Numerical Study of Two-Dimensional Volterra Integral Equations by RDTM and Comparison with DTM

Abstract

The two-dimensional Volterra integral equations are solved using more recent semianalytic method, the reduced differential transform method (the so-called RDTM), and compared with the differential transform method (DTM). The concepts of DTM and RDTM are briefly explained, and their application to the two-dimensional Volterra integral equations is studied. The results obtained by DTM and RDTM together are compared with exact solution. As an important result, it is depicted that the RDTM results are more accurate in comparison with those obtained by DTM applied to the same Volterra integral equations. The numerical results reveal that the RDTM is very effective, convenient, and quite accurate compared to the other kind of nonlinear integral equations. It is predicted that the RDTM can be found widely applicable in engineering sciences.

1. Introduction

Mathematical modeling of many problems in science, engineering, physics, and other disciplines leads to linear and nonlinear integrodifferential equations (IDE). The great use of mathematical models including integrodifferential equations is one of the main reasons obtaining the solutions of this kind of problems (see, e.g., [1–3] and the references therein). So, it is very important to get some information about the analytical solutions of these problems because these solutions give significant information about the character of the modeled event. But, in some cases, it is more difficult to obtain analytical solutions of these models. These are usually difficult to solve analytically, and in many cases the solution must be approximated. To approximate the solutions of these models, in recent years several numerical approaches have been proposed.

In this paper, we consider the following Volterra type of integral equation [4, 5]: 𝑢(𝑥,𝑡)−^𝑡₀^𝑥₀𝐾𝑥,𝑡,𝑦,𝑧,𝑢𝑦,𝑧𝑑𝑦𝑑𝑧=𝑓(𝑥,𝑡), (1) where 𝐾 and 𝑓 are continuous functions and 𝐾 has the following form: 𝐾𝑥,𝑡,𝑦,𝑧,𝑢𝑦,𝑧=^𝑚_𝑖=0𝑝_𝑖(𝑥,𝑡)𝑞_𝑖𝑦,𝑧,𝑢𝑦,𝑧. (2) The one-dimensional Volterra type of integral equation has been solved by many numerical methods, such as collocation methods [1], Taylor-series expansion methods [2], Gauss-type quadratures method [3], spectral methods [6], Chebyshev polynomial method [7], Tau method [8], sine-cosine wavelets method [9], Monte Carlo method [10], and Haar functions method [11].

But in two-dimensional cases, a small amount of work has been done (see, e.g., [12–14]). Very recently, Tari et al. in [4] employed the classic differential transform method for solving two-dimensional Volterra type of integral equations (1), and Jang in [5] improved the proofs of the presented theorems by Tari et al. in [4]. They derived fundamental properties of the differential transforms of some kernel functions 𝐾 in Volterra integral equations.

However, the classic differential transform method, introduced by Zhou [15], is based on the definition of the differential transform, which is a Taylor series. Thus, it requires a cumbersome calculation to obtain the basic properties of the differential transforms. Some of DTM applications are mentioned in [16–21].

Recently, Keskin and Oturanç introduced a reduced form of DTM as reduced DTM (RDTM) and applied it to approximate some PDE [22] and factional PDEs [23]. More recently, Abazari and Ganji [24] extended RDTM to study the partial differential equation with proportional delay in 𝑡 and shrinking in 𝑥 and showed that, as a special advantage of RDTM rather than DTM, the reduced differential transform recursive equations produce exactly all the Poisson series coefficients of solutions based on the initial condition as weighted function, whereas the differential transform recursive equations produce exactly all the Taylor series coefficients of solutions.

Here, we suggest the RDTM, for the approximating of the solutions of the two-dimensional Volterra integral equations (1) with the same kernel functions in [4, 5]. In order to demonstrate the effectiveness of the RDTM, the illustrative examples for the same kernel function of references [5] are presented. These examples show that the RDTM produces exactly all the POISSON series coefficients (see Remark 5) of the exact solutions, whereas, the classic DTM produces exactly all the TAYLOR series coefficients of the exact solutions. As an important result, notwithstanding the simplicity and robustness of RDTM, it is depicted that the RDTM results are more accurate in comparison with those obtained by classic DTM.

2. Basic Definitions

With reference to the articles [16–21], the basic definitions of two-dimensional differential transform method (DTM) and their reduced form (RDTM) are introduced in the following two subsections, respectively.

