Reproducing Kernel Method for Solving Nonlinear Differential-Difference Equations

Abstract

On the basis of reproducing kernel Hilbert spaces theory, an iterative algorithm for solving some nonlinear differential-difference equations (NDDEs) is presented. The analytical solution is shown in a series form in a reproducing kernel space, and the approximate solution 𝑢_𝑛,𝑚 is constructed by truncating the series to 𝑚 terms. The convergence of 𝑢_𝑛,𝑚 to the analytical solution is also proved. Results obtained by the proposed method imply that it can be considered as a simple and accurate method for solving such differential-difference problems.

1. Introduction

Differential-difference equations play a crucial role in modelling of much physical phenomena such as particle vibrations in lattices, currents in electrical networks, pulses in biological chains, discretization in solid state, quantum physics, textile engineering, stratified hydrostatic flows, and so on. See, for example [1–14].

Recently, there have been lots of efforts in giving exact or approximate solutions of NDDEs. For instance, Zhu [4] extended the Exp-function method for solving NDDEs. In this method, exact solutions are sought in the form of an exponential type rational function in which both the numerator and denominator are polynomials of exponential functions. Mokhtari [5] applied the variational iteration method in which a correction functional is established by a general Lagrange multiplier, which can be identified optimally via the variational theory. The method gives rapidly convergent successive approximations of the exact solution. Qian et al. [6] extended the multilinear variable separation approach to a special differential-difference equation. Baldwin et al. [7] presented an algorithm to find exact travelling wave solutions of differential-difference equations in terms of the tanh function and found kink-type solutions in many spatially discrete nonlinear models. Wu et al. [8] and Wang et al. [9] extended the Adomian decomposition method to solve NDDEs. The technique is based on a decomposition of the solution of NDDE in a series of functions. Each term of the series is obtained from a polynomial generated from an expansion of an analytic function into a power series. Many authors [10–14] applied the homotopy perturbation method to solve NDDEs. The method, which is based on homotopy, constructs a continuous mapping of an initial guess through an auxiliary linear operator, and an auxiliary parameter is used to ensure the convergence of solution series.

The theory of reproducing kernels [15] was used for the first time at the beginning of the 20th century by S. Zaremba in his work on boundary value problems for harmonic and biharmonic functions. This theory has been successfully applied to fractal interpolation [16], solving ordinary differential equations [17–20] and partial differential equations [21, 22]. The book [23] provides excellent overviews of the existing reproducing kernel methods for solving various model problems such as integral and integrodifferential equations.

In this study, a general technique is proposed for solving some NDDEs in the reproducing kernel space. The main idea is to construct the reproducing kernel space satisfying the conditions for determining solution of the NDDEs. The analytical solution is represented in the form of series through the function value at the right side of the equation. For illustration, we apply this method to the Volterra lattice equation, the discretized mKdV lattice equation and the discrete sine-Gordon equation. The advantages of the approach lie in the following facts. The approximate solution 𝑢_𝑛,𝑚(𝑡) converges uniformly to the analytical solution 𝑢_𝑛(𝑡). The method is mesh-free, easily implemented and capable in treating various boundary conditions. Since the method needs no time discretization, there is no matter, in which time the approximate solution is computed, from the both elapsed time and stability problem, point of views. Also we can evaluate the approximate solution 𝑢_𝑛,𝑚(𝑡) for fixed 𝑚 once and use it over and over.

In the next section we describe how to solve a NDDE through the reproducing kernel method and verify convergence of the approximate solution to the exact solution. Several numerical results are presented in Section 3. The last section is a brief conclusion.

2. Construction of the Method

Consider the following NDDE 𝐿𝑢_𝑛(𝑡)=𝑁𝑡,𝑢_𝑛+𝑝1,…,𝑢_{𝑛+𝑝𝑑},𝑡∈(0,𝑇], (2.1) where 𝐿 is a linear differential operator of one-order derivative in 𝑡, 𝑁 is a nonlinear differential operator and 𝑛,𝑝_𝑖∈ℤ. Since nonhomogeneous initial condition can be reduced readily to the homogeneous one, we only consider the following homogeneous initial condition 𝑢_𝑛(0)=0. (2.2) We assume that (2.1)-(2.2) has a unique solution.

