial conditions(38)w1kx,0=g1,kx,w2kx,0=g2,kx,⋮wnkx,0=gn,kx,for k=0,1,…,m−1. Then, for i=1,2,…,n, the series solution wix,t=∑k=0∞φi,kx,t is obtained using the iteration formula:(39)Γαk+r+1Γαk+1Wi,k+1αx+ℛWi,kx+NWi,kx=ℋi,kx,which converges to a solution of equation (37) if ∀k∈ℕ∪0 and ∃0<λi<1 such that φi,k+1≤λiφi,k.?


## 5. Applications

The principal aim of this section is to apply the reduced differential transform method on the following examples to illustrate the accuracy of the presented method.Example 1.
We first consider the following linear ordinary differential equation,(40)w″t+wt=0,0<t≤1,subject to the initial conditions:(41)w0=0,w′0=1.
According to the operations of differential transformation given in Table1 for equation (40), we obtain the following recurrent relation:(42)k+1k+2Wk+2+Wk=0,and from initial conditions (42), we write(43)W0=0,W1=1.
Substituting the above equations in equation (43), we drive the following results:(44)W2=0,W3=−13!,W4=0,W5=15!,⋯.
Hence, the solution in the series form is as follows:(45)w˜nt=∑k=0∞Wktk=W0+W1t+W2t2+⋯=t−t33!+t55!−⋯,which converges efficiently to the exact solution wt=sint.Example 2.
As the second example, we consider the following nonlinear ordinary differential equation:(46)w′t=w2t+1,0<t≤1,which subjects to the initial condition:(47)w0=0.
According to the operations of differential transformation given in Table1 for equation (46), we obtain the following recurrent relation:(48)k+1Wk+1=∑r=0kWrWk−r+δk,and from initial condition (47), we write(49)W0=0.
Substituting the above equation in equation (48), we drive the following results:(50)W1=1,W2=0,W3=13,W4=0,W5=215,….
Hence, the solution in series form is as follows:(51)w˜nt=∑k=0∞Wktk=W0+W1t+W2t2+⋯=t+13t3+215t5+⋯,which converges efficiently to the exact solution wt=tant.Example 3.
As the third example, we consider the following Burger’s equation [35]:(52)∂w∂t+w∂w∂x=∂2w∂x2,0≤t<12,which subjects to the initial condition:(53)wx,0=2x.
According to the operations of differential transformation given in Table1 for equation (52), we obtain the following recurrent relation:(54)k+1Wk+1x+∑r=0kWrx∂∂xWk−rx=∂2∂x2Wkx,and from initial condition (53), we write(55)W0x=2x.
Substituting the above equation in equation (54), we drive the following results:(56)W1x=−4x,W2x=8x,W3x=−16x,W4x=32x,….
Hence, the solution in the series form is as follows:(57)w˜nx,t=∑k=0∞Wkxtk=W0x+W1xt+W2xt2+⋯=2x1−2t+2t2−⋯,which converges efficiently to the exact solution wx,t=2x/1+2t.Example 4.
As the fourth example, we consider the following nonlinear Klein–Gordon equation [36]:(58)∂2w∂t2−∂2w∂x2+w2=−xcost+x2cos2t,which subjects to the initial conditions:(59)wx,0=x,∂∂twx,0=0.
According to the operations of differential transformation given in Table1 for equation (58), we obtain the following recurrent relation:(60)k+1k+2Wk+2x−∂2∂x2Wkx+∑r=0kWrxWk−rx=−xcoskπ/2k!+12x2δk+2kcoskπ/2k!,and from initial conditions (59), we write(61)W0x=x,W1x=0.
Substituting the above equation in equation (60), we drive the following results:(62)W2x=−x2!,W3x=0,W4x=x4!,W5x=0,….
Hence, the solution in the series form is as follows:(63)w˜nx,t=∑k=0∞Wkxtk=W0x+W1xt+W2xt2+⋯=x1−t22!+t44!−⋯,which converges efficiently to the exact solution wx,t=xcost.Example 5.
As the fifth example, we consider the following homogeneous nonlinear time-fractional gas dynamics equation as [37, 38](64)∂αw∂tα+w∂w∂x−w+w2=0,which subjects to the initial condition:(65)wx,0=e−x.
According to the operations of differential transformation given in Table1 for equation (64), we obtain the following recurrent relation:(66)Γαk+1+1Γαk+1Wk+1αx+∑r=0kWrx∂∂xWk−rx−Wk+∑r=0kWrxWk−rx=0,and from initial conditions (65), we write(67)W0αx=e−x.