2.1. Two-Dimensional DTM

Consider a function of two variables 𝑤(𝑥,𝑡), and suppose that it can be represented as a product of two single-variable functions, that is, 𝑤(𝑥,𝑡)=𝑓(𝑥)𝑔(𝑡). On the basis of the properties of the one-dimensional differential transform, the function 𝑤(𝑥,𝑡) can be represented as 𝑤(𝑥,𝑡)=^∞_𝑖=0𝐹(𝑖)𝑥^𝑖∞_𝑗=0𝐺𝑗𝑡^𝑗=^∞∞_𝑖=0𝑗=0𝑊𝑖,𝑗𝑥^𝑖𝑡^𝑗, (3) where 𝑊(𝑖,𝑗)=𝐹(𝑖)𝐺(𝑗) is called the spectrum of 𝑤(𝑥,𝑡).

The basic definitions and operations for two-dimensional differential transform are introduced as follows.

Definition 1. If 𝑤(𝑥,𝑡) is analytic and continuously differentiable with respect to time 𝑡 in the domain of interest, then 𝑊()=1𝑚,𝑛𝜕𝑚!𝑛!𝜕𝑥^𝑚𝜕^𝑚+𝑛𝑡^𝑛𝑤(𝑥,𝑡)_{𝑥=𝑥0𝑡=𝑡0}, (4) where the spectrum function 𝑊(𝑚,𝑛) is the transformed function, which is also called 𝑇-function in brief.

The differential inverse transform of  𝑊(𝑘,ℎ) is defined as 𝑤(𝑥,𝑡)=^∞∞_𝑚=0𝑛=0𝑊(𝑚,𝑛)𝑥−𝑥₀^𝑚𝑡−𝑡₀^𝑛. (5) Combining (4) and (5), it can be obtained that 𝑤(𝑥,𝑡)=^∞∞_𝑚=0𝑛=01𝜕𝑚!𝑛!𝜕𝑥^𝑚𝜕^𝑚+𝑛𝑡^𝑛𝑤(𝑥,𝑡)_{𝑥=𝑥0𝑡=𝑡0}×𝑥−𝑥₀^𝑚𝑡−𝑡₀^𝑛. (6) When (𝑥₀,𝑡₀) are taken as (0,0), then (5) can be expressed as 𝑤(𝑥,𝑡)=^∞∞_𝑚=0𝑛=0𝑊(𝑚,𝑛)𝑥^𝑚𝑡^𝑛. (7) In real applications, the function 𝑤(𝑥,𝑡) is represented by a finite series of (7) that can be written as 𝑤_𝑀,𝑁(𝑥,𝑡)=^𝑀𝑁_𝑚=0𝑛=0𝑊(𝑚,𝑛)𝑥^𝑚𝑡^𝑛+𝑅_𝑀,𝑁(𝑥,𝑡), (8) and (7) implies that 𝑅_𝑀,𝑁(𝑥,𝑡)=∑^∞_𝑚=𝑀+1∑^∞_𝑛=𝑁+1𝑊(𝑚,𝑛)𝑥^𝑚𝑡^𝑛 is negligibly small. Usually, the values of 𝑀 and 𝑁 are decided by convergency of the series coefficients.

From the above definitions, it can be found that the concept of the two-dimensional differential transform is derived from the two-dimensional Taylor series expansion. With (4) and (5), the fundamental mathematical operations performed using the two-dimensional differential transform may be readily obtained, and these are listed in Table 1. (See [4, 5, 15, 16].)

Table 1 

The fundamental operations of two-dimensional differential transform method.

Recently, Jang [5] extended the two-dimensional DTM on (1) as follows.

Theorem 2. Assume that 𝑈(𝑚,𝑛), 𝑉(𝑚,𝑛), 𝐻(𝑚,𝑛), and 𝐺(𝑚,𝑛) are the differential transforms of the functions 𝑢(𝑥,𝑡), 𝑣(𝑥,𝑡), ℎ(𝑥,𝑡), and 𝑔(𝑥,𝑡), respectively; then we have the following:(a)if 𝑔(𝑥,𝑡)=∫^𝑡₀∫^𝑥₀𝑢(𝑦,𝑧)𝑑𝑦𝑑𝑧, then 𝐺()()𝐺()=1𝑚,0=𝐺0,𝑛=0,𝑚,𝑛=0,1,…,𝑚,𝑛𝑈()𝑚𝑛𝑚−1,𝑛−1,𝑚,𝑛=1,2,…, (9)(b)if 𝑔(𝑥,𝑡)=∫^𝑡₀∫^𝑥₀𝑢(𝑦,𝑧)𝑣(𝑦,𝑧)𝑑𝑦𝑑𝑧, then 𝐺()()𝐺()=1𝑚,0=𝐺0,𝑛=0,𝑚,𝑛=0,1,…,𝑚,𝑛𝑚𝑛^𝑛−1_ℓ=0^𝑚−1_𝑘=0𝑈(𝑘,ℓ)𝑉(𝑚−𝑘−1,𝑛−ℓ−1),𝑚,𝑛=1,2,…, (10)(c)if 𝑔(𝑥,𝑡)=ℎ(𝑥,𝑡)∫^𝑡₀∫^𝑥₀𝑢(𝑦,𝑧)𝑑𝑦𝑑𝑧, then 𝐺()()𝐺()=1𝑚,0=𝐺0,𝑛=0,𝑚,𝑛=0,1,…,𝑚,𝑛𝑚𝑛^𝑛−1_ℓ=0^𝑚−1_𝑘=0𝐻()𝑉()𝑘,ℓ𝑚−𝑘−1,𝑛−ℓ−1(),𝑚−𝑘)(𝑛−ℓ𝑚,𝑛=1,2,…. (11)

