Extension of Zhou's Method to Neutral Functional-Differential Equation with Proportional Delays

Abstract

The differential transform method (DTM) is a reliable method applied by providing new theorems to develop exact and approximate solutions of neutral functional-differential equation (NFDE) with proportional delays. The results obtained with the proposed methods are in good agreement with one obtained by other methods. The advantages of this technique are illustrated. It is easy to see that the DTM is very accurate and easy to implement in finding analytical solutions of wide classes of linear and nonlinear NFDEs.

1. Introduction

The neutral functional-differential equation (NFDE) is 𝑢(𝑡)+𝑎(𝑡)𝑢𝜎_𝑚(𝑡)^(𝑚)=𝛽𝑢(𝑡)+^𝑚−1_𝑘=0𝑏_𝑘(𝑡)𝑢^𝑘𝜎_𝑘(𝑡)+𝑓(𝑡),𝑡≥0,(1.1) under the conditions ^𝑚⁻¹_𝑘=0𝑐_𝑖𝑘𝑢^(𝑘)(0)=𝜆_𝑖,𝑖=0,1,…,𝑚−1,(1.2) where 𝑎,𝑏_𝑘,𝜎_𝑘 are analytical functions; 𝛽,𝑐_𝑖𝑘 and 𝜆_𝑖∈𝒞. A classical case [1] 𝜎_𝑘(𝑡)=𝑡−𝜏_𝑘,𝑘=0,1,…,𝑚,(1.3) where 𝜏_𝑘 is positive. Another interesting case [2] is 𝜎_𝑘(𝑡)=𝑞_𝑘𝑡,𝑘=0,1,…,𝑚,(1.4) where 0<𝑞_𝑘<1. Both cases are playing an interesting role in many applications (see [1, 2] and references therein). In recent years, there has been a growing interest in the numerical treatment of NFDE, some of which are the Adams method [3], continuous Runge-Kutta methods [4], segmented Tau approximation [5], Homotopy perturbation method [6], one-leg 𝜃-methods [7, 8], and variational iteration method [9].

In this paper we consider the following neutral functional-differential equations with proportional delays.

Problem 1. Consider𝑢^(𝑡)=𝛽𝑢(𝑡)−𝑎(𝑡)𝑢^𝑞_ℓ𝑡+𝑓𝑢𝑡,(𝑢0()𝑡)=,𝑢𝑢₀𝑞_𝑖𝑡,𝑢^𝑞_𝑖𝑡,𝑖=1,2,...,ℓ−1,.(1.5)

Problem 2. Consider𝑢^(𝑚)(𝑡)=𝛽𝑢(𝑡)−𝑎(𝑡)𝑢^(𝑚)𝑞_ℓ𝑡+𝑓𝑡,𝑢(𝑡),𝑢𝑞_𝑖𝑡,𝑢^𝑞_𝑖𝑡,𝑢^𝑞_𝑖𝑡,...,𝑢^(𝑚−1)𝑞_𝑖𝑡_𝑚−1,_𝑘=0𝑐_𝑗𝑘𝑢^(𝑘)(0)=𝜆_𝑗,𝑗=0,1,2,...,𝑚−1,(1.6) where 𝑎,𝑓 are analytical functions; 𝛽,𝑐_𝑗𝑘 and 𝜆_𝑗 are real or complex constants; 0<𝑞_𝑖<1,𝑖=1,2,…,ℓ.

The basic motivation of this work is to extend the differential transform method (DTM) by presenting and proving new theorems to create the exact or approximate solutions to a high degree of accuracy to the Problems 1 and 2. The DTM is a numerical-analytical technique that was first proposed by Zhou (1986) [10], who solved problems in electric circuit analysis. Since then, DTM was successfully applied for a large variety problems. For example, differential-difference equations [11], Volterra integral equation with separable kernels [12], MHD boundary-layer equations [13], linear and nonlinear systems of partial differential equations [14], and nonlinear oscillators with fractional nonlinearities [15]. To the best of our knowledge differential transform method has not be used by any researcher before to solve NFDE. By this method it is possible to obtain highly accurate results when compared with existing results from variational iteration method [9] and homotopy perturbation method [6].