Substituting the above equation in equation (66), we drive the following results:(68)W1αx=e−xΓα+1,W2αx=e−xΓ2α+1,W3αx=e−xΓ3α+1,….
Hence, the solution in series form is as follows:(69)w˜nx,t=∑k=0∞Wkαxtαk=W0αx+W1αxtα+W2αxt2α+⋯=e−x1+1Γα+1tα+1Γ2α+1t2α+⋯,which for α=1 converges efficiently to the exact solution wx,t=e−x+t.Example 6.
As the sixth example, we consider the following system of inhomogeneous linear PDEs [33]:(70)∂w∂t−∂v∂x−w+v=−2,∂v∂t+∂w∂x−w+v=−2,which subjects to the initial condition:(71)wx,0=1+ex,vx,0=−1+ex.
According to the operations of differential transformation given in Table1 for equation (70), we obtain the following recurrent relation:(72)k+1Wk+1x−∂∂xVkx−Wkx+Vkx=−2δk,k+1Vk+1x+∂∂xWkx−Wkx+Vkx=−2δk,and from initial conditions (71), we write(73)W0x=1+ex,V0x=−1+ex.
Substituting the above equations in equation (72), we drive the following results:(74)W1x=ex,W2x=12!ex,W3x=13!ex,W4x=14!ex,…,V1x=−ex,V2x=12!ex,V3x=−13!ex,V4x=14!ex,….
Hence, the solutions in the series form are as follows:(75)w˜nx,t=∑k=0∞Wkxtk=W0x+W1xt+W2xt2+⋯=1+ex1+t+t22!+⋯,v˜nx,t=∑k=0∞Vkxtk=V0x+V1xt+V2xt2+⋯=−1+ex1−t+t22!−⋯,which converges efficiently to the exact solutions wx,t=1+ex+t and vx,t=−1+ex−t.Example 7.
As the seventh example, we consider the following system of nonlinear PDEs:(76)∂w∂t+∂w∂xv−w∂v∂x−w=0,∂v∂t−w∂v∂x+∂w∂xv+v=0,which subjects to the initial conditions:(77)wx,0=ex,vx,0=ex.
According to the operations of differential transformation given in Table1 for equation (76), we obtain the following recurrent relation:(78)k+1Wk+1x+∑r=0k∂∂xWrxVk−r−∑r=0kWr∂∂xVk−rx−Wkx=0,k+1Vk+1x−∑r=0kWr∂∂xVk−rx+∑r=0k∂∂xWrxVk−r+Vkx=0,and from initial conditions (77), we write(79)W0x=ex,V0x=ex.
Substituting the above equations in equation (78), we drive the following results:(80)W1x=ex,W2x=12!ex,W3x=13!ex,W4x=14!ex,…,V1x=−ex,V2x=12!ex,V3x=−13!ex,V4x=14!ex,….
Hence, the solutions in the series form are as follows:(81)w˜nx,t=∑k=0∞Wkxtk=W0x+W1xt+W2xt2+⋯=ex1+t+t22!+⋯,v˜nx,t=∑k=0∞Vkxtk=V0x+V1xt+V2xt2+⋯=ex1−t+t22!−⋯,which converges efficiently to the exact solutions wx,t=ex+t and vx,t=ex−t.Example 8.
Lastly, we consider the following system of homogeneous linear FPDEs [39, 40]:(82)∂αw∂tα−∂v∂x+v+w=0,∂βv∂tβ−∂w∂x+v+w=0,which subject to the initial conditions:(83)wx,0=sinhx,vx,0=coshx.
According to the operations of differential transformation given in Table1 for equation (82), we obtain the following recurrent relation:(84)Γαk+1+1Γαk+1Wk+1αx−∂∂xVkx+Vkx+Wkx=0,Γβk+1+1Γβk+1Vk+1βx−∂∂xWkx+Vkx+Wkx=0,and from initial conditions (83), we write(85)W0αx=sinhx,V0βx=coshx.
Substituting the above equations in equation (84), we drive the following results:(86)W1αx=−coshxΓα+1,W2αx=sinhxΓ2α+1,W3αx=−coshxΓ3α+1,…,V1βx=−sinhxΓβ+1,V2βx=coshxΓ2β+1,V3βx=−sinhxΓ3β+1,….