Proof. See [5].

Theorem 3. Assume that 𝑈(𝑚,𝑛), 𝑉(𝑚,𝑛), and 𝐺(𝑚,𝑛) are the differential transforms of the functions 𝑢(𝑥,𝑡), 𝑣(𝑥,𝑡), and 𝑔(𝑥,𝑡), respectively; then we have the following:(a)if 𝑔(𝑥,𝑡)=∫^𝑡₀∫^𝑥₀(𝑢(𝑦,𝑧)/𝑣(𝑦,𝑧))𝑑𝑦𝑑𝑧, then 𝐺(𝑚,0)=𝐺(0,𝑛)=0,𝑚,𝑛=0,1,…,_{𝑛−1ℓ=0𝑚−1𝑘=0}()𝑉()𝐺()(),𝑚−𝑘+1)(𝑛−ℓ+1𝑘,ℓ𝑚−𝑘+1,𝑛−ℓ+1=𝑈𝑚,𝑛 (12)(b)if 𝑔(𝑥,𝑡)=(1/𝑣(𝑥,𝑡))∫^𝑡₀∫^𝑥₀𝑢(𝑦,𝑧)𝑑𝑦𝑑𝑧, then 𝐺(𝑚,0)=𝐺(0,𝑛)=0,𝑚,𝑛=0,1,…,_{𝑛−1ℓ=0𝑚−1𝑘=0}𝑉()𝐺()=1𝑘,ℓ𝑚−𝑘+1,𝑛−ℓ+1()𝑈().𝑚+1)(𝑛+1𝑚,𝑛 (13)

Proof. See [5].

2.2. Two-Dimensional Reduced DTM (RDTM)

Consider a function of two variables 𝑤(𝑥,𝑡), and suppose that it can be represented as a product of two single-variable functions, that is, 𝑤(𝑥,𝑡)=𝑓(𝑥)𝑔(𝑡). Based on the properties of one-dimensional differential transform, the function 𝑤(𝑥,𝑡) can be represented as 𝑤(𝑥,𝑡)=^∞_𝑖=0𝐹(𝑖)𝑥^𝑖∞_𝑗=0𝐺𝑗𝑡^𝑗=^∞∞_𝑖=0𝑗=0𝑊𝑖,𝑗𝑥^𝑖𝑡^𝑗, (14) where 𝑊(𝑖,𝑗)=𝐹(𝑖)𝐺(𝑗) is called the spectrum of 𝑤(𝑥,𝑡).

Remark 4. The poisson function series generates a multivariate Taylor series expansion of the input expression 𝑤, with respect to the variables 𝑋, to order 𝑛, using the variable weights 𝑊.

Remark 5. The relationship introduced in (14) is the poisson series form of the input expression 𝑤(𝑥,𝑡), with respect to the variables 𝑥 and 𝑡, to order 𝑁, using the variable weights 𝑊_𝑘(𝑥).

Similarly on previous section, the basic definitions of two-differential reduced differential transformation are introduced as follows.

Definition 6. If 𝑤(𝑥,𝑡) is analytical function in the domain of interest, then the spectrum function 𝑊_𝑘(𝑥)=1𝜕𝑘!𝜕𝑡^𝑘𝑘𝑤(𝑥,𝑡)_𝑡=𝑡0 (15) is the reduced transformed function of 𝑤(𝑥,𝑡).

