The Combined RKM and ADM for Solving Nonlinear Weakly Singular Volterra Integrodifferential Equations

Abstract

The reproducing kernel method (RKM) and the Adomian decomposition method (ADM) are applied to solve 𝑛th-order nonlinear weakly singular Volterra integrodifferential equations. The numerical solutions of this class of equations have been a difficult topic to analyze. The aim of this paper is to use Taylor’s approximation and then transform the given 𝑛th-order nonlinear Volterra integrodifferential equation into an ordinary nonlinear differential equation. Using the RKM and ADM to solve ordinary nonlinear differential equation is an accurate and efficient method. Some examples indicate that this method is an efficient method to solve 𝑛th-order nonlinear Volterra integro-differential equations.

1. Introduction

In this paper, we consider the following 𝑛th-order nonlinear weakly singular Volterra integro-differential equation of the following form [1–4]: ^𝑛_𝑖=0𝑎_𝑖(𝑥)𝑢^(𝑖)(𝑥)=𝑓(𝑥)+^𝑥_𝑎𝐾(𝑡,𝑥)𝑢^𝑚(𝑡)𝑑𝑡,𝑎⩽𝑥⩽𝑏, (1.1) where 𝑚>1, 𝑓(𝑥) is a given function, 𝑢(𝑥) is the unknown function, and 𝐾(𝑡,𝑥) is the kernel of the integro equation. We usually assume that the function 𝑢(𝑥) and 𝑓(𝑥) are continuous or square integrable on [𝑎,𝑏].

Some problems of mathematical physics are described in terms of (1.1) which has been studied by different methods including the spline collocation method [5], piecewise polynomials [6], Haar wavelets [7], the homotopy perturbation method (HPM) [8, 9], the wavelet-Galerkin method [10], Taylor polynomials [11], the Tau method [12], the sinc-collocation method [13], the combined Laplace transform-Adomian decomposition method [14], and the Adomian’s asymptotic decomposition method [15] to determine exact and approximate solutions. But to our knowledge there is still no viable analytic approach for solving weakly singular Volterra integro-differential equations. The present work is motivated by the desire to obtain approximate solution to 𝑛th-order nonlinear weakly singular Volterra integro-differential equation, where the integrand is weakly singular in the sense that its integral is continuous at the singular point, that is, its kernel 𝐾(𝑡,𝑥)=1/(𝑥−𝑡)^𝛼 is singular as 𝑡→𝑥.

Reproducing kernel theory has important application in numerical analysis, differential equation, probability and statistics and so on [16, 17]. And the RKM has been applied successfully to solving linear and nonlinear problems [18–20].

The rest of the paper is organized as follows. In the Section 2, transforming (1.1) into an differential equation by Taylor’s approximation. In Section 3, the RKM is introduced. Applying RKM and ADM to solving (1.1) is discussed in Section 4. The numerical examples are presented in Section 5. Finally, a brief conclusion is stated in last section.