2. Basic Idea of Differential Transform Method

The differential transform of the 𝑘th derivative of a function 𝑢(𝑡) is defined as follows: 𝑈(𝑘)=1𝑘!𝑑𝑑𝑡^𝑘𝑘𝑢(𝑡)_𝑡=𝑡0.(2.1) The original and transformed functions are denoted throughout this paper by small and capital letters, respectively. The differential inverse transform of 𝑈(𝑘) is defined as 𝑢(𝑡)=^𝑁_𝑘=0𝑈(𝑘)𝑡−𝑡₀^𝑘,for𝑁⟶∞.(2.2) And from (2.1) and (2.2): 𝑢(𝑡)=^𝑁_𝑘=01𝑘!𝑑𝑑𝑡^𝑘𝑘𝑢(𝑡)_𝑡=𝑡0𝑡−𝑡₀^𝑘,for𝑁⟶∞,(2.3) which is actually the Taylor series expansion of 𝑢(𝑡) about the point 𝑡₀.

The fundamental mathematical operations performed by one-dimensional differential transform can be obtained from (1.5) and (1.6) in Table 1 (also, see [10]).

Table 1 

Fundamental operations of differential transformation.

3. Main Results

In the following theorems, we find the differential transformations of given functions. These results are very useful in our approach for solving NFDEs.

Theorem 3.1. Suppose that 𝑈(𝑘),𝐹(𝑘), and 𝐺(𝑘) are the differential transformations of the functions 𝑢(𝑡),𝑓(𝑡), respectively, and 𝑔(𝑡) and 0<𝑞,𝑞_𝑖<1, for 𝑖=1,2,…,𝑚: (I)if 𝑢(𝑡)=𝑓(𝑞𝑡), then 𝑈(𝑘)=𝑞^𝑘𝐹(𝑘), (II)if 𝑢(𝑡)=𝑑^𝑛𝑓(𝑞𝑡)/𝑑(𝑞𝑡)^𝑛, then 𝑈(𝑘)=𝑞^𝑘((𝑘+𝑛)!/𝑘!)𝐹(𝑘+𝑛), (III)if 𝑢(𝑡)=𝑔(𝑡)𝑑^𝑛𝑓(𝑞𝑡)/𝑑(𝑞𝑡)^𝑛, then 𝑈(𝑘)=^𝑘_ℓ=0𝑞^𝑘−ℓ(𝑘−ℓ+𝑛)(𝑘−ℓ)!𝐺(ℓ)𝐹(𝑘−ℓ+𝑛),(3.1)(IV)if 𝑢(𝑡)=(𝑑^𝑛/𝑑(𝑞₁𝑡)^𝑛)[𝑓(𝑞₁𝑡)](𝑑^𝑚/𝑑(𝑞₂𝑡)^𝑚)[𝑔(𝑞₂𝑡)], then 𝑈(𝑘)=^𝑘_ℓ=0𝑞^ℓ1𝑞^2𝑘−ℓ(ℓ+𝑛)!(𝑘−ℓ+𝑚)!(𝑘−ℓ)!(ℓ)!𝐹(ℓ+𝑛)𝐺(𝑘−ℓ+𝑚),(3.2)(V)if 𝑢(𝑡)=(𝑑^𝑛1/𝑑(𝑞₁𝑡)^𝑛1)[𝑓₁(𝑞₁𝑡)](𝑑^𝑛2/𝑑(𝑞₂𝑡)^𝑛2)[𝑓₂(𝑞₂𝑡)]⋯(𝑑^𝑛𝑚−1/𝑑(𝑞_𝑚−1𝑡)^𝑛𝑚−1)[𝑓_𝑚−1(𝑞_𝑚−1𝑡)(𝑑^𝑛𝑚/𝑑(𝑞_𝑚𝑡)^𝑛𝑚)[𝑓_𝑚(𝑞_𝑚𝑡)], then 𝑈(𝑘)=^𝑘_{ℓ𝑚−1ℓ=0}^𝑚−1_{ℓ𝑚−2=0}⋯^ℓ3_ℓ2ℓ=0²_ℓ1=0𝑞^ℓ1₁𝑞^ℓ2−ℓ1₂⋯𝑞^{ℓ𝑚−1−ℓ𝑚−2}_𝑚−1𝑞^{ℓ𝑘−ℓ𝑚−1}_𝑚×ℓ¹+𝑛₁!ℓ₁!ℓ²−ℓ₁+𝑛₂!ℓ₂−ℓ₁!⋯ℓ^𝑚−1−ℓ_𝑚−2+𝑛_𝑚−1!ℓ_𝑚−1−ℓ_𝑚−2!𝑘−ℓ^𝑚−1+𝑛_𝑚!×𝐹₁𝑘−ℓ_𝑚−1!ℓ₁+𝑛₁𝐹₂ℓ₂−ℓ₁+𝑛₂⋯𝐹_𝑚−1ℓ_𝑚−1−ℓ_𝑚−1+𝑛_𝑚−1𝐹_𝑚𝑘−ℓ_𝑚−1+𝑛_𝑚.(3.3)