Similarly on previous sections, the lowercase 𝑤(𝑥,𝑡) respects the original function while the uppercase 𝑊_𝑘(𝑥) stands for the reduced transformed function. The differential inverse transform of 𝑊_𝑘(𝑥) is defined as 𝑤(𝑥,𝑡)=^∞_𝑘=0𝑊_𝑘(𝑥)𝑡−𝑡₀^𝑘. (16) Combining (15) and (16), it can be obtained that 𝑤(𝑥,𝑡)=^∞_𝑘=01𝜕𝑘!𝜕𝑡^𝑘𝑘𝑤(𝑥,𝑡)_𝑡=𝑡0𝑡−𝑡₀^𝑘. (17) In real applications, the function 𝑤(𝑥,𝑡) is represented by a finite series of (16), around 𝑡₀=0, and can be written as 𝑤_𝑛(𝑥,𝑡)=^𝑛_𝑘=0𝑊_𝑘(𝑥)𝑡^𝑘+𝑅_𝑛(𝑥,𝑡), (18) and (18) implies that 𝑅_𝑛(𝑥,𝑡)=∑^∞_𝑘=𝑛+1𝑊_𝑘(𝑥)𝑡^𝑘 is negligibly small. Usually, the values of 𝑛 and 𝑚 are decided by convergency of the series coefficients. From the above proposition, it can be found that the concept of the reduced two-dimensional differential transform is derived from the two-dimensional differential transform method. With (15) and (16), the fundamental mathematical operations performed by reduced two-dimensional differential transform can readily be obtained and listed in Table 2.

Table 2 

The fundamental operations of two-dimensional RDTM.

Similarly on previous subsection, we can extend the RDTM on Volterra integral equations (1) as follow.

Theorem 7. Assume that 𝑈_𝑘(𝑥), 𝑉_𝑘(𝑥), 𝐻_𝑘(𝑥), and 𝑊_𝑘(𝑥) are the reduced differential transforms of the functions 𝑢(𝑥,𝑡), 𝑣(𝑥,𝑡), ℎ(𝑥,𝑡), and 𝑤(𝑥,𝑡), respectively; then we have the following:(a)if 𝑤(𝑥,𝑡)=∫^𝑡_𝑡0∫^𝑥_𝑥0𝑢(𝑦,𝑧)𝑣(𝑦,𝑧)𝑑𝑦𝑑𝑧, then 𝑊_𝑘(𝑥)=1𝑘^𝑥_𝑥0^𝑘−1_𝑟=0𝑈_𝑟𝑦𝑉_{𝑘−𝑟−1}𝑦𝑑𝑦,𝑘=1,2,…, (19)(b)if 𝑤(𝑥,𝑡)=ℎ(𝑥,𝑡)∫^𝑡_𝑡0∫^𝑥_𝑥0𝑢(𝑦,𝑧)𝑑𝑦𝑑𝑧, then 𝑊_𝑘(𝑥)=^𝑘_𝑟=01𝐻𝑘−𝑟^𝑟(𝑥)^𝑥_𝑥0𝑈_{𝑘−𝑟−1}𝑦𝑑𝑦,𝑘=1,2,…. (20)

Proof. (a) According to the fundamental operations of two-dimensional RDTM listed in Table 2 and from Leibnitz formula, we get 𝜕^𝑘𝜕𝑡^𝑘𝑤()=𝜕𝑥,𝑡^𝑘𝜕𝑡^𝑘𝑡_𝑡0^𝑥_𝑥0𝑢𝑦,𝑧𝑣𝑦,𝑧𝑑𝑦𝑑𝑧=^𝑥_𝑥0𝜕^𝑘−1𝜕𝑡^𝑘−1𝑢𝑦,𝑡𝑣𝑦,𝑡𝑑𝑦=^𝑥_𝑥0^𝑘−1_𝑟=0𝑟𝑘−1𝜕^𝑟𝜕𝑡^𝑟𝑢×𝜕𝑦,𝑡_{𝑘−𝑟−1}𝜕𝑡^{𝑘−𝑟−1}𝑣𝑦,𝑡𝑑𝑦, (21) therefore 𝜕^𝑘𝜕𝑡^𝑘𝑤(𝑥,𝑡)_𝑡=𝑡0=^𝑥_𝑥0^𝑘−1_𝑟=0𝑟()!𝑘−1𝑟!𝑘−𝑟−1×𝑈_𝑟𝑦𝑉_{𝑘−𝑟−1}𝑦=()!𝑑𝑦𝑘−1^𝑥_𝑥0^𝑘−1_𝑟=0𝑈_𝑟𝑦𝑉_{𝑘−𝑟−1}𝑦𝑑𝑦, (22) and then, from using (15), for 𝑘=1,2,…, we get 𝑊_𝑘(𝑥)=1𝑘^𝑥_𝑥0^𝑘−1_𝑟=0𝑈_𝑟𝑦𝑉_{𝑘−𝑟−1}𝑦𝑑𝑦. (23) (b) Analogous to part (a), we get 𝜕^𝑘𝜕𝑡^𝑘𝑤()=𝜕𝑥,𝑡^𝑘𝜕𝑡^𝑘ℎ(𝑥,𝑡)^𝑡_𝑡0^𝑥_𝑥0𝑢=𝑦,𝑧𝑑𝑦𝑑𝑧_𝑘𝑟=0𝑘𝑟𝜕^𝑟𝜕𝑡^𝑟ℎ(𝑥,𝑡)^𝑥_𝑥0𝜕^{𝑘−𝑟−1}𝜕𝑡^{𝑘−𝑟−1}𝑢𝑦,𝑡𝑑𝑦, (24) therefore 𝜕^𝑘𝜕𝑡^𝑘𝑤(𝑥,𝑡)_𝑡=𝑡0=^𝑘_𝑟=0𝑘𝑟𝑟!(𝑘−𝑟−1)!𝐻_𝑟(𝑥)×^𝑥_𝑥0𝑈_{𝑘−𝑟−1}𝑦=𝑑𝑦,_𝑘𝑟=0𝑘!𝐻𝑘−𝑟^𝑟(𝑥)^𝑥_𝑥0𝑈_{𝑘−𝑟−1}𝑦𝑑𝑦, (25) and then from using (15), for 𝑘=1,2,…, we get 𝑊_𝑘(𝑥)=^𝑘_𝑟=01𝐻𝑘−𝑟^𝑟(𝑥)^𝑥_𝑥0𝑈_{𝑘−𝑟−1}𝑦𝑑𝑦. (26)

