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

Moosavi Noori, Seyyedeh Roodabeh; Taghizadeh, Nasir

doi:https://doi.org/10.1155/2021/6696414

International Journal of Differential Equations

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Research Article | Open Access

Volume 2021 | Article ID 6696414 | https://doi.org/10.1155/2021/6696414

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

Seyyedeh Roodabeh Moosavi Noori¹and Nasir Taghizadeh¹

Academic Editor: Mostafa Eslami

Received16 Oct 2020

Revised29 Mar 2021

Accepted19 Apr 2021

Published30 Apr 2021

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.

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 Section 2, 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 variables and assume that it can be expressed as the product of two different variable functions, i.e., . The function can be displayed due to the properties of differential transform as follows:where is the -dimensional spectrum function of the original function .

Definition 1. The reduced differential transform function of can be yield in the following form:where is analytic and differentiated continuously function with regard to space and time , in the domain of interest.

Definition 2. The reduced differential inverse transform of is determined asAfterward, consolidating equations (2) and (3) yieldsBy 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 aswith considering the following initial condition:where , and indicate linear and nonlinear operators which have partial derivatives, and is a nonhomogeneous term.
After applying the RDTM definition on both sides of equation (5), we can write the following iteration formula:where , , , and are the reduced differential transform functions of , , , and , respectively.
Implementing the aforesaid method to the initial condition (6), we haveTo discover the remaining iteration, we plug equation (8) into (7) by simple reiterative calculation, and we tend to get the subsequent values. Afterwards, the inverse transformation of the set of values admits the -terms approximation solution as follows:Thus, the exact solution of the considered problem can be gained byTable 1 contains the basic mathematical operations carried out by RDTM.

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 time :where , . The important results are proposed in the following theorems.

Theorem 1. If , then the series solution , stated in equation (11), .(i)It is convergent if such that (ii)It is divergent if such that

Theorem 1 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. Let be the Banach space of all continuous functions on with the norm . Also, assume that , where is a positive number. Define the sequence of partial sums asWe want to show that is a Cauchy sequence in this Banach space. To reach this goal, we takeTherefore, for any , , making use of (13) and the triangle inequality successively, we haveand because , we obtainHence, is a Cauchy sequence in the Banach space . Thus, the series solution , 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 haveAs a result, the series is divergent.

Remark 2. If the series of the nonlinear equation (5) converges, then it is an exact solution.

Theorem 2. Suppose that the series solution , where , converges to the solution . If the truncated series is used as an approximation to the solution , then the maximum absolute truncated error is estimated as

Proof. According to Theorem 1, we have the inequality equation (14) as follows:for . Also, since , in the numerator, we have ; therefore, the inequality equation (18) can be reduced toIt is clear when , . Thus, inequality equation (17) is obtained and the theorem is proved.

In summary, Theorems 1 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 such that , . In other words, if we define, for every , the parameters,then the series solution of equation (5) converges to an exact solution , when . In addition, the maximum absolute truncation error, as discussed in Theorem 2, is estimated to bewhere .

Remark 3. The first finite terms have no effect the convergence of the series solution. In other words, if the first finite s, , are not less than one and for , then the series solution of equation (5) converges to an exact solution. Because, according to Theorem 1, we haveand since , for and fixed , we get . In this case, the convergence of the RDTM approach depends on , for .

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:where , is a linear operator, is a nonlinear operator, and is a nonhomogeneous term, with the initial conditions , . According to the operations of differential transformation, the series solution is obtained using the following iteration formula:which converges to a solution of equation (23) if , such that .

4.2. Partial Differential Equations (PDE)

Let us write the nonlinear partial differential equation as the following:where , is a linear operator, is a nonlinear operator, and is a nonhomogeneous term, with the initial conditions , . According to the operations of differential transformation, the series solution is obtained using the following iteration formula:which converges to a solution of equation (25) if , such that .

4.3. Fractional Partial Differential Equations (FPDE)

Let us write the nonlinear fractional partial differential equation as follows:where , is the Caputo fractional derivative of order , is a linear operator, is a nonlinear operator, and is a nonhomogeneous term, with the initial conditions , .

Definition 3. The Caputo fractional derivative operator is defined aswhere and are the order of the derivative and the initial value of function , 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

Properties of Riemann–Liouville fractional integral operator can be found in [33, 34].

Definition 5. Let function be analytic and differentiated continuously with respect to space and time in the domain of interest, and the fractional reduced differential transform function (FRDTM) iswhere , the -dimensional spectrum function is the transformed function, and is gamma function is defined as

Definition 6. The fractional reduced differential inverse transform of is determined as follows:Afterward, combining equations (30) and (32), we writewhich in practical application can be approximated by a finite series:Thus, the exact solution to the problem can be obtained by

In case of , 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 is obtained using the following iteration formula:which converges to a solution of equation (27) if and such that .