Proof. (I), (II) The proof follows immediately by substituting 𝑢(𝑡) into (2.1).
(III) By using the definition of DTM (2.1), we have 𝑈(𝑘)=1𝑘!𝑑𝑑𝑡^𝑘𝑘𝑔(𝑡)𝑑(𝑑𝑞^𝑛𝑡)^𝑛𝑓(𝑞𝑡)_𝑡=𝑡0=1𝑘!⎡⎢⎢⎣^𝑘_ℓ=0⎛⎜⎜⎝ℓ𝑘⎞⎟⎟⎠𝑑_ℓ𝑑𝑡^ℓ𝑔(𝑡)𝑑^𝑘−ℓ𝑑𝑡^𝑘−ℓ𝑑(𝑑𝑞^𝑛𝑡)^𝑛𝑓(𝑞𝑡)⎤⎥⎥⎦_𝑡=𝑡0,(3.4) and from (II), we have 𝑈(𝑘−ℓ)=1(𝑘−ℓ)!𝑑𝑑𝑡^{𝑘−ℓ𝑘−ℓ}𝑑^𝑛𝑑(𝑞𝑡)^𝑛𝑓(𝑞𝑡)_𝑡=𝑡0=𝑞^𝑘−ℓ(𝑘−ℓ+𝑛)!(𝑘−ℓ)!𝐹(𝑘−ℓ+𝑛).(3.5) By utilizing this value, we get 𝑈(𝑘)=^𝑘_ℓ=0𝑞^𝑘−ℓ(𝑘−ℓ+𝑛)!(𝑘−ℓ)!𝐺(ℓ)𝐹(𝑘−ℓ+𝑛).(3.6)
(IV) By using the definition of DTM (2.1), we have 𝑈(𝑘)=1𝑘!𝑑𝑑𝑡^𝑘𝑘𝑑(𝑞𝑑^𝑛₁𝑡)^𝑛𝑓𝑞₁𝑡𝑑^𝑚𝑑(𝑞₂𝑡)^𝑚𝑓𝑞₂𝑡_𝑡=𝑡0=1𝑘!⎡⎢⎢⎣^𝑘_ℓ=0⎛⎜⎜⎝ℓ𝑘⎞⎟⎟⎠𝑑_ℓ𝑑𝑡^ℓ𝑑(𝑞𝑑^𝑛₁𝑡)^𝑛𝑓𝑞₁𝑡𝑑^𝑘−ℓ𝑑𝑡^𝑘−ℓ𝑑(𝑑𝑞^𝑚_2𝑡)^𝑚𝑔𝑞₂𝑡⎤⎥⎥⎦_𝑡=𝑡0,(3.7) then from (II), we have 𝑈(ℓ)=1ℓ!𝑑𝑑𝑡^ℓℓ𝑑^𝑛𝑑𝑞₁𝑡^𝑛𝑓𝑞₁𝑡=𝑞^ℓ1(ℓ+𝑛)!𝑈(𝑘−ℓ)=1ℓ!𝐹(ℓ+𝑛),(𝑘−ℓ)!