Theorem 8. Assume that 𝑈_𝑘(𝑥), 𝑉_𝑘(𝑥), and 𝑊_𝑘(𝑥) are the reduced differential transforms of the functions 𝑢(𝑥,𝑡), 𝑣(𝑥,𝑡), and 𝑤(𝑥,𝑡), respectively; then we have the following:(a)if 𝑤(𝑥,𝑡)=∫^𝑡_𝑡0∫^𝑥_𝑥0(𝑢(𝑦,𝑧)/𝑣(𝑦,𝑧))𝑑𝑦𝑑𝑧, then 𝑈_𝑘(𝑥)=^𝑘_𝑟=0(𝑟+1)𝜕𝑊_𝑟+1(𝑥)𝑉𝜕𝑥^𝑘−𝑟𝑦,𝑘=0,1,2,…, (27)(b)if 𝑤(𝑥,𝑡)=(1/𝑣(𝑥,𝑡))∫^𝑡_𝑡0∫^𝑥_𝑥0𝑢(𝑦,𝑧)𝑑𝑦𝑑𝑧, then 𝑘^𝑘_𝑟=0𝑊_𝑟(𝑥)𝑉_𝑘−𝑟(𝑥)=^𝑥_𝑥0𝑈_𝑘−1𝑦𝑑𝑦,𝑘=1,2,…. (28)

Proof. (a) By following the same manner as in the Theorem 7, we get 𝑢()=𝜕𝑥,𝑡²𝑤(𝑥,𝑡)𝑣(),𝜕𝑥𝜕𝑡𝑥,𝑡 (29) then 𝜕^𝑘𝜕𝑡^𝑘𝑢()=𝜕𝑥,𝑡^𝑘𝜕𝑡^𝑘𝜕²𝑤(𝑥,𝑡)𝑣(=𝜕𝑥𝜕𝑡𝑥,𝑡)_𝑘𝑟=0𝑘𝑟𝜕^𝑟+2𝜕𝑥𝜕𝑡^𝑟+1𝑤()𝜕𝑥,𝑡^𝑘−𝑟𝜕𝑡^𝑘−𝑟𝑣𝑦,𝑡, (30) therefore 𝑘!𝑈_𝑘(𝑥)=^𝑘_𝑟=0𝑘𝑟(𝑟+1)!(𝑘−𝑟)!𝜕𝑊_𝑟+1(𝑥)𝑉𝜕𝑥^𝑘−𝑟(𝑥), (31) and then from using (15), for 𝑘=0,1,2,…, we get 𝑈_𝑘(𝑥)=^𝑘_𝑟=0(𝑟+1)𝜕𝑊_𝑟+1(𝑥)𝑉𝜕𝑥^𝑘−𝑟(𝑥). (32) (b) Analogous to part (a), we get 𝑤(𝑥,𝑡)𝑣(𝑥,𝑡)=^𝑡_𝑡0^𝑥_𝑥0𝑢𝑦,𝑧𝑑𝑦𝑑𝑧, (33) then 𝜕^𝑘𝜕𝑡^𝑘{𝑤(𝑥,𝑡)𝑣(𝑥,𝑡)}=^𝑥_𝑥0𝜕^𝑘−1𝜕𝑡^𝑘−1𝑢𝑦,𝑡𝑑𝑦, (34) therefore ^𝑘_𝑟=0𝑘𝑟𝜕^𝑟𝜕𝑡^𝑟𝑤()𝜕𝑥,𝑡^𝑘−𝑟𝜕𝑡^𝑘−𝑟𝑣(𝑥,𝑡)=^𝑥_𝑥0𝜕^𝑘−1𝜕𝑡^𝑘−1𝑢𝑦,𝑡𝑑𝑦, (35) and then from using (15), for 𝑘=1,2,…, we get 𝑘!^𝑘_𝑟=0𝑊_𝑟(𝑥)𝑉_𝑘−𝑟(𝑥)=(𝑘−1)!^𝑥_𝑥0𝑈_𝑘−1𝑦𝑑𝑦, (36) and therefore 𝑘^𝑘_𝑟=0𝑊_𝑟(𝑥)𝑉_𝑘−𝑟(𝑥)=^𝑥_𝑥0𝑈_𝑘−1𝑦𝑑𝑦. (37)

