Abstract
Ordinary differential equations describe several phenomena in different fields of engineering and physics. Our aim is to use the reproducing kernel Hilbert space method (RKHSM) to find a solution to some ordinary differential equations (ODEs) that are described by using the global derivative. In this research, we used the RKHSM to construct new numerical solutions for nonlinear ODEs with global derivative. The used method systematically produces analytic and approximate solutions in the series’s form. We tested three applications for showing the performance of the RKHSM.
1. Introduction
In the last decades, the rate of change has been increasingly used for understanding the instantaneous changes that arise in widespread fields. Thinking of the derivative as representing a rate of change is very useful when solving physics problems. The derivative plays a fundamental role in forming the ordinary differential equations (ODEs) that are of great importance because of their ability to describe numerous phenomena in physics, such as electrical networks, oscillating and vibrating systems, satellite orbits, and chemical reactions. Finding the ODEs’ solutions is the key to understanding nature, but it is hard and sometimes impossible to get the exact solutions of most real-life ODEs, especially the nonlinear ones. And for such a case, one resorts to numerical methods.
The RKHSM is a widely used numerical method for solving nonlinear ODEs (NODEs). This method which was proposed in 1908 [1] is an effective numerical method for complex nonlinear problems without discretization. Many researchers applied it to solve several types of equations [2–14]. The principal advantages of this method are (1)the feature that is it is easy to be applied, especially because it is meshfree(2)its capability to deal with diverse complex differential equations(3)the uniform convergence between the numerical and exact solutions as well as their derivatives
This research aims to provide a new convenient method using the reproducing kernel (RK) theory for obtaining the solution of some nonlinear ODEs that are described by using the global derivative.
In this paper and for the first time, the RKHSM is used for constructing numerical solutions for the nonlinear ODEs with global derivative.
The next section shows some basic definitions and theorems concerning RK theory and global derivative. The description of the RKHSM and its application to the proposed problem are presented in the third section. The RKHSM’s effectiveness and the solutions’ accuracy are validated through three applications in the fourth section. Finally, the conclusion is given.
2. Preliminaries
This section covers the theory required to understand the RKHSM we will apply to solve some important nonlinear ODEs with global derivative.
Definition 1. A global derivative of a differentiable function is [15] in which the function is an increasing nonzero.
Remark 2. If the function is differentiable then [15]
Remark 3. The global derivative covers the following three cases that we are going to deal with throughout the numerical part: (1)Case 1: let us choose Hence, the classical derivative is a special case of global derivative. (2)Case 2: let us choose Hence, the fractal derivative is a special case of global derivative. (3)Case 3: let us choose
Definition 4. A function which satisfies (1) for all (2) for all and for all is called a reproducing kernel of is a Hilbert space over
Definition 5. We set [16]. An inner product on is and its norm is denoted by for all
Theorem 6. The function is the reproducing kernel function of
For the proof of this theorem, see [17].
Definition 7. We set [16]. An inner product on is and its norm is denoted by for all
Theorem 8. The function is the reproducing kernel function of
For the proof of this theorem, see [17].
3. Solution Methodology
We now consider the 1st-order nonlinear ODE, where is the global derivative, is the unknown, is a function of and and is a constant.
To apply the RKHSM, let us begin with making a change of variable to homogenize the initial condition :
Replacing by in (14) gives where is a nonlinear function of and
The second step is to define a linear operator such that
We use this linear operator to get
The next step is to build an orthogonal function system of Let where (i) represents the RK function of (ii)The set is dense in (iii) is the adjoint of
Now, to find we need to use Gram-Schmidt’s process: where denotes the function system in obtained by
And the coefficients can be found by where
Theorem 9. Suppose is dense in then is the complete system of
Proof. We know that So, for each fixed it follows Since and is dense on the interval we have Then, that gives
Lemma 10. Assume then where and
Proof. we have Using the expression of we can reach Consequently, where Then Lemma 10 follows from (31).
Theorem 11. Assume is dense in and problem (18) has a solution that should be unique on Therefore, the solution of (18) is While the solution of (14) is
Proof. Firstly, the fact that is a complete orthonormal basis in allows us to write
with
Secondly, by replacing by its formula (32) in the transformation (15), we get
We now write the RKHSM’s solution as
The space is a Hilbert space, hence
which means that converges to in the norm.