Definition 2.1. 𝑊[0,𝑇]={𝑢(𝑡)∣𝑢(𝑡),𝑢′(𝑡) are absolutely continuous in [0,𝑇],𝑢^′′(𝑡)∈𝐿²[0,𝑇], 𝑢(0)=0}. The inner product and norm in 𝑊[0,𝑇] are defined, respectively, by ⟨𝑢,𝑣⟩_𝑊=¹_𝑖=0𝑢^(𝑖)(0)𝑣^(𝑖)(0)+^𝑇₀𝑢^′′(𝑡)𝑣^′′(𝑡)[],‖𝑢‖𝑑𝑡,𝑢,𝑣∈𝑊0,𝑇_𝑤=⟨𝑢,𝑢⟩_𝑊,𝑢∈𝑊[0,𝑇]. (2.3)

Remark 2.2. The space 𝑊[0,𝑇] is a reproducing kernel space, and its reproducing kernel is given by 𝐾_𝑠(𝑡)=𝑠𝑠𝑡+2𝑡²−16𝑡³𝑠𝑡≤𝑠,𝑠𝑡+²21𝑡−6𝑠³𝑡>𝑠. (2.4)

Definition 2.3. 𝑊[0,𝑇]={𝑢(𝑡)∣𝑢(𝑡) is absolutely continuous in [0,𝑇],𝑢′(𝑡)∈𝐿²[0,𝑇]}. The inner product and norm in 𝑊[0,𝑇] are defined, respectively, by ⟨𝑢,𝑣⟩_𝑊=𝑢(0)𝑣(0)+^𝑇₀𝑢^′(𝑡)𝑣^′(𝑡)𝑊[],‖𝑢‖𝑑𝑡,𝑢,𝑣∈0,𝑇_𝑊=⟨𝑢,𝑢⟩_𝑊,𝑢∈𝑊[0,𝑇]. (2.5)

Remark 2.4. The space 𝑊[0,𝑇] is a reproducing kernel space and its reproducing kernel is given by 𝐺_𝑠(𝑡)=1+𝑡𝑡≤𝑠,1+𝑠𝑡>𝑠. (2.6)

Remark 2.5. The definition of the spaces 𝑊[0,𝑇] and 𝑊[0,𝑇] is suitable for our numerical examples and in general 𝑊[0,𝑇] is defined on the basis of the operator 𝐿, and 𝑊[0,𝑇] must be defined based on the operator 𝑁.

In order to represent the analytical solution of the model problem, we can assume that 𝐿∶𝑊[0,𝑇]→𝑊[0,𝑇] is an invertible bounded linear operator, choose a countable dense subset {𝑡_𝑖}^∞_𝑖=1 in [0,𝑇] and define 𝜙_𝑖(𝑡)=𝐺_𝑡𝑖(𝑡),𝜓_i(𝑡)=𝐿^∗𝜙_𝑖(𝑡), (2.7) where 𝐿^∗ is the adjoint operator of 𝐿. The orthonormal system {𝜓_𝑖(𝑡)}^∞_𝑖=1 of 𝑊[0,𝑇] can be derived from the Gram-Schmidt orthogonalization process of {𝜓_𝑖(𝑡)}^∞_𝑖=1 as 𝜓_𝑖(𝑡)=^𝑖_𝑘=1𝛽_𝑖𝑘𝜓_𝑘(𝑡). (2.8)

Theorem 2.6. Suppose that {𝑡_𝑖}^∞_𝑖=1 is dense in [0,𝑇], then {𝜓_𝑖(𝑡)}^∞_𝑖=1 is a complete system in 𝑊[0,𝑇] and 𝜓_𝑖(𝑡)=𝐿_𝑠𝐾_𝑠(𝑡)|_{𝑠=𝑡𝑖}.

Proof. See [22].