4.4. Systems of Fractional Partial Differential Equations

Let us write the following system of nonlinear fractional partial differential equations:where , , are linear operators, are nonlinear operators, and are the nonhomogeneous terms, with the initial conditionsfor . Then, for , the series solution is obtained using the iteration formula:which converges to a solution of equation (37) if and such that .?

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,subject to the initial conditions:According to the operations of differential transformation given in Table 1 for equation (40), we obtain the following recurrent relation:and from initial conditions (42), we writeSubstituting the above equations in equation (43), we drive the following results:Hence, the solution in the series form is as follows:which converges efficiently to the exact solution .

Example 2. As the second example, we consider the following nonlinear ordinary differential equation:which subjects to the initial condition:According to the operations of differential transformation given in Table 1 for equation (46), we obtain the following recurrent relation:and from initial condition (47), we writeSubstituting the above equation in equation (48), we drive the following results:Hence, the solution in series form is as follows:which converges efficiently to the exact solution .

Example 3. As the third example, we consider the following Burger’s equation [35]:which subjects to the initial condition:According to the operations of differential transformation given in Table 1 for equation (52), we obtain the following recurrent relation:and from initial condition (53), we writeSubstituting the above equation in equation (54), we drive the following results:Hence, the solution in the series form is as follows:which converges efficiently to the exact solution .

Example 4. As the fourth example, we consider the following nonlinear Klein–Gordon equation [36]:which subjects to the initial conditions:According to the operations of differential transformation given in Table 1 for equation (58), we obtain the following recurrent relation:and from initial conditions (59), we writeSubstituting the above equation in equation (60), we drive the following results:Hence, the solution in the series form is as follows:which converges efficiently to the exact solution

Example 5. As the fifth example, we consider the following homogeneous nonlinear time-fractional gas dynamics equation as [37, 38]which subjects to the initial condition:According to the operations of differential transformation given in Table 1 for equation (64), we obtain the following recurrent relation:and from initial conditions (65), we writeSubstituting the above equation in equation (66), we drive the following results:Hence, the solution in series form is as follows:which for converges efficiently to the exact solution .

Example 6. As the sixth example, we consider the following system of inhomogeneous linear PDEs [33]:which subjects to the initial condition:According to the operations of differential transformation given in Table 1 for equation (70), we obtain the following recurrent relation:and from initial conditions (71), we writeSubstituting the above equations in equation (72), we drive the following results:Hence, the solutions in the series form are as follows:which converges efficiently to the exact solutions and .

Example 7. As the seventh example, we consider the following system of nonlinear PDEs:which subjects to the initial conditions:According to the operations of differential transformation given in Table 1 for equation (76), we obtain the following recurrent relation:and from initial conditions (77), we writeSubstituting the above equations in equation (78), we drive the following results:Hence, the solutions in the series form are as follows:which converges efficiently to the exact solutions and .

Example 8. Lastly, we consider the following system of homogeneous linear FPDEs [39, 40]:which subject to the initial conditions:According to the operations of differential transformation given in Table 1 for equation (82), we obtain the following recurrent relation:and from initial conditions (83), we writeSubstituting the above equations in equation (84), we drive the following results:Hence, the solutions in the series form are as follows:which for converges efficiently to the exact solutions and

Results for Examples 1–8 are reported in Figures 1–8 and Tables 2–9, respectively. In these tables, the terms , , and stand for exact solution, th order approximate solution of RDTM, and their absolute error, respectively.

(a)

(b)

(a)

(b)

(a)

(b)

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 Theorem 1, 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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