3. Numerical Results of DTM and RDTM

In this section, the reduced differential transform technique is described to solve a class of Volterra integral equations (1) with kernel functions of (2). In order to demonstrate the effectiveness of the RDTM, the illustrative examples for the same kernel function of [5] are presented. In each example, the numerical results of DTM, RDTM, and their comparisons with exact solution are given in separate tables. The results of the test examples show that the RDTM results are more powerful than DTM results.

Example 9. In the first example, consider the following two-dimensional Volterra integral equation [5]: 𝑢(𝑥,𝑡)−^𝑡₀^𝑥₀𝑢𝑦,𝑧=(()).2+sin𝑦+𝑧𝑑𝑦𝑑𝑧𝑥−𝑡)(4+2sin𝑥+𝑡−𝑥𝑡 (38)(a)DTM: Jang [5] solved this equation by using DTM and obtained the following five-term DTM solution: 𝑢_5,5(𝑥,𝑡)=4𝑥+2𝑥²−𝑥⁴3+−4−2𝑥³3+𝑥⁵𝑡+𝑥15−2+⁴𝑡12²+2𝑥3−𝑥⁵𝑡180³+13−𝑥²𝑡12⁴+−𝑥+𝑥15³𝑡180⁵. (39)(b)RDTM: from Volterra integral equation (38), it is easy to see that the 𝑢(𝑥,0)=𝑥(4+2sin(𝑥)), and therefore RDTM version is 𝑈₀(𝑥)=𝑥(4+2sin(𝑥)). (40)By applying the RDTM properties listed in Theorem 8, on Volterra integral equation (38), for 𝑘=0,1,2,…, we get 𝑈_𝑘(𝑥)=^𝑘_𝑟=0()𝑑𝑟+1×𝑈𝑑𝑥_𝑟+1(𝑥)−^𝑟+1_ℓ=0𝑥𝛿_ℓ,0−𝛿_ℓ,1×4𝛿_{𝑟+1−ℓ,0}+2sin(𝑥+((𝑟+1−ℓ)𝜋/2))(𝑟+1−ℓ)!−𝑥𝛿_{𝑟+1−ℓ,1}×2𝛿_{𝑘−𝑟,0}()𝜋+sin𝑥+𝑘−𝑟2, (41) where 𝑈_𝑖(𝑥) is the reduced differential transform of 𝑢(𝑥,𝑡). After expanding the RDTM recurrence equations (41), with initial value of (40), for 𝑘=0,1,2,3,4, the first five terms of  𝑈_𝑘(𝑥) are obtained as follows: 𝑈₁(𝑥)(𝑥)(𝑥),𝑈=2𝑥cos−4−2sin₂(𝑥)(𝑥)(𝑥),𝑈=−𝑥sin−2cos₃(𝑥)(𝑥)−1=sin3(𝑥),𝑈𝑥cos₄(𝑥)=13(𝑥)+1cos(𝑥),𝑈12𝑥sin₅(𝑥)=1(𝑥)−160𝑥cos(𝑥).12sin (42) In the same manner, the rest of the components can be obtained by using the recursive equations (41). Substituting the quantities (41) in (18), the approximation solution of Volterra integral equation (38) in the Poisson series form is 𝑈₅()((𝑥+((𝑥)(𝑥𝑡+((𝑥)(𝑥𝑡𝑥,𝑡=2𝑥2+sin))2𝑥cos−4−2sin))−𝑥sin−2cos))²+(𝑥)−(𝑥)sin𝑥cos3𝑡³+cos(𝑥)3+𝑥sin(𝑥)𝑡12⁴+𝑥cos(𝑥)−(𝑥)60sin𝑡12⁵, (43) which is the same as the first five terms of the Poisson series of the exact solution 𝑢(𝑥,𝑡)=2(𝑥−𝑡)(2+sin(𝑥+𝑡)). The numerical results obtained with RDTM are presented in Table 3, in comparison with the classic DTM solution of [5] and the exact solution 𝑢(𝑥,𝑡)=2(𝑥−𝑡)(2+sin(𝑥+𝑡)), for some points of the intervals 0≤𝑥≤1 and 0≤𝑡≤1.