Theorem 2.7. If {𝑡_𝑖}^∞_𝑖=1 is dense in [0,𝑇], then the analytical solution of (2.1)-(2.2) will be 𝑢_𝑛(𝑡)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑁𝑡_𝑘,𝑢_𝑛+𝑝1𝑡_𝑘,…,𝑢_{𝑛+𝑝𝑑}𝑡_𝑘𝜓_𝑖(𝑡). (2.9)

Proof. Since {𝜓_𝑖(𝑡)}^∞_𝑖=1 is a complete system in 𝑊[0,𝑇], we have 𝑢_𝑛(𝑡)=^∞_𝑖=1𝑢_𝑛(𝑡),𝜓_𝑖(𝑡)_𝑊𝜓_𝑖(𝑡)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑢_𝑛(𝑡),𝜓_𝑘(𝑡)_𝑊𝜓_𝑖(𝑡)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑢_𝑛(𝑡),𝐿^∗𝜙_𝑘(𝑡)_𝑊𝜓_𝑖(𝑡)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝐿𝑢_𝑛(𝑡),𝜙_𝑘(𝑡)_𝑊𝜓_𝑖(𝑡)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝐿𝑢_𝑛(𝑡),𝐺_𝑡𝑘(𝑡)_𝑊𝜓_𝑖(𝑡)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝐿𝑢_𝑛𝑡_𝑘𝜓_𝑖(𝑡)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑁𝑡_𝑘,𝑢_𝑛+𝑝1𝑡_𝑘,…,𝑢_{𝑛+𝑝𝑑}𝑡_𝑘𝜓_𝑖(𝑡). (2.10)

Now the approximate solution 𝑢_𝑛,𝑚(𝑡) can be obtained by the 𝑚-term intercept of the analytical solution 𝑢_𝑛(𝑡), that is, 𝑢_𝑛,𝑚(𝑡)=^𝑚𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑁𝑡_𝑘,𝑢_𝑛+𝑝1𝑡_𝑘,…,𝑢_{𝑛+𝑝𝑑}𝑡_𝑘𝜓_𝑖(𝑡). (2.11) Obviously, 𝑢_𝑛,𝑚(𝑡)−𝑢_𝑛(𝑡)_𝑊⟶0,(𝑚⟶∞). (2.12) In the sequel, for evaluating (2.11), we choose nonnegative integer 𝑚 and put the initial function 𝑢_𝑛,0(𝑡)=0. Then the approximate solution is defined by 𝑢_𝑛,𝑚(𝑡)=^𝑚_𝑖=1𝐵_𝑖𝜓_𝑖(𝑡), (2.13) where 𝐵_𝑖=^𝑖_𝑘=1𝛽_𝑖𝑘𝑁𝑡_𝑘,𝑢_{𝑛+𝑝1,𝑘−1}𝑡_𝑘,…,𝑢_{𝑛+𝑝𝑑,𝑘−1}𝑡_𝑘. (2.14)

Lemma 2.8. If 𝑢−→_𝑛,𝑚^‖.‖𝑢_𝑛, ‖𝑢_𝑛,𝑚‖ is bounded in (2.13), 𝑡_𝑚→𝑠 and 𝑁(𝑡,𝑢_𝑛+𝑝1(𝑡),…,𝑢_{𝑛+𝑝𝑑}(𝑡)) is continuous, then 𝑁(𝑡_𝑚,𝑢_{𝑛+𝑝1,𝑚−1}(𝑡_𝑚),…,𝑢_{𝑛+𝑝𝑑,𝑚−1}(𝑡_𝑚))→𝑁(𝑠,𝑢_𝑛+𝑝1(𝑠),…,𝑢_{𝑛+𝑝𝑑}(𝑠)).