2. Taylor’s Approximation

Consider the following 𝑛th-order nonlinear weakly singular Volterra integro-different equation: ^𝑛_𝑖=0𝑎_𝑖(𝑥)𝑢^(𝑖)(𝑥)=𝑓(𝑥)+^𝑥_𝑎𝐹(𝑢(𝑡))(𝑥−𝑡)^𝛼𝑑𝑡,𝑎⩽𝑥⩽𝑏,0<𝛼<1. (2.1) We have by setting 𝑦(𝑡)=𝐹(𝑢(𝑡))_𝑛𝑖=0,𝑎_𝑖(𝑥)𝑢^(𝑖)(𝑥)=𝑓(𝑥)+^𝑥_𝑎𝑦(𝑡)(𝑥−𝑡)^𝛼𝑑𝑡,𝑎⩽𝑥⩽𝑏,0<𝛼<1. (2.2)(2.3) Rewriting (2.3) as ^𝑛_𝑖=0𝑎_𝑖(𝑥)𝑢^(𝑖)(𝑥)=𝑓(𝑥)+^𝑥_𝑎𝑦(𝑡)−𝑦(𝑥)+𝑦(𝑥)(𝑥−𝑡)^𝛼𝑑𝑡, (2.4) where the solution under the integral has been replaced by 𝑦(𝑡)−𝑦(𝑥)+𝑦(𝑥). Thus, ^𝑛_𝑖=0𝑎_𝑖(𝑥)𝑢^(𝑖)(𝑥)=𝑓(𝑥)+𝑦(𝑥)^𝑥_𝑎1(𝑥−𝑡)^𝛼𝑑𝑡+^𝑥_𝑎𝑦(𝑡)−𝑦(𝑥)(𝑥−𝑡)^𝛼𝑑𝑡. (2.5) So that ^𝑛_𝑖=0𝑎_𝑖(𝑥)𝑢^(𝑖)(𝑥)(𝑥)(𝑥)()=𝑓+𝑦𝑥−𝑎^1−𝛼+1−𝛼^𝑥_𝑎𝑦(𝑡)−𝑦(𝑥)(𝑥−𝑡)^𝛼𝑑𝑡, (2.6) or equivalently ^𝑛_𝑖=0𝑎_𝑖(𝑥)𝑢^(𝑖)(𝑥)(𝑥)(𝑥)()=𝑓+𝑦𝑥−𝑎^1−𝛼−1−𝛼^𝑥_𝑎(𝑥−𝑡)^1−𝛼𝑦(𝑡)−𝑦(𝑥)𝑡−𝑥𝑑𝑡. (2.7) We use the following Taylor’s approximation of degree 𝑛 of 𝑦(𝑡) about 𝑡=𝑥: 𝑦(𝑡)≈𝑦(𝑥)+(𝑡−𝑥)𝑦^′(𝑥)+(𝑡−𝑥)²𝑦2!^′′(𝑥)()+⋯+𝑡−𝑥^𝑛𝑦𝑛!^(𝑛)(𝑥). (2.8) Thus, 𝑦(𝑡)−𝑦(𝑥)≈(𝑡−𝑥)𝑦^′(𝑥)+(𝑡−𝑥)²𝑦2!^′′(𝑥)()+⋯+𝑡−𝑥^𝑛𝑦𝑛!^(𝑛)(𝑥). (2.9) Substituting the approximate relation (2.9) into the right hand side of (2.7) yields ^𝑛_𝑖=0𝑎_𝑖(𝑥)𝑢^(𝑖)(𝑥)=𝑓(𝑥)+^𝑛_𝑖=0(−1)^𝑖(𝑥−𝑎)^{𝑖+1−𝛼}()𝑦𝑖!𝑖+1−𝛼^(𝑖)(𝑥). (2.10) Therefore, (2.1) can be approximated by the 𝑛th-order nonlinear differential equation (2.10).

3. Analysis of Reproducing Kernel Hilbert Space

Definition 3.1 (reproducing kernel Space 𝑊_𝑛+1[𝑎,𝑏], see [17]). 𝑊_𝑛+1[𝑎,𝑏]=𝑢(𝑥)∣𝑢^(𝑛)(𝑥)[],𝑢isanabsolutelycontinuousrealvaluefunctionin𝑎,𝑏_(𝑛+1)(𝑥)∈𝐿²[𝑎,𝑏],𝑥∈[𝑎,𝑏] (3.1) and endowed it with the inner product and norm, respectively, ⟨𝑢(𝑥),𝑣(𝑥)⟩_𝑊𝑛+1=^𝑛_𝑖=0𝑢^(𝑖)(𝑎)𝑣^(𝑖)(𝑎)+^𝑏_𝑎𝑢^(𝑛+1)(𝑥)𝑣^(𝑛+1)(𝑥)𝑑𝑥,‖𝑢‖_𝑊𝑛+1=⟨𝑢,𝑢⟩^1/2. (3.2)