Table 3 

Comparisons of the exact solution 𝑢(𝑥,𝑡)=2(𝑥−𝑡)(2+sin(𝑥+𝑡)), with 𝑈_5,5(𝑥,𝑡) obtained by classic DTM [5] and 𝑈₅(𝑥,𝑡) obtained by RDTM at some test points (𝑥,𝑡) in Example 9.

Example 10. In the second example, consider the following two-dimensional Volterra integral equation [5]: 𝑢(𝑥,𝑡)−2^𝑡₀^𝑥₀𝑒^𝑦−𝑧𝑢()𝑒𝑦,𝑧𝑑𝑦𝑑𝑧=sin𝑥+𝑡^𝑥−𝑡+1−𝑒^−𝑡sin(𝑡)−𝑒^𝑥sin(𝑥). (44)(a)DTM: the approximation solution of this equation is also obtained by DTM in [5] as follows: 𝑢_5,5()=𝑥𝑥,𝑡𝑥−³6+𝑥⁵+𝑥1201−²2+𝑥⁴𝑡+−𝑥242+𝑥³−𝑥24⁵𝑡240²+−16+𝑥²−𝑥12⁴𝑡144³+𝑥−𝑥24³+𝑥144⁵𝑡2880⁴+1−𝑥120²+𝑥240⁴𝑡2880⁵. (45)(b)RDTM: it is easy to see that the 𝑢(𝑥,0)=sin(𝑥), and therefore RDTM version is 𝑈₀(𝑥)=sin(𝑥). (46)By applying the RDTM on nonlinear Volterra integral equation (44), for 𝑘=1,2,…, we get 𝑈_𝑘(𝑥)−^𝑘_𝑟=0sin(𝑥+𝑟𝜋/2)()𝑟!−1^𝑘−𝑟𝑒^𝑥(𝑘−𝑟)!+𝛿_{𝑘−𝑟,0}−^𝑘_𝑟=0(−1)^𝑟sin((𝑘−𝑟)𝜋/2)𝑟!(𝑘−𝑟)!−𝛿_𝑘,0𝑒^𝑥(𝑥=2sin)𝑘^𝑥₀^𝑘−1_𝑟=0𝑒^𝑦(−1)^𝑟𝑈𝑟!^{𝑘−𝑟−1}𝑦𝑑𝑦, (47) where 𝑈_𝑖(𝑥) is the reduced differential transform of 𝑢(𝑥,𝑡). After expanding the RDTM recurrence equations (47), with initial value of (46), for 𝑘=1,2,3,4,5, the first five terms of 𝑈_𝑘(𝑥) are obtained as follows: 𝑈₁(𝑥)(𝑥),𝑈=cos₂(𝑥)1=−2(𝑥),𝑈sin₃(𝑥)1=−6(𝑥),𝑈cos₄(𝑥)=1(𝑥),𝑈24sin₅(𝑥)1=−(𝑥).120cos (48) In the same manner, the rest of the components were obtained by using the recursive equations (47). Substituting the quantities (48) in (18), the approximation solution of Volterra integral equation (44) in the Poisson series form is 𝑈₅()(𝑥)(𝑥)(𝑥)𝑥,𝑡=sin+cos𝑡−sin2𝑡²−cos(𝑥)6𝑡³+sin(𝑥)𝑡24⁴−cos(𝑥)𝑡120⁵, (49) which is the same as the first five terms of the Poisson series of the exact solution 𝑢(𝑥,𝑡)=sin(𝑥+𝑡). The numerical results obtained with reduced DTM are presented in Table 4, in comparison with the classic DTM solution of [5] and the exact solution 𝑢(𝑥,𝑡)=sin(𝑥+𝑡), for some points of the intervals 0≤𝑥≤1 and 0≤𝑡≤1.

Table 4 

Comparisons of the exact solution 𝑢(𝑥,𝑡)=sin(𝑥+𝑡), with 𝑈_5,5(𝑥,𝑡) obtained by classic DTM [5] and 𝑈₅(𝑥,𝑡) obtained by reduced DTM at some test points (𝑥,𝑡) in Example 10.