Proof. Note that 𝑢_{𝑛+𝑝𝑖}(𝑡)=𝑢_{𝑛+𝑝𝑖}(𝑠),𝐾_𝑡(𝑠)_𝑊≤𝑢_{𝑛+𝑝𝑖}(𝑠)_𝑊𝐾_𝑡(𝑠)_𝑊≤𝑐₀𝑢_{𝑛+𝑝𝑖}(𝑠)_𝑊,𝜕𝑢_{𝑛+𝑝𝑖}(𝑡)=𝑢𝜕𝑡^{𝑛+𝑝𝑖}(𝑠),𝜕𝐾_𝑡(𝑠)𝜕𝑡_𝑊≤𝑢_{𝑛+𝑝𝑖}(𝑠)_𝑊𝜕𝐾_𝑡(𝑠)𝜕𝑡_𝑊≤𝑐₁𝑢_{𝑛+𝑝𝑖}(𝑠)_𝑊,𝑢_{𝑛+𝑝𝑖,𝑚−1}(𝑠)−𝑢_{𝑛+𝑝𝑖}(𝑠)=⟨𝑢_{𝑛+𝑝𝑖,𝑚−1}(𝑡)−𝑢_{𝑛+𝑝𝑖}(𝑡),𝐾_𝑠(𝑡)⟩_𝑊≤𝑢_{𝑛+𝑝𝑖,𝑚−1}(𝑡)−𝑢_{𝑛+𝑝𝑖}(𝑡)_𝑊𝐾_𝑡(𝑠)_𝑊≤𝑐₀𝑢_{𝑛+𝑝𝑖,𝑚−1}(𝑡)−𝑢_{𝑛+𝑝𝑖}(𝑡)_𝑊. (2.15) On the other hand, we have 𝑢_{𝑛+𝑝𝑖,𝑚−1}𝑡_𝑚−𝑢_{𝑛+𝑝𝑖}(𝑠)=𝑢_{𝑛+𝑝𝑖,𝑚−1}𝑡_𝑚−𝑢_{𝑛+𝑝𝑖,𝑚−1}(𝑠)+𝑢_{𝑛+𝑝𝑖,𝑚−1}(𝑠)−𝑢_{𝑛+𝑝𝑖}(𝑠≤𝜕_𝑡)𝑢_{𝑛+𝑝𝑖,𝑚−1}𝜂𝑡_𝑚−𝑠+𝑢_{𝑛+𝑝𝑖,𝑚−1}(𝑠)−𝑢_{𝑛+𝑝𝑖}(𝑠). (2.16) From 𝑢−→_𝑛,𝑚^‖.‖𝑢_𝑛 and former statements, it follows that 𝑢_{𝑛+𝑝𝑖,𝑚−1}(𝑠)−𝑢_{𝑛+𝑝𝑖}(𝑠)⟶0,𝜕_𝑡𝑢_{𝑛+𝑝𝑖,𝑚−1}𝜂≤𝑐₁𝑢_{𝑛+𝑝𝑖,𝑚−1𝑤}. (2.17) Therefore, 𝑢_{𝑛+𝑝𝑖,𝑚−1}𝑡_𝑚−𝑢_{𝑛+𝑝𝑖}(𝑠)⟶0,as𝑚⟶∞, (2.18) because of the boundedness of ‖𝑢_𝑛,𝑚‖. The continuity of 𝑁(𝑡,𝑢_𝑛+𝑝1(𝑡),…,𝑢_{𝑛+𝑝𝑑}(𝑡)) implies that 𝑁𝑡_𝑚,𝑢_{𝑛+𝑝1,𝑚−1}𝑡_𝑚,…,𝑢_{𝑛+𝑝𝑑,𝑚−1}𝑡_𝑚⟶𝑁𝑠,𝑢_𝑛+𝑝1(𝑠),…,𝑢_{𝑛+𝑝𝑑}(𝑠),as𝑚⟶∞. (2.19)

Theorem 2.9. Suppose that ‖𝑢_𝑛,𝑚‖ is bounded in (2.13), and (2.1)-(2.2) has a unique solution. If {𝑡_𝑖}^∞_𝑖=1 is dense in [0,𝑇], then the 𝑚-term approximate solution 𝑢_𝑛,𝑚(𝑡) derived from the above method converges to the analytical solution 𝑢_𝑛(𝑡) of (2.1)-(2.2) and 𝑢_𝑛(𝑡)=^∞_𝑖=1𝐵_𝑖𝜓_𝑖(𝑡), (2.20) where 𝐵_𝑖 is given by (2.14).