Theorem 3.2. The space 𝑊_𝑛+1[𝑎,𝑏] is a reproducing kernel space. That is, there exists a function 𝑅_𝑥(𝑦), for each fixed 𝑥∈[𝑎,𝑏],𝑅_𝑥(𝑦)∈𝑊_𝑛+1[𝑎,𝑏], and for any 𝑢(𝑦)∈𝑊_𝑛+1[𝑎,𝑏], satisfying 𝑢𝑦,𝑅_𝑥𝑦_𝑊𝑛+1=𝑢(𝑥), (3.3) the reproducing kernel 𝑅_𝑥(𝑦) can be denoted by 𝑅_𝑥𝑦=^2𝑛+2_𝑖∑₌₁𝑎_𝑖(𝑥)𝑦^𝑖−1,𝑦⩽𝑥,_2𝑛+2𝑖∑₌₁𝑏_𝑖(𝑥)𝑦^𝑖−1,𝑦>𝑥, (3.4) where 𝑎_𝑖(𝑥),𝑏_𝑖(𝑥)(𝑖=1,2,…,2𝑛+2) are known coefficients.

Theorem 3.3. Let 𝑊_𝑛+1[𝑎,𝑏] be a reproducing kernel space and 𝑢_𝑚(𝑥),𝑢(𝑥)∈𝑊_𝑛+1[𝑎,𝑏],𝑚=1,2,…. If 𝑢_𝑚(𝑥) converges to 𝑢(𝑥) in the sense of ‖⋅‖_𝑊𝑛+1, then 𝑢_𝑚^(𝑘)(𝑥) converges to 𝑢^(𝑘)(𝑥)(𝑘=0,1,…,𝑛) uniformly.

Property 1. If 𝑊_𝑛+1[𝑎,𝑏] is a reproducing kernel space, the reproducing kernel function 𝑅_𝑥(𝑦) in 𝑊_𝑛+1[𝑎,𝑏] is unique.

Definition 3.4 (reproducing kernel Space 𝑊₁[𝑎,𝑏], see [17]). 𝑊₁[]=𝑢(𝑥)(𝑥)[],𝑢𝑎,𝑏∣𝑢isanabsolutelycontinuousrealvaluefunctionin𝑎,𝑏_′(𝑥)∈𝐿²[𝑎,𝑏],𝑥∈[𝑎,𝑏] (3.5) and endowed it with the inner product and norm, respectively, ⟨𝑢(𝑥),𝑣(𝑥)⟩_𝑊1=𝑢(𝑎)𝑣(𝑎)+^𝑏_𝑎𝑢^′(𝑥)𝑣^′(𝑥)𝑑𝑥,‖𝑢‖_𝑊1=⟨𝑢,𝑢⟩^1/2. (3.6)

There exists a unique reproducing kernel function 𝑄(𝑥,𝑦)∈𝑊₁[𝑎,𝑏], and 𝑄(𝑥,𝑦) can be denoted by 𝑄_𝑥𝑦=1+𝑦,𝑦⩽𝑥,1+𝑥,𝑦>𝑥. (3.7)

The method of obtaining 𝑄_𝑥(𝑦), the coefficients of the reproducing kernel 𝑅_𝑥(𝑦), and the proof of Theorems 3.2 and 3.3 are given in [17].

4. Combined ADM and RKM

4.1. Representation of the Inverse Operator

Here, we propose a new differential operator, as follows: ℒ𝑢(𝑥)=^𝑛_𝑖=0𝑎_𝑖(𝑥)𝑢^(𝑖)(𝑥), (4.1) letting 𝑁(𝑥,𝑢(𝑥))=^𝑛_𝑖=0(−1)^𝑖(𝑥−𝑎)^{𝑖+1−𝛼}()𝑦𝑖!𝑖+1−𝛼^(𝑖)(𝑥), (4.2) where 𝑦(𝑥)=𝐹(𝑢(𝑥)), then we convert (2.10) as follows: ℒ𝑢(𝑥)=𝑓(𝑥)+𝑁(𝑥,𝑢(𝑥)),𝑎⩽𝑥⩽𝑏. (4.3) We now introduce how to determine the inverse operator 𝐿⁻¹ of 𝐿. Obviously, ℒ∶𝑊_𝑛+1[𝑎,𝑏]→𝑊₁[𝑎,𝑏] is a bounded linear operator.