References

Y. Keskin, Selcuk University, Konya, Turkish, 2010, Ph.D. thesis.
Y. Keskin and G. Oturanc, “Reduced differential transform method for partial differential equations,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 6, pp. 741–749, 2009.
View at: Publisher Site | Google Scholar
Y. Keskin and G. Oturanc, “The reduced differential transform method: a new approach to fractional partial differential equations,” Nonlinear Science Letters A, vol. 1, no. 2, pp. 207–217, 2010.
View at: Google Scholar
Y. Keskin and G. Oturanc, “The reduced differential transformation method for solving linear and nonlinear wave equations,” Iranian Journal of Science and Technology, vol. 34, no. 2, pp. 113–122, 2010.
View at: Google Scholar
M. A. Abdou, “Fractional reduced differential transform method and its applications,” Journal of Nonlinear Sciences and Numerical Simulation, vol. 26, pp. 55–64, 2018.
View at: Google Scholar
M. O. Al-Amr, “New applications of reduced differential transform method,” Alexandria Engineering Journal, vol. 53, no. 1, pp. 243–247, 2014.
View at: Publisher Site | Google Scholar
M. S. Mohamed and K. A. Gepreel, “Reduced differential transform method for nonlinear integral member of Kadomtsev-Petviashvili hierarchy differential equations,” Journal of the Egyptian Mathematical Society, vol. 25, no. 1, pp. 1–7, 2017.
View at: Publisher Site | Google Scholar
R. K. Saeed and A. A. Mustafa, Numerical Solution of FisherKPP Equation by Using Reduced Differential Transform Method, AIP Publishing, Melville, NY, USA, 2017.
V. K. Srivastava, M. K. Awasthi, and S. Kumar, “Analytical approximations of two and three dimensional time-fractional telegraphic equation by reduced differential transform method,” Egyptian Journal of Basic and Applied Sciences, vol. 1, no. 1, pp. 60–66, 2014.
View at: Publisher Site | Google Scholar
R. Abazari, “Numerical simulation of coupled nonlinear Schrödinger equation by RDTM and comparison with DTM,” Journal of Applied Sciences, vol. 11, no. 20, pp. 3454–3463, 2011.
View at: Publisher Site | Google Scholar
O. Acan, O. Firat, and Y. Keskin, “Conformable variational iteration method, conformable fractional reduced differential transform method and conformable homotopy analysis method for nonlinear fractional partial differential equations,” Waves Random Complex Media, vol. 30, no. 2, pp. 250–268, 2018.
View at: Publisher Site | Google Scholar
Z. Cui, Z. Mao, S. Yang, and P. Yu, “Approximate analytical solutions of fractional perturbed diffusion equation by reduced differential transform method and the homotopy perturbation method,” Mathematical Problems in Engineering, vol. 2013, Article ID 186934, 2013.
View at: Publisher Site | Google Scholar
P. K. Gupta, “Approximate analytical solutions of fractional Benney-Lin equation by reduced differential transform method and the homotopy perturbation method,” Computers & Mathematics with Applications, vol. 61, no. 9, pp. 2829–2842, 2011.
View at: Publisher Site | Google Scholar
A. Saravanan and N. Magesh, “A comparison between the reduced differential transform method and the Adomian decomposition method for the Newell-Whitehead-Segel equation,” Journal of the Egyptian Mathematical Society, vol. 21, no. 3, pp. 259–265, 2013.
View at: Publisher Site | Google Scholar
J. Biazar and M. Eslami, “Differential transform method for quadratic riccati differential equation,” International Journal of Nonlinear Science, vol. 9, no. 4, pp. 444–447, 2010.
View at: Google Scholar
B. K. Singh and Mahendra, “A numerical computation of a system of linear and nonlinear time dependent partial differential equations using reduced differential transform method,” International Journal of Differential Equations, vol. 2016, Article ID 4275389, 8 pages, 2016.
View at: Publisher Site | Google Scholar
N. Taghizadeh and S. R. Moosavi Noori, “Exact solutions of the cubic nonlinear Schrodinger equation with a trapping potential by reduced differential transform method,” Mathematical Sciences Letters, vol. 5, no. 3, pp. 1–5, 2016.
View at: Publisher Site | Google Scholar
N. Taghizadeh and S. R. M. Noori, “Reduced differential transform method for solving parabolic-like and hyperbolic-like equations,” SeMA Journal, vol. 74, no. 4, pp. 559–567, 2017.
View at: Publisher Site | Google Scholar
O. Acan, M. Mohamed Al Qurashi, and D. Baleanu, “Reduced differential transform method for solving time and space local fractional partial differential equations,” Journal of Nonlinear Sciences and Applications, vol. 10, no. 10, pp. 5230–5238, 2017.
View at: Publisher Site | Google Scholar
M. Garg, P. Manohar, and S. L. Kalla, “Generalized differential transform method to space-time fractional telegraph equation,” International Journal of Differential Equations, vol. 2011, Article ID 548982, 9 pages, 2011.
View at: Publisher Site | Google Scholar