Proof. At first, we prove the convergence of 𝑢_𝑛,𝑚(𝑡). From (2.13), we infer that 𝑢_𝑛,𝑚+1(𝑡)=𝑢_𝑛,𝑚(𝑡)+𝐵_𝑚+1𝜓_𝑚+1(𝑡). (2.21) The orthonormality of {𝜓_𝑖}^∞_𝑖=1 yields that 𝑢_𝑛,𝑚+1²=𝑢_𝑛,𝑚²+𝐵²_𝑚+1=⋯=^𝑚+1_𝑖=1𝐵²_𝑖. (2.22) In terms of (2.22), it holds that ‖𝑢_𝑛,𝑚+1‖≥‖𝑢_𝑛,𝑚‖. Due to the condition that ‖𝑢_𝑛,𝑚‖ is bounded, ‖𝑢_𝑛,𝑚‖ is convergent and there exists a constant 𝑐 such that ∑^∞_𝑖=1𝐵²_𝑖=𝑐. If 𝑙>𝑚, and then 𝑢_𝑛,𝑙−𝑢_𝑛,𝑚²=𝑢_𝑛,𝑙−𝑢_{𝑛,𝑙−1}+𝑢_{𝑛,𝑙−1}−𝑢_{𝑛,𝑙−2}+⋯+𝑢_𝑛,𝑚+1−𝑢_𝑛,𝑚²=𝑢_𝑛,𝑙−𝑢_{𝑛,𝑙−1}²+𝑢_{𝑛,𝑙−1}−𝑢_{𝑛,𝑙−2}²+⋯+𝑢_𝑛,𝑚+1−𝑢_𝑛,𝑚². (2.23) On account of ‖𝑢_𝑛,𝑙−𝑢_{𝑛,𝑙−1}‖²=𝐵²_𝑙, consequently, we have 𝑢_𝑛,𝑙−𝑢_𝑛,𝑚²=^𝑙_𝑖=𝑚+1𝐵²_𝑖⟶0as𝑚⟶∞. (2.24) The completeness of 𝑊[0,𝑇] shows that 𝑢_𝑛,𝑚→𝑢_𝑛 as 𝑚→∞. Now, we prove that 𝑢_𝑛 is the solution of (2.1)-(2.2). Taking limits in (2.13), we get 𝑢_𝑛(𝑡)=^∞_𝑖=1𝐵_𝑖𝜓_𝑖(𝑡). (2.25) Note that 𝐿𝑢_𝑛(𝑡)=^∞_𝑖=1𝐵_𝑖𝐿𝜓_𝑖(𝑡),𝐿𝑢_𝑛𝑡_𝑙=^∞_𝑖=1𝐵_𝑖𝐿𝜓_𝑖𝑡_𝑙=^∞_𝑖=1𝐵_𝑖𝐿𝜓_𝑖(𝑡),𝜙_𝑙(𝑡)_𝑊=^∞_𝑖=1𝐵_𝑖𝜓_𝑖(𝑡),𝐿^∗𝜙_𝑙(𝑡)_𝑊=^∞_𝑖=1𝐵_𝑖𝜓_𝑖(𝑡),𝜓_𝑙(𝑡)_𝑊. (2.26)(2.27) Therefore, consider ^𝑖_𝑙=1𝛽_𝑖𝑙𝐿𝑢_𝑛𝑡_𝑙=^∞_𝑖=1𝐵_𝑖𝜓_𝑖(𝑡),^𝑖_𝑙=1𝛽_𝑖𝑙𝜓_𝑙(𝑡)_𝑊=^∞_𝑖=1𝐵_𝑖𝜓_𝑖(𝑡),𝜓_𝑙(𝑡)_𝑊=𝐵_𝑙. (2.28) In view of (2.14), we have 𝐿𝑢_𝑛𝑡_𝑙=𝑁𝑡_𝑙,𝑢_{𝑛+𝑝1,𝑙−1}𝑡_𝑙,…,𝑢_{𝑛+𝑝𝑑,𝑙−1}𝑡_𝑙. (2.29) Since {𝑡_𝑖}^∞_𝑖=1 is dense in [0,𝑇], for each 𝑠∈[0,𝑇], there exist a subsequence {𝑡_𝑚𝑗}^∞_𝑗=1 such that 𝑡_𝑚𝑗⟶𝑠𝑗⟶∞. (2.30) We have known that 𝐿𝑢_𝑛𝑡_𝑚𝑗=𝑁𝑡_𝑚𝑗,𝑢_{𝑛+𝑝1,𝑚𝑗−1}𝑡_𝑚𝑗,…,𝑢_{𝑛+𝑝𝑑,𝑚𝑗−1}𝑡_𝑚𝑗. (2.31) Let 𝑗→∞, by Lemma 2.8 and the continuity of 𝑁, we have 𝐿𝑢_𝑛(𝑠)=𝑁𝑠,𝑢_𝑛+𝑝1(𝑠),…,𝑢_{𝑛+𝑝𝑑}(𝑠), (2.32) which indicates that 𝑢_𝑛 satisfies (2.1)-(2.2).