We choose {𝑥_𝑖}^∞_𝑖=1 as any dense set in [𝑎,𝑏], and let 𝜓_𝑥(𝑦)=ℒ^∗𝑄_𝑥(𝑦), where ℒ^∗ is the conjugate operator of ℒ and 𝑄_𝑥(𝑦) is given by (3.7). Furthermore, for simplicity, let 𝜓_𝑖(𝑥) denote 𝜓_𝑥𝑖(𝑥), namely, 𝜓_𝑖(𝑥)=^def𝜓_𝑥𝑖(𝑥)=ℒ^∗𝑄_𝑥𝑖(𝑥). (4.4)

Now, several Lemmas are given.

Lemma 4.1. {𝜓_𝑖(𝑥)}^∞_𝑖=1 is the complete system of 𝑊_𝑛+1[𝑎,𝑏].

Proof. For 𝑢(𝑥)∈𝑊_𝑛+1[𝑎,𝑏], let ⟨𝑢(𝑥),𝜓_𝑖(𝑥)⟩=0(𝑖=1,2,…), that is, 𝑢(𝑥),ℒ^∗𝑄_𝑥𝑖(𝑥)=(ℒ𝑢)𝑥_𝑖=0. (4.5)
Note that {𝑥_𝑖}^∞_𝑖=1 is the dense set in [𝑎,𝑏], therefore (ℒ𝑢)(𝑥)=0. It follows that 𝑢(𝑥)=0 from the existence of ℒ⁻¹.

Lemma 4.2. The following formula holds 𝜓_𝑖(𝑥)=ℒ_𝜂𝑅_𝑥𝜂𝑥_𝑖, (4.6) where the subscript 𝜂 of operator ℒ_𝜂 indicates that the operator ℒ applies to functions of 𝜂.

Proof. Consider 𝜓_𝑖(𝑥)=𝜓_𝑖(𝜉),𝑅_𝑥(𝜉)_{𝑊𝑛+1[𝑎,𝑏]}=ℒ^∗𝑄_𝑥𝑖(𝜉),𝑅_𝑥(𝜉)_{𝑊𝑛+1[𝑎,𝑏]}=𝑄_𝑥𝑖(𝜉),ℒ_𝜂𝑅_𝑥𝜂(𝜉)_{𝑊1[𝑎,𝑏]}=ℒ_𝜂𝑅_𝑥𝜂𝑥_𝑖. (4.7) This completes the proof.

The orthonormal system {𝜓_𝑖(𝑥)}^∞_𝑖=1 of 𝑊_𝑛+1[𝑎,𝑏] can be derived from Gram-Schmidt orthogonalization process of {𝜓_𝑖(𝑥)}^∞_𝑖=1, 𝜓_𝑖(𝑥)=^𝑖_𝑘=1𝛽_𝑖𝑘𝜓_𝑘(𝑥),𝛽_𝑖𝑖>0,𝑖=1,2,…, (4.8) where 𝛽_𝑖𝑘 are orthogonal coefficients.

Theorem 4.3. If the inverse operator 𝐿⁻¹, exists and {𝑥_𝑖}^∞_𝑖=1 is dense on [0,1], then the inverse operator 𝐿⁻¹ can be determined as 𝑢(𝑥)=𝐿⁻¹𝑓(𝑥)+𝑁(𝑥,𝑢(𝑥))=^∞_𝑖=1𝐵_𝑖𝜓_𝑖(𝑥), (4.9) where 𝐵_𝑖=∑^𝑖_𝑘=1𝛽_𝑖𝑘(𝑓(𝑥_𝑘)+𝑁(𝑥_𝑘,𝑢(𝑥_𝑘))).