Theorem 12. (1) converges uniformly to (2) converges uniformly to
Proof. For the first result, we need to estimate the term on the left below:
where is a constant.
Following the same way, we get
due to the uniform boundedness of we have
where is a positive constant.
Therefore
4. A Numerical Experiment
This section is the numerical part that assures the efficiency of the proposed method by testing three examples. The rate of convergence of the presented method is as follows [18]: where
Now, how to apply the RKHSM can be summarized in the following procedure:
Step 1. Fix
Step 2. Set .
Step 3. Calculate the orthogonalization coefficients using (22).
Step 4. Set .
Step 5. Choose an initial guess .
Step 6. Set .
Step 7. Set .
Step 8. .
Step 9. If set Go to Step 7. Else stop.
where and is the number of collocation points.
Example 1. Taking the following linear ODE with global derivative: As the initial condition is homogeneous. We can then directly define a bounded linear operator as
Taking collocation points in which The approximate solution for Example 1 is found using the RKHSM for different cases of the function in the global derivative when equals and For each case, the results are compared with the exact solution. Figure 1 shows the exact solution and the RKHSM’s solution with The absolute error of this case is plotted in Figure 2. In Figure 3, we compared the exact solution with the RKHSM’s solution when with , and its absolute error is given in Figure 4, whereas in Figures 5 and 6, we depicted the obtained results for and together. Figures 7 and 8 are where the results of the last case of are given. We can see from these figures that the graphs’ behavior is very similar. To highlight more comparisons between the RKHSM and the exact solution, we gave the rate of convergence for in Table 1, and we drew the absolute error for each case through the figures presented. What we can observe here is that the RKHSM’s solution is very close to the exact one. And this confirms that the proposed method is effective.
Example 2. Taking the following linear ODE with global derivative: As the initial condition is homogeneous. We can then directly define a bounded linear operator as
Taking collocation points in which The approximate solution for Example 2 is found using the RKHSM for different cases of the function in the global derivative when equals and For each case, the results are compared with the exact solution of each case. Figure 9 shows the exact solution and the RKHSM’s solution with The absolute error of this case is plotted in Figure 10. In Figure 11, we compared the exact solution with the RKHSM’s solution when with , and its absolute error is given in Figure 12, whereas in Figures 13 and 14, we depicted the obtained results for and together. Figures 15 and 16 are where the results of the last case of are given. We can see from these figures that the graphs’ behavior is very similar. To highlight more comparisons between the RKHSM and the exact solution, we gave the rate of convergence for in Table 2, and we drew the absolute error for each case through the figures presented. What we can observe here is that the RKHSM’s solution is very close to the exact one. And this confirms that the proposed method is effective.
Example 3. Taking the following linear ODE with global derivative: As the initial condition is homogeneous. We can then directly define a bounded linear operator as
Taking collocation points in which The approximate solution for Example 3 is found using the RKHSM for different cases of the function in the global derivative when equals and For each case, the results are compared with the exact solution of each case. Figure 17 shows the exact solution and the RKHSM’s solution with The absolute error of this case is plotted in Figure 18. In Figures 19–21, we compared the exact solution with the RKHSM’s solution when with Figures 22 and 23 are where the results of the last case of are given. We can see from these figures that the graphs’ behavior is very similar. To highlight more comparisons between the RKHSM and the exact solution, we gave the rate of convergence for in Table 3, and we drew the absolute error for each case through the figures presented. What we can observe here is that the RKHSM’s solution is very close to the exact one. And this confirms that the proposed method is effective.
5. Conclusion
In this paper, an efficient method, named the reproducing kernel Hilbert space method, is applied successfully for solving nonlinear ODEs described by using the global derivative. The accuracy and applicability of the RKHSM are validated by computing the numerical solutions at many grid points. The results show that the RKHSM is a powerful method to deal with many other nonlinear problems that arise in a large variety of physical problems with different types of derivatives.
Data Availability
All of the necessary data and the implementation details have been included in the manuscript.
Conflicts of Interest
The authors declare that they have no conflicts of interest.