𝑑𝑑𝑡^{𝑘−ℓ𝑘−ℓ}𝑑^𝑚𝑑𝑞₂𝑡^𝑚𝑔𝑞₂𝑡=𝑞^2𝑘−ℓ(𝑘−ℓ+𝑚)!(𝑘−ℓ)!𝐺(𝑘−ℓ+𝑚).(3.8) By utilizing these values 𝑈(𝑘)=^𝑘_ℓ=0𝑞^ℓ1𝑞^2𝑘−ℓ(ℓ+𝑛)!(𝑘−ℓ+𝑚)!(𝑘−ℓ)!(ℓ)!𝐹(ℓ+𝑛)𝐺(𝑘−ℓ+𝑚).(3.9)
(V) Let the differential transform of 𝑑^𝑛𝑖𝑓_𝑖/𝑑(𝑞_𝑖𝑡)^𝑛𝑖 at 𝑡=𝑡₀ for 𝑖=1,2,...,𝑚 be 𝐻_𝑖(𝑘), then by using operations of differential transformation given in Table 1, we have 𝑈(𝑘)=^𝑘_{ℓ𝑚−1ℓ=0}^𝑚−1_{ℓ𝑚−2=0}⋯^ℓ3_ℓ2ℓ=0²_ℓ1=0𝐻₁ℓ₁𝐻₂ℓ₂−ℓ₁⋯𝐻_𝑚−1ℓ_𝑚−1−ℓ_𝑚−1𝐻𝑘−ℓ_𝑚−1,(3.10) and from (II), we have 𝐻₁ℓ₁=𝑞^ℓ1₁ℓ₁+𝑛₁!ℓ₁!𝐹¹ℓ₁𝐻₂+𝑛₁,ℓ₂−ℓ₁=𝑞^ℓ2−ℓ1₂ℓ₂−ℓ₁+𝑛₂!ℓ₂−ℓ₁!𝐹²𝐻⋮ℓ₂−ℓ₁+𝑛₁,(3.11)_𝑚−1ℓ_𝑚−1−ℓ_𝑚−2=𝑞^{ℓ𝑚−1−ℓ𝑚−2}_𝑚−1ℓ_𝑚−1−ℓ_𝑚−2+𝑛_𝑚−1!ℓ_𝑚−1−ℓ_𝑚−21!𝐹^𝑚−1𝐻_𝑚ℓ_𝑚−1−ℓ_𝑚−2+𝑛_𝑚−1,𝑘−ℓ_𝑚−1=𝑞^{𝑘−ℓ𝑚−1}_𝑚𝑘−ℓ_𝑚−1+𝑛_𝑚!𝑘−ℓ_𝑚−1!𝐹^𝑚𝑘−ℓ_𝑚−1+𝑛_𝑚.(3.12) Substituting those values into (3.10), we obtain (3.3).

