On the basis of reproducing kernel Hilbert spaces theory, an iterative algorithm for solving some nonlinear differentialdifference equations (NDDEs) is presented. The analytical solution is shown in a series form in a reproducing kernel space, and the approximate solution
Differentialdifference 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 [
Recently, there have been lots of efforts in giving exact or approximate solutions of NDDEs. For instance, Zhu [
The theory of reproducing kernels [
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 sineGordon equation. The advantages of the approach lie in the following facts. The approximate solution
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
Consider the following NDDE
The space
The space
The definition of the spaces
In order to represent the analytical solution of the model problem, we can assume that
Suppose that
See [
If
Since
Now the approximate solution
If
Note that
Suppose that
At first, we prove the convergence of
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
Consider the Volterra equation of the form
Relative errors of


RE ( 
RE ( 
RE ( 

0.1  3.33333 



0.2  2.85714 



0.3  2.50000 



0.4  2.22222 



0.5  2.00000 



0.6  1.81818 



0.7  1.66667 



0.8  1.53846 



0.9  1.42857 



1  1.33333 



Consider the discrete mKdV equation
With the initial condition
Relative errors of

RE ( 
RE ( 

0.1 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


0.8 


0.9 





Consider the discrete sineGordon equation of the form
With the initial condition
Relative errors of

RE ( 
RE ( 

0.1 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


0.8 


0.9 





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