3. Numerical Results

To test the accuracy of the proposed method, three examples are treated in this section. The results are compared with the exact solutions. All experiments are done by taking 𝑇=1.

Example 3.1. Consider the Volterra equation of the form 𝜕𝑢_𝑛𝜕𝑡=𝑢_𝑛𝑢_𝑛−1−𝑢_𝑛+1. (3.1) With the initial condition 𝑢_𝑛(0)=𝑛, it is easy to check that the exact solution will be 𝑢_𝑛(𝑡)=𝑛.1+2𝑡 (3.2) Taking 𝑛=4, relative error (RE) of numerical results for 𝑚=16, 𝑚=50, and 𝑚=150 are reported in Table 1. Accuracy of approximate solutions is getting better as 𝑚 increases.

Table 1 

Relative errors of 𝑢_4,𝑚(𝑡), for Example 3.1.

Example 3.2. Consider the discrete mKdV equation 𝜕𝑢_𝑛=𝜕𝑡𝛼−𝑢²_𝑛𝑢_𝑛+1−𝑢_𝑛−1. (3.3)
With the initial condition 𝑢_𝑛(0)=√−𝛼tanh(𝑘/2)(1+cosh(𝑘))sech(𝑘𝑛−2), where 𝑘 and 𝛼 are arbitrary constants, the exact solution will be 𝑢_𝑛(𝑡)=√𝑘−𝛼tanh2((𝑘𝑘1+cosh))sech𝑘𝑛+2𝛼tanh2(1+cosh(𝑘))𝑡−2. (3.4) Taking 𝑚=50, 𝛼=−0.1 and 𝑘=0.3 and choosing 𝑛=10 and 𝑛=15, we have listed relative error of approximate solutions at some nodal points in Table 2. Numerical results are in good agreement with the exact solutions.

Table 2 

Relative errors of 𝑢_𝑛,50(𝑡), for Example 3.2.

Example 3.3. Consider the discrete sine-Gordon equation of the form 𝜕𝑢_𝑛+1−𝜕𝑡𝜕𝑢_𝑛𝑢𝜕𝑡=sin_𝑛+1−𝑢_𝑛. (3.5)
With the initial condition 𝑢_𝑛(0)=2arctan(tanh(𝑘𝑛+𝛿)) where 𝛿 and 𝑘≠0 are arbitrary constants, the exact solution will be 𝑢_𝑛(𝑡)1=2arctantanh𝑘𝑛−(𝑘).2tanh𝑡+𝛿 (3.6) Taking 𝑚=50, 𝛿=9 and 𝑘=0.3 and choosing 𝑛=5 and 𝑛=10, we have given relative error of approximate solutions at some nodal points in Table 3. Numerical results are in good agreement with the exact solutions.

Table 3 

Relative errors of 𝑢_𝑛,50(𝑡), for Example 3.3.

4. Conclusion

In this work, we proposed an algorithm for solving a class of nonlinear differential-difference equations on the basis of reproducing kernel spaces. Results of numerical examples show that the present method is an accurate and reliable analytical-numerical technique for solving such differential-difference equations. The method is shown to be of good convergence, simple in principle, and easy to program.