It is attempted to provide the stability and convergence analysis of the reproducing kernel space method for solving the Duffing equation with with boundary integral conditions. We will prove that the reproducing space method is stable. Moreover, after introducing the method, it is shown that it has convergence order two.

Reproducing kernel space method is a very powerful method for solving linear and nonlinear equation such as initial or boundary differential equation and integral equations [

it is accurate, with needless effort to achieve the results,

it is possible to pick any point in the interval of integration and as well the approximate solutions and their derivatives will be applicable,

the method does not require discretization of the variables, and it is not affected by computation round off errors and one is not faced with necessity of large computer memory and time,

it is of global nature in terms of the solutions obtained as well as its ability to solve other mathematical, physical, and engineering problems.

Duffing equation springs from modeling some different branches of sciences and engineerings such as chemical engineering, thermoelasticity, periodic orbit extraction, nonlinear mechanical oscillators, and prediction of diseases [

To approximate the solution of the forced Duffing equation (

Section

In this section, we recall some basics which have been taken from [

One has

Also we need the following.

The inner product and norm in

The function space

The reproducing kernel

It has been proven that the reproducing kernel space

Here, we study the convergence order of the RKSM for solving (

To apply the RKSM, first of all, an orthogonal system of functions is constructed. Let

One has

Consider

If

According to [

If

It is worth nothing that when

If

To obtain the approximate solution

The main contribution of [

Let the conditions of the Theorem

Because of the properties of reproducing kernel definition and assumptions, we have

In what follows, we provide a priori and a posteriori error estimations.

Suppose that

By Lemma

Very similar to the above argument, we have the following.

Let the conditions of Theorem

Similar to Theorem

Suppose that

Now, we deal with the stability of RKHS method for the solution of

If

Consider the problem

Suppose that

Similarly, we have the following theorem.

Consider the problem

The authors declare that there is no conflict of interests regarding the publication of this paper.

Also, the authors acknowledge Hamedan Branch of Isalmic Azad University for their support during conducting this research.