J. G. Liu, X. J. Yang, Y. Y. Feng, and P. Cui, “On the (N + 1)-dimensional local fractional reduced differential transform method and its applications,” Mathematical Methods in the Applied Sciences, vol. 43, no. 15, pp. 8856–8866, 2020.
View at: Publisher Site | Google Scholar
S. Saha Ray, “Numerical solutions and solitary wave solutions of fractional KdV equations using modified fractional reduced differential transform method,” Computational Mathematics and Mathematical Physics, vol. 53, no. 12, pp. 1870–1881, 2013.
View at: Publisher Site | Google Scholar
M. Valizadeh, Y. Mahmoudi, and F. Dastmalchi Saei, “Application of natural transform method to fractional pantograph delay differential equations,” Journal of Mathematics, vol. 2019, Article ID 3913840, 9 pages, 2019.
View at: Publisher Site | Google Scholar
M. Bakhshi and M. Asghari-Larimi, “Three dimensional differential transform method for solving nonlinear three-dimensional Volterra integral equations,” Journal of Mathematics and Computer Science, vol. 4, no. 2, pp. 246–256, 2012.
View at: Publisher Site | Google Scholar
S. R. Moosavi Noori and N. Taghizadeh, “Application of reduced differential transform method for solving two-dimensional Volterra integral equations of the second kind,” Applications and Applied Mathematics, vol. 14, pp. 1003–1019, 2019.
View at: Google Scholar
A. Ziqan, S. Armiti, and I. Suwan, “Solving three-dimensional Volterra integral equation by the reduced differential transform method,” International Journal of Applied Mathematical Research, vol. 5, no. 2, pp. 103–106, 2016.
View at: Publisher Site | Google Scholar
A. Arikoglu and I. Ozkol, “Solutions of integral and integro-differential equation systems by using differential transform method,” Computers & Mathematics with Applications, vol. 56, no. 9, pp. 2411–2417, 2008.
View at: Publisher Site | Google Scholar
S. R. Moosavi Noori and N. Taghizadeh, “Study on solving two-dimensional linear and nonlinear Volterra partial integro-differential equations by reduced differential transform method,” Applications and Applied Mathematics, vol. 15, no. 1, pp. 394–407, 2020.
View at: Google Scholar
A. Tari and S. Shahmorad, “Differential transform method for the system of two-dimensional nonlinear Volterra integro-differential equations,” Computers & Mathematics with Applications, vol. 61, no. 9, pp. 2621–2629, 2011.
View at: Publisher Site | Google Scholar
M. Caputo, “Linear models of dissipation whose Q is almost frequency independent–II,” Geophysical Journal International, vol. 13, no. 5, pp. 529–539, 1967.
View at: Publisher Site | Google Scholar
M. Caputo, Elasticita e Dissipazione. Zanichelli, Bologna, Italy, 1969.
M. Caputo and F. Mainardi, “Linear models of dissipation in anelastic solids,” La Rivista del Nuovo Cimento, vol. 1, no. 2, pp. 161–198, 1971.
View at: Publisher Site | Google Scholar
H. F. Ahmed, M. S. M. Bahgat, and M. Zaki, “Numerical approaches to system of fractional partial differential equations,” Journal of the Egyptian Mathematical Society, vol. 25, no. 2, pp. 141–150, 2017.
View at: Publisher Site | Google Scholar
I. Podlubny, Fractional Dierential Equations. Vol. 198 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1999.
D. Zeidan, C. K. Chau, T. T. Lu, and W. Q. Zheng, “Mathematical studies of the solution of Burgers’ equations by Adomian decomposition method,” Mathematical Methods in the Applied Sciences, pp. 1–18, 2019.
View at: Publisher Site | Google Scholar
M. S. H. Chowdhury and I. Hashim, “Application of homotopy-perturbation method to Klein-Gordon and sine-Gordon equations,” Chaos, Solitons & Fractals, vol. 39, pp. 1928–1935, 2009.
View at: Publisher Site | Google Scholar
A. Akgül, A. Cordero, and J. R. Torregrosa, “Solutions of fractional gas dynamics equation by a new technique,” Mathematical Methods in the Applied Sciences, vol. 43, no. 3, pp. 1349–1358, 2019.
View at: Publisher Site | Google Scholar
H. Thabet and S. Kendre, “Analytical solutions for conformable space-time fractional partial differential equations via fractional differential transform,” Chaos, Solitons & Fractals, vol. 109, pp. 238–245, 2018.
View at: Publisher Site | Google Scholar
H. Jafari, M. Nazari, D. Baleanu, and C. M. Khalique, “A new approach for solving a system of fractional partial differential equations,” Computers & Mathematics with Applications, vol. 66, no. 5, pp. 838–843, 2013.
View at: Publisher Site | Google Scholar
H. Jafari and S. Seifi, “Solving a system of nonlinear fractional partial differential equations using homotopy analysis method,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 5, pp. 1962–1969, 2009.
View at: Publisher Site | Google Scholar

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Copyright © 2021 Seyyedeh Roodabeh Moosavi Noori and Nasir Taghizadeh. 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.

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