Proof. From (4.9), it holds that 𝑢(𝑥)=^∞_𝑖=1𝑢(𝑥),𝜓_𝑖(𝑥)_𝑊𝑛+1𝜓_𝑖(𝑥)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑢(𝑥),ℒ^∗𝑄_𝑥𝑘(𝑥)_𝑊𝑛+1𝜓_𝑖(𝑥)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘ℒ𝑢(𝑥),𝑄_𝑥𝑘(𝑥)_𝑊1𝜓_𝑖(𝑥)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑓(𝑥)+𝑁(𝑥,𝑢(𝑥)),𝑄_𝑥𝑘(𝑥)_𝑊1𝜓_𝑖(𝑥)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑓𝑥_𝑘+𝑁𝑥_𝑘,𝑢𝑥_𝑘𝜓_𝑖(𝑥). (4.10) The proof of the theorem is complete.

From Theorem 4.3, obviously, 𝐿⁻¹ is determined.

4.2. Decomposition Method

By applying 𝐿⁻¹ to both sides of (4.3), we have 𝑢(𝑥)=𝐿⁻¹𝑓(𝑥)+𝐿⁻¹𝑁(𝑥,𝑢(𝑥)). (4.11) The ADM introduces the solution 𝑢(𝑥) and the nonlinear function 𝑁(𝑥,𝑢) by infinite series 𝑢(𝑥)=^∞_𝑖=0𝑢_𝑖(𝑥),𝑁(𝑥,𝑢(𝑥))=^∞_𝑖=0𝐴_𝑖(𝑥), (4.12)(4.13) where 𝐴_𝑛 are Adomian polynomials for the nonlinear term 𝑁(𝑥,𝑢(𝑥)) and can be found from the following formula: 𝐴_𝑛=1𝑑𝑛!𝑑𝜆^𝑛𝑛𝑓^∞_𝑛=0𝜆^𝑖𝑢_𝑖𝜆0,𝑛⩾0. (4.14) Substituting (4.12) and (4.13) into (4.11) yields ^∞_𝑖=0𝑢_𝑖(𝑥)=𝐿⁻¹𝑓(𝑥)+𝐿^∞−1_𝑖=0𝐴_𝑖(𝑥). (4.15) According to the ADM, the components 𝑢_𝑖(𝑥) can be determined as 𝑢₀(𝑥)=𝐿⁻¹𝑓(𝑥),𝑢_𝑖+1(𝑥)=𝐿⁻¹𝐴_𝑖(𝑥),𝑖⩾0, (4.16) which gives 𝑢₀(𝑥)=^∞_𝑗=0𝐵_0𝑗𝜓_𝑗(𝑥),𝑢_𝑖+1(𝑥)=^∞_𝑗=0𝐵_(𝑖+1)𝑗𝜓_𝑗(𝑥),𝑖⩾0, (4.17) where 𝐵_0𝑗=∑^𝑗_𝑘=1𝛽_𝑗𝑘𝑓(𝑥_𝑘), 𝐵_𝑖𝑗=∑^𝑗_𝑘=1𝛽_𝑗𝑘𝐴_𝑖−1(𝑥_𝑘),𝑖⩾1.

From (4.17), we can determine the components 𝑢_𝑖(𝑥), and hence the series solution of 𝑢(𝑥) in (4.12) can be immediately obtained.

We obtain approximate solution of the following equation: 𝑈_𝑛(𝑥)=^𝑛_𝑖=0𝑢_𝑖(𝑥). (4.18)

The following use some examples to demonstrate the effectiveness of the algorithm.

5. Numerical Examples

To illustrate the applicability and effectiveness of our method, we consider the following examples. Symbolic and numerical computations performed by using Mathematica 5.0.

Example 5.1. Consider the following first-order nonlinear weakly singular Volterra integro-differential equation: 𝑢^′(𝑥)+𝑝(𝑥)𝑢(𝑥)=𝑓(𝑥)+^𝑥_𝑎𝐹(𝑢(𝑡))(𝑥−𝑡)^𝛼𝑑𝑡,𝑎⩽𝑥⩽𝑏,0<𝛼<1 (5.1) with 𝛼=1/2, 𝑎=0, 𝑢(0)=0 and 𝑢(𝑥)=𝑥(𝑥−1).