Hence, the solutions in the series form are as follows:(87)w˜nx,t=∑k=0∞Wkαxtαk=W0αx+W1αxtα+W2αxt2α+W3αxt3α+⋯=sinhx+−1tαΓα+1coshx+−12t2αΓ2α+1sinhx+−13t3αΓ3α+1coshx+⋯,v˜nx,t=∑k=0∞Vkβxtβk=V0βx+V1βxtβ+V2βxt2β+V3βxt3β+⋯=coshx+−1tβΓβ+1sinhx+−12t2βΓ2β+1coshx+−13t3βΓ3β+1sinhx+⋯.which for α=β=1 converges efficiently to the exact solutions wx,t=sinhx−t and vx,t=coshx−t.Results for Examples1–8 are reported in Figures 1–8 and Tables 2–9, respectively. In these tables, the terms wE,vE, wn,R,vn,R, and ew,ev stand for exact solution, nth order approximate solution of RDTM, and their absolute error, respectively.Figure 1
Comparison of the exact solution (blue) and the approximate solutions (red) of Example1.Figure 2
Comparison of the exact solution (blue) and the approximate solutions (red) of Example2.Figure 3
Comparison of the exact solution (blue) and the approximate solutions (red) of Example3.Figure 4
Comparison of the exact solution (blue) and the approximate solutions (red) of Example4.Figure 5
Comparison of the exact solution (blue) and the approximate solutions (red) of Example5 for α=1.Figure 6
Comparison of the exact solution (blue) and the approximate solutions (red) of Example6. (a) The solution of wnx,t. (b) The solution of vnx,t.
(a)(b)Figure 7
Comparison of the exact solution (blue) and the approximate solutions (red) of Example7. (a) The solution of wnx,t. (b) The solution of vnx,t.
(a)(b)Figure 8
Comparison of the exact solution (blue) and the approximate solutions (red) of Example8 for α=1. (a) The solution of wnx,t. (b) The solution of vnx,t.
(a)(b)Table 2
Numerical result of Example1.
tuEu7,Ru9,Rn=7,eun=9,eu0000000.20.19866933080.19866933090.19866933097.02 × 10−125.6 × 10−120.40.38941834230.38941834150.38941834227.12 × 10−106.36 × 10−120.60.56464247340.56464244570.56464247352.77 × 10−87.57 × 10−110.80.71735609090.71735572320.71735609313.68 × 10−72.13 × 10−910.84147098480.84146825390.84147100962.73 × 10−62.49 × 10−8Table 3
Numerical result of Example2.
tuEu7,Ru9,Rn=7,eun=9,eu0000000.20.20271003550.20271002420.20271003541.14 × 10−81.77 × 10−100.40.42279321870.42278708820.42279282126.13 × 10−63.97 × 10−70.60.68413680830.68387876570.68409915982.58 × 10−43.76 × 10−50.81.0296385571.0256752961.0286105693.96 × 10−31.03 × 10−311.5574077251.5206349201.5425044093.68 × 10−21.49 × 10−2Table 4
Numerical result of Example3.
x,tuEu7,Ru9,Rn=7,eun=9,eu(0, 0)00000(0.2, 0.1)0.33333333340.333332480.33333329928.53 × 10−73.41 × 10−8(0.4, 0.2)0.57142857140.571054080.57136865283.74 × 10−45.9 × 10−5(0.6, 0.3)0.750.737402880.74546503681.26 × 10−24.53 × 10−3(0.8, 0.4)0.88888888890.739758080.79344517121.49 × 10−19.54 × 10−2Table 5
Numerical result of Example4.
x,tuEu6,Ru10,Rn=6,eun=10,eu(0.1, 0)0.10.10.100(0.3, 0.2)0.29401997330.29401997330.29401997343.33 × 10−115.24 × 10−11(0.5, 0.4)0.46053049700.46053048890.46053049708.11 × 10−91.29 × 10−12(0.7, 0.6)0.57773493040.57773464000.57773493042.9 × 10−73.36 × 10−11(0.9, 0.8)0.62703603840.62703232000.62703603833.72 × 10−61.16 × 10−10Table 6
Numerical result of Example5.
x,tuEu7,Ru10,Rn=7,eun=10,eu(0.1, 0)0.90483741800.90483741800.904837418000(0.3, 0.2)0.90483741800.90483741870.90483741872.34 × 10−112.47 × 10−11(0.5, 0.4)0.90483741800.90483740790.90483741821.03 × 10−83.13 × 10−11(0.7, 0.6)0.90483741800.90483719640.90483741842.21 × 10−72.38 × 10−11(0.9, 0.8)0.90483741800.90483556300.90483741731.85 × 10−69.49 × 10−10Table 7
Numerical result of Example6.