4. Illustrative Examples

In this part, we will apply the DTM to solve NFDE with proportional delays.

The numerical solutions of Examples 4.3, 4.4, and 4.5 have been calculated by variational iteration method [9] and homotopy perturbation method [6], which did not yield the exact solutions. However, applying DTM gives the exact solutions of those examples, as we will show later.

Example 4.1 (see [6, 9]). Consider the following first-order NFDE with proportional delay: 𝑢^(𝑡)=−𝑢(𝑡)+12𝑢𝑢2(𝑡0)+=211.𝑢^2𝑡,0<𝑡<1,(4.1) Taking the differential transform of (4.1) as given in (2.1), we get (𝑘+1)𝑈(𝑘+1)=−𝑈(𝑘)+1212^𝑘𝑈(𝑘)+1212^𝑘(𝑘+1)𝑈(𝑘+1),(4.2) which can be rewritten as follows: 𝑈(𝑘+1)=−𝑈(𝑘)(𝑘+1).(4.3) The differential transform of the initial condition of 𝑢(𝑡) at 𝑡₀=0, is 𝑈(0)=1, form (4.3) for 𝑘=0,1,…,8, we can get 𝑈(1)=−1,𝑈(2)=1𝑈(5)=−12!,𝑈(3)=−31!,𝑈(4)=41!,5!,𝑈(6)=61!,𝑈(7)=−71!,𝑈(8)=81!,(4.4) substituting these values into (2.2), to get 𝑢(𝑡)=1−𝑡+𝑡²2!−3𝑡³!+4𝑡⁴!−5𝑡⁵!+6𝑡⁶!−7𝑡⁷!+8𝑡⁸!.(4.5) The closed form of the above solution, when 𝑁→∞ is 𝑢(𝑡)=𝑒^−𝑡, which is the exact solution. In Table 2 the absolute errors of DTM for 𝑁=7,8, VIM [9] with eight terms and HPM [6] with eight terms (Table 3) are compared.

Table 2 

Comparison of the absolute errors for Example 4.1.

Table 3 

Comparison of the absolute errors for Example 4.2.

Example 4.2 (see [6, 9]). Consider the first-order NFDE with proportional delay 𝑢^(𝑡)=−𝑢(𝑡)+0.1𝑢(0.8𝑡)+0.5𝑢𝑢^((00.)8=𝑡)0+.(0.32𝑡−0.5)𝑒^−0.8𝑡+𝑒^−𝑡,𝑡≥0,(4.6) Let The differential transforms of 𝑡𝑒^−0.8𝑡, 𝑒^−0.8𝑡, and 𝑒^−𝑡 at 𝑡₀=0 be 𝛿₁(𝑘),𝛿₂(𝑘), and 𝛿₃(𝑘), respectively: 𝛿₁(𝑘)=⎧⎪⎨⎪⎩0(−,1)^𝑘−1(0.8)^𝑘−1𝑘=0,𝛿𝑘−1!,𝑘≠0,₂(𝑘)=(−1)^𝑘(0.8)^𝑘𝛿𝑘!₃,(𝑘)=(−1)^𝑘𝑘!.(4.7) We can obtain the differential transform of (4.6) as 𝑈(𝑘+1)=−𝑈(𝑘)1−0.1(0.8)^𝑘+0.32𝛿₁(𝑘)−0.5𝛿₂(𝑘)+𝛿₃(𝑘)(𝑘+1)1−0.5(0.8)^𝑘.(4.8) At 𝑡₀=0, the initial condition transformed to 𝑈(0)=0, so from (4.8), we have 𝑈(1)=1,𝑈(2)=−1,𝑈(3)=1𝑈(5)=12,𝑈(4)=−16,24,𝑈(6)=−1120,𝑈(7)=7210,𝑈(8)=−50140.(4.9) Substituting these values into (2.2), we get 𝑢(𝑡)=𝑡−𝑡²+12𝑡³−16𝑡⁴+124𝑡⁵−1120𝑡⁶+1720𝑡⁷−15040𝑡⁸.(4.10) The closed form of the above solution, when 𝑁→∞, is 𝑢(𝑡)=𝑡𝑒^−𝑡, which is the exact solution. In Table 2 we compare the absolute errors of DTM for 𝑁=7,8, VIM [9] with eight terms, and HPM [6] with eight terms.