Let 𝐹(𝑢(𝑥))=𝑢², 𝑝(𝑥)=(16/315)𝑥^5/2(21+4𝑥(4𝑥−9))+1, 𝑓(𝑥)=𝑥²+𝑥−1.

On [0,1] select 100 points and get the approximate solution 𝑈₂(𝑥)=∑²_𝑛=0𝑢_𝑛(𝑥), the results are shown in Figure 1.

Figure 1 

The approximate values 𝑈2(𝑥) in comparison with its exact values 𝑥(𝑥−1).

Example 5.2. Consider the following second-order nonlinear weakly singular Volterra integro-differential equation: 𝑢^′′(𝑥)+𝑝(𝑥)𝑢^′(𝑥)+𝑞(𝑥)𝑢(𝑥)=𝑓(𝑥)+^𝑥_𝑎𝐹(𝑢(𝑡))(𝑥−𝑡)^𝛼𝑑𝑡,𝑎⩽𝑥⩽𝑏,0<𝛼<1 (5.2) with 𝛼=1/3, 𝑎=0, 𝑢(0)=𝑢(1)=0 and 𝑢(𝑥)=𝑥(𝑥−1).

Let 𝐹(𝑢(𝑥))=𝑢³, 𝑝(𝑥)=𝑥²+1, 𝑞(𝑥)=(243/52360)𝑥^11/3(−119+9𝑥(9𝑥²−30𝑥+34)), 𝑓(𝑥)=2𝑥³+ 𝑥²+2𝑥+3.

On [0,1] select 100 points and get the approximate solution 𝑈₁(𝑥)=∑¹_𝑛=0𝑢_𝑛(𝑥), the results are shown in Table 1.

Table 1 

Comparison of the errors of 𝑢(𝑥) and 𝑈₁(𝑥).

Example 5.3. Consider the following third-order nonlinear weakly singular Volterra integro-differential equation: 𝑢^′′′(𝑥)+𝑝(𝑥)𝑢^′′(𝑥)+𝑞(𝑥)𝑢^′(𝑥)+𝑟(𝑥)𝑢(𝑥)=𝑓(𝑥)+^𝑥_𝑎𝐹(𝑢(𝑡))(𝑥−𝑡)^𝛼𝑑𝑡,𝑎⩽𝑥⩽𝑏,0<𝛼<1 (5.3) with 𝛼=1/2, 𝑎=0, 𝑢(0)=𝑢(1)=0, 𝑢^′(0)=0 and 𝑢(𝑥)=𝑥²(𝑥−1).

Let 𝐹(𝑢(𝑥))=𝑢², 𝑝(𝑥)=𝑥, 𝑞(𝑥)=𝑥−1, 𝑟(𝑥)=(256/45045)𝑥^9/2(143+20𝑥(6𝑥−13)), 𝑓(𝑥)=3𝑥³+ 𝑥²+6.

On [0,1] select 100 points and get the approximate solution 𝑈₂(𝑥)=∑²_𝑛=0𝑢_𝑛(𝑥), the results are shown in Figure 2.

Figure 2 

The approximate values 𝑈2(𝑥) in comparison with its exact values 𝑥2(𝑥−1).

6. Conclusion

In this paper, we have reduced the solution of nonlinear weakly singular Volterra integro-differential equations to the solution of ordinary nonlinear differential equations by removing the singularity using an appropriate Taylor’s approximation. Then we have demonstrated the solution of these ordinary nonlinear differential equations by RKM and ADM. The ADM is an accurate and efficient method to solve nonlinear weakly singular Volterra integro-differential equations.

Acknowledgments

Research supported by the National Natural Science Foundation of China (61071181), the Educational Department Scientific Technology Program of Heilongjiang Province(12512133 and 12521148) and the Scientific Innovation Project for Graduate of Heilongjiang Province (YJSCX2012-184HLJ).