Example 11. In the third example, consider the following two-dimensional Volterra integral equation [5]: 𝑢(𝑥,𝑡)−𝑒^{𝑡−𝑥𝑡}₀^𝑥₀𝑢()𝑒𝑦,𝑧𝑑𝑦𝑑𝑧=sinh𝑥+𝑡^𝑡−𝑥+1−𝑒^𝑡−𝑥(sinh(𝑥)−sinh(𝑡)). (50)(a)DTM: the approximation solution of this equation is also obtained by DTM in [5] as follows 𝑢_5,5()=𝑥𝑥,𝑡−𝑥−³6−𝑥⁵+𝑥1201+²2+𝑥⁴𝑡+−𝑥242−𝑥³−𝑥12⁵𝑡240²+16+𝑥²+𝑥12⁴𝑡144³+−𝑥−𝑥24³−𝑥144⁵𝑡2880⁴+1+𝑥120²+𝑥240⁴𝑡2880⁵. (51)(b)RDTM: it is easy to see that the 𝑢(𝑥,0)=−sinh(𝑥), and therefore RDTM version is 𝑈₀(𝑥)=−sinh(𝑥). (52)By applying the RDTM on nonlinear Volterra integral equation (50), for 𝑘=1,2,…, we get ^𝑘_𝑟=0𝑈_𝑟(𝑥)+𝛿_𝑟,01+𝑒^−𝑥2+1()2𝑟!1+−1^𝑟𝑒^𝑥−1+2^𝑟𝑒^−𝑥+𝑒^−2𝑥(−1)^𝑘−𝑟()!𝑒𝑘−𝑟^𝑥=1𝑘^𝑥₀𝑈_𝑘−𝑟𝑦𝑑𝑦, (53) where 𝑈_𝑖(𝑥) is the reduced differential transform of 𝑢(𝑥,𝑡). After expanding the RDTM recurrence equations (53), with initial value of (52), for 𝑘=1,2,3,4,5, the first five terms of 𝑈_𝑘(𝑥) are obtain as follows: 𝑈₁(𝑥)(𝑥),𝑈=cosh₂(𝑥)1=−2(𝑥),𝑈sinh₃(𝑥)=16(𝑥),𝑈cosh₄(𝑥)=1(𝑥),𝑈24sinh₅(𝑥)=1(𝑥).120cosh (54) In the same manner, the rest of the components were obtained by using the recursive equations (47). Substituting the quantities (48) in (18), the approximation solution of Volterra integral equation (44) in the Poisson series form is 𝑈₅()(𝑥)(𝑥)(𝑥)𝑥,𝑡=−sinh+cosh𝑡−sinh2𝑡²+cosh(𝑥)6𝑡³−sinh(𝑥)𝑡24⁴+cosh(𝑥)𝑡120⁵, (55) which is same as the first five terms of the Poisson series of the exact solution 𝑢(𝑥,𝑡)=sinh(𝑡−𝑥). The numerical results obtained with reduced DTM are presented in Table 5, in comparison with the classic DTM solution of [5] and the exact solution 𝑢(𝑥,𝑡)=sinh(𝑡−𝑥), for some points of the intervals 0≤𝑥≤1 and 0≤𝑡≤1.

Table 5 

Comparisons of the exact solution 𝑢(𝑥,𝑡)=sinh(𝑡−𝑥), with 𝑈_5,5(𝑥,𝑡) obtained by classic DTM [5] and 𝑈₅(𝑥,𝑡) obtained by reduced DTM at some test points (𝑥,𝑡) in Example 11.

4. Conclusions

In this study, we presented the definition and operation of both two-dimensional differential transformation method (DTM) and their reduced form, the so-called reduced-DTM (RDTM) for finding the solutions of a class of Volterra integral equations. For illustration purposes, we consider three different examples. It is worth pointing out that both DTM and RDTM have convergence for the solutions; actually, the accuracy of the series solution increases when the number of terms in the series solution is increased. From the computational process of DTM and RDTM, we find that the RDTM is easier to apply. In other words, it is obvious that DTM has very complicated computational process rather than RDTM. The RDTM reduces the computational difficulties of the DTM and all the calculations can be made with simple manipulations MATLAB. Actually, as a special advantage of RDTM rather than DTM, the reduced differential transform recursive equations produce exactly all the Poisson series coefficients of solutions, whereas the differential transform recursive equations produce exactly all the Taylor series coefficients of solutions. The reliability of the RDTM and the reduction in the size of computational domain give this method a wider applicability. For small value of 𝑥, 𝑡, in Tables 3, 4, and 5, we find that the RDTM has a smaller error than DTM. Also, for large values of 𝑥, 𝑡, we may increase the accuracy of the series solution by computing more terms, which is quite easy using MATLAB.