Example 4.3 (see[6, 9]). Consider the following second-order NFDE with proportional delay: 𝑢^(𝑡)=𝑢^𝑡𝑢(20)=−121,𝑡𝑢^𝑢2𝑡_+2,0<𝑡<1,(0)=0.(4.11) The differential transform for (4.11) is found as (𝑘+1)(=𝑘+12)𝑈(𝑘+2)2^𝑘(𝑘+1)𝑈(𝑘+1)−12^𝑘_ℓ=012^𝑘−ℓ(𝑘−ℓ+2)!(𝑘−ℓ)!𝛿(ℓ−1)𝑈(𝑘−ℓ+2)+2𝛿(𝑘),(4.12) form the initial condition we can get 𝑈(0)=1 and 𝑈(1)=0. Form (4.12), we get 𝑈(𝑘)=10,,𝑘𝑘=>22,.(4.13) Then, by using (2.2), 𝑢(𝑡)=1+𝑡², which is the exact solution.

Example 4.4 (see [6, 9]). Consider the second-order NFDE with proportional delay: 𝑢^(𝑡)=34𝑢(𝑡)+𝑢2𝑡+𝑢^2𝑡+21𝑢^2𝑡−𝑡²𝑢(0)=𝑢_−𝑡+1,0<𝑡<1,(0)=0.(4.14) The differential transform for (4.14) at 𝑡₀=0 is given by (𝑘+1)(𝑘+2)𝑈(𝑘+2)=34𝑈(𝑘)+21^𝑘𝑈(𝑘)+12^𝑘(𝑘+1)𝑈(𝑘+1)+1221^𝑘×(𝑘+2)𝑈(𝑘+2)−𝛿(𝑘−2)−𝛿(𝑘−1)+𝛿(𝑘(𝑘),+1)(4.15) and can be rewritten as 𝑈(𝑘+2)=3/4+1/2^𝑘𝑈(𝑘)+1/2^𝑘(𝑘+1)𝑈(𝑘+1)−𝛿(𝑘−2)−𝛿(𝑘−1)+𝛿(𝑘)(𝑘+1)(𝑘+2)1−1/2^𝑘+1.(4.16)
Form the initial condition we can get 𝑈(0)=𝑈(1)=0, and form (4.16), we get 𝑈(𝑘)=10,,𝑘𝑘=>22,.(4.17) Then, by using (2.2), 𝑢(𝑡)=𝑡², which is the exact solution.

Example 4.5 (see [6, 9]). Consider the following third-order NFDE with proportional delays: 𝑢^(𝑡)=𝑢(𝑡)+𝑢^𝑡2+𝑢^3𝑡+21𝑢^4𝑡−𝑡⁴−𝑡³𝑢(0)=0,𝑢_2−43𝑡²+21𝑡,(0)=0,𝑢^(0)=0.(4.18) The differential transform of (4.18) can be written as 𝑈(𝑘)+12^𝑘(𝑘+1)𝑈(𝑘+1)+13^𝑘𝑈(𝑘+3)=−𝛿(𝑘−4)−(1/2)𝛿(𝑘−(𝑘3+)−1)((4𝑘/3+)𝛿2)(𝑘𝑈−(𝑘2+)+2)21𝛿(𝑘−1)(𝑘+1)(𝑘+2)(𝑘+3)1−(1/2)1/4^𝑘.(4.19)
Form the initial condition we can get 𝑈(0)=𝑈(1)=𝑈(2)=0, so from the (4.19), we get 𝑈(𝑘)=01,,𝑘𝑘≠=44,.(4.20) Substituting (4.20) in (2.2) gives 𝑢(𝑡)=𝑡⁴, which is the exact solution.

5. Conclusion

In this study, we extended DTM to the solution of NFDE with proportional delays. New theorems are presented with their proofs. All examples results show that the DTM is more effective than VIM and HPM for solving NFDE with proportional delays. We believe that the ease of implementation and efficiency of the DTM gives it much wider applicability.