x,tuEu7,Ru10,Rn=7,eun=10,eu(0, 0.1)2.1051709182.1051709182.1051709187.54 × 10−117.57 × 10−11(0.2, 0.3)2.6487212712.6487212672.6487212702.57 × 10−95.14 × 10−10(0.4, 0.5)3.4596031113.4596029593.4596031111.52 × 10−76.81 × 10−10(0.6, 0.7)4.6692966684.6692938444.6692966662.82 × 10−62.09 × 10−9(0.8, 0.9)6.4739473926.4739210206.4739473702.64 × 10−52.06 × 10−8x,tvEv7,Rv10,Rn=7,evn=10,ev(0, 0.1)−0.951625820e − 1−0.951625819e − 1−0.951625819e − 13.57 × 10−113.59 × 10−11(0.2, 0.3)−0.951625820e − 1−0.951625840e − 1−0.951625822e − 12.01 × 10−98.46 × 10−11(0.4, 0.5)−0.951625820e − 1−0.951627186e − 1−0.951625816e − 11.37 × 10−72.19 × 10−10(0.6, 0.7)−0.951625820e − 1−0.951649979e − 1−0.951625810e − 12.42 × 10−66.6 × 10−10(0.8, 0.9)−0.951625820e − 1−0.951841642e − 1−0.951625658e − 12.16 × 10−51.62 × 10−8Table 8
Numerical result of Example7.
x,tuEu7,Ru10,Rn=7,eun=10,eu(0, 0.3)1.3498588081.3498588051.3498588072.11 × 10−94.23 × 10−10(0.2, 0.4)1.8221188001.8221187801.8221188012.06 × 10−81.55 × 10−10(0.4, 0.5)2.4596031112.4596029592.4596031111.52 × 10−76.81 × 10−10(0.6, 0.6)3.3201169233.3201161093.3201169238.14 × 10−71.14 × 10−9(0.8, 0.7)4.4816890704.4816856214.4816890683.45 × 10−61.37 × 10−9x,tvEv7,Rv10,Rn=7,evn=10,ev(0, 0.3)0.74081822070.74081821910.74081822061.59 × 10−91.92 × 10−11(0.2, 0.4)0.81873075310.81873073400.81873075301.91 × 10−81.32 × 10−10(0.4, 0.5)0.90483741800.90483728140.90483741841.37 × 10−72.19 × 10−10(0.6, 0.6)10.999999288417.11 × 10−76.62 × 10−11(0.8, 0.7)1.1051709181.1051679671.1051709192.95 × 10−64.99 × 10−10Table 9
Numerical result of Example8.
x,tuEu7,Ru9,Rn=7,eun=9,eu(0.1, 0.1)0003.5 × 10−113.5 × 10−11(0.2, 0.3)−0.1001667500−0.1001667505−0.10016675024.17 × 10−101.73 × 10−10(0.3, 0.5)−0.2013360025−0.2013360265−0.20133600262.4 × 10−88.6 × 10−11(0.4, 0.7)−0.3045202934−0.3045207633−0.30452029614.7 × 10−72.88 × 10−9(0.5, 0.9)−0.4107523258−0.4107567262−0.41075236794.4 × 10−64.1 × 10−8x,tvEv7,Rv9,Rn=7,evn=9,ev(0.1, 0.1)1114.76 × 10−114.76 × 10−11(0.2, 0.3)1.0050041681.0050041661.0050041681.18 × 10−94.67 × 10−10(0.3, 0.5)1.0200667561.0200666551.0200667561.01 × 10−78.15 × 10−10(0.4, 0.7)1.0453385141.0453370061.0453385061.51 × 10−67.99 × 10−9(0.5, 0.9)1.0810723721.0810607841.0810722651.16 × 10−51.05 × 10−7


## 6. Conclusions

In this work, the convergence of the reduced differential transform method (RDTM) for solving the linear and nonlinear ordinary, partial, fractional differential equations, and their systems is discussed. The main strength of the RDTM is its fast convergence, and under the assumption of Theorem1, the method is convergent to the exact solution of the problem. The sufficient condition for convergence of the method and an error estimate has been addressed. For the efficiency of the RDTM, the form of the initial approximation is very important. We note that the RDTM solutions computed via a simple algorithm and without involving the perturbation, linearization, or discretization provide a solution in both numerical and analytical manner. RDTM can be applied in most of the biological, physical, engineering, etc. models as an alternative for obtaining reliable and fastest convergence and useful approximations. Thus, it can be concluded that the RDTM is a simple and powerful tool for solving functional equations.



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