Abstract
This paper proposes a numerical method for solving fractional relaxation-oscillation equations. A relaxation oscillator is a type of oscillator that is based on how a physical system returns to equilibrium after being disrupted. The primary equation of relaxation and oscillation processes is the relaxation-oscillation equation. The fractional derivatives in the relaxation-oscillation equations under consideration are defined in the -Caputo sense. The numerical method relies on a novel type of operational matrix method, namely, the -Haar wavelet operational matrix method. The operational matrix approach has a lower computational complexity. The proposed scheme simplifies the main problem to a set of linear algebraic equations. Numerical examples demonstrate the validity and applicability of the proposed technique.
1. Introduction
The history of fractional or noninteger order differential and integral operators can be traced back to the origins of integer order calculus [1]. In recent years, fractional differential equations have attracted a lot of attention. In fields such as damping laws, diffusion processes, and other physical phenomena, fractional differential equations have proven to be adequate models. Since the majority of fractional differential equations do not have analytical solutions, we must use an approximate method. Many studies have analyzed solution techniques for fractional differential equation such as the collocation method, Adomian decomposition method, variational iteration method, tau method, and operational matrix method [2–12].
A relaxation oscillator is a type of oscillator that is based on a physical system’s ability to return to equilibrium after being disrupted. The main equation of relaxation and oscillation processes is the relaxation-oscillation (R-O) equation. A relaxation equation in its standard form is given by where is a real number and is a given function. Equation (1) can be used to represent a variety of physical processes, such as the Maxwell model, which uses a spring and a dashpot in succession to explain the characteristics of a viscoelastic material. A simple physical process with a regulated phase shift is described by the standard oscillation equation. The equation that defines the oscillation of a system corresponding to an external force has the simplest linear form as where is the oscillator’s natural frequency. In the relaxation and oscillation models given in Equations (1) and (2), fractional derivatives are used to depict slow relaxation and damped oscillation (see [13, 14]). The fractional relaxation-oscillation differential equation (FRODE) is given by having the following initial conditions: , if , and , if , where denotes a fractional differential operator of order .
The numerical investigation of FRODEs has received a lot of interest recently. The numerical solution of problem (3) (with ) was investigated in [15] by taking into account the positive fractional and fractal derivatives. The authors of [16] employed a Taylor matrix method to find the numerical solution of problem (3) by taking into account the Caputo fractional derivative. This approach is based on a fractional version of Taylor’s formula, which was first proposed in [17]. The numerical solution of problem (3) is achieved by the optimal homotopy asymptotic approach in [18], where the fractional derivative is given in the Caputo sense. In [19], a trapezoidal approximation of the fractional integral is used to get the numerical solution of problem (3) with Caputo fractional derivative. To solve problem (3), [20] proposes a generalized wavelet collocation operational matrix approach based on the Haar wavelet (HW), where the fractional derivative is represented in the Caputo sense. Inspired by the above-mentioned studies, this paper focuses on a numerical solution of the fractional differential equation of the form with initial conditions , and is a natural number. is the -Caputo fractional derivative of order , and
These problems are studied in [21] by using an operational matrix of -shifted Legendre polynomials.
As far as we know, there is no open literature article dealing with the numerical treatment of FRODEs involving the -Caputo fractional derivative employing HW. Therefore, the basic aim of this paper is to provide a numerical technique for solving -FRODEs that arise in physics. Our method is based on a new type of operational matrix of fractional integration called the -HW operational matrix. We provide a rigorous verification of convergence for the suggested method. Furthermore, numerical experiments are presented to demonstrate the convergence of the procedure by comparing the exact values to numerical approximation.
Different types of orthogonal polynomials, including Chebyshev polynomials [22], Legendre polynomials [23], and Laguerre and Hermite polynomials [24], have been utilized with the operational matrix of integer order integration. Many authors then expanded it to the fractional case, as seen in [25–31] and the references therein. Only Caputo or Riemann-Liouville fractional derivatives were examined in all of the above listed papers.
The following is a description of this paper’s structure.
In Section 2, we go through the basic fundamentals of fractional calculus. We introduce HW and function approximation using HW in Section 3. In Section 4, we construct an explicit formula for the -fractional integration of the HW and the -Haar wavelet operational matrix. Section 5 discusses the numerical scheme as well as the method’s convergence.
2. Preliminaries
We will go over some definitions of -fractional integral and differential operators in this section.
Let the function be integrable, is a positive real number, is a natural number, , and , where is increasing.
Definition 1 (see [32–34]). The -Riemann-Liouvile (-RL) fractional integral operator of order is defined by
The -RL differential operator of fractional is defined by
Definition 2 (see [21, 35, 36]). Let be a positive real number, a natural number, , and , where is increasing. The -Caputo differential operator of fractional-order is defined by where , for and when .
2.1. Function Approximation by Haar Wavelet
HW are the simplest wavelets with a compact support among the various wavelet families. These wavelets have been shown to be an effective method for numerical function approximation. The Haar functions, which are orthogonal, contain only one wavelet during some subinterval of time and remain zero elsewhere.
The th HW, , where in the HW family is defined as [37] where , is the dilation parameter and is the translation parameter. is the max resolution level of HW. The parameters , , and are related by the equation ; is called the wavelet number. Equation (9) holds true for .
For and , the scaling functions for HW family are defined, respectively, by
A square integrable function on , can be approximated by HW in the following way: where is defined by the inner product of and and represents the inner product. The first terms are employed for function approximation, that is, which can be written in the matrix notation as where is the coefficient matrix determined by and is the vector of Haar functions.
3. -HW Operational Matrix
The fractional-ordered -RL integral of the HW is defined by where .
The -HW operational matrix is computed in the interval [0, 1] for and . The numerical and exact integration of the function for and different values of is plotted in Figure 1.
4. Error Analysis
Caputo-type FDEs have recently been investigated in context of error analysis in [38]. In addition, the convergence analysis of solution of nonlinear Fredholm integral equations by HW is given in [39].
Using the -Caputo fractional differential operator, we estimated the max error, demonstrating the efficiency of the -HW approach for -FDEs.
Theorem 3. Let be continuous on and suppose that so that , where , and is approximated by , then we have
Proof. can be approximated by HW as
Here, is given by
Let the approximation of be which is defined by
in which .
Therefore,
This gives
The sequence being orthogonal, we get , which represents the identity matrix of order .☐
Therefore, from Equation (21) we have
Equation (18) gives
Employing mean value theorem of integration where so that we arrive at
Therefore,
Applying the -Caputo fractional differential operator along with the facts that is increasing and , we get
As and also are increasing, so
Therefore,
According to the mean value theorem, such that , we get which implies that
Putting together equations (22) and (32), we have which implies that
Let , (34) can also be written as
To compute the error bound, we need the value of .
So we will estimate first. Since is continuous and bounded on , so is and it is approximated by where and .
Integration of (36) gives
Similarly,
Continuing in the same manner, we arrive at
Taking , and putting it in (39), we have
The matrix form of (40) is as
The linear system in (41) determines the value of the vector ; by putting this value in (36), can be obtained .
Let , then can be computed for the equidistant points , then is the approximation of .
Theorem 4. Assume that , computed from -HW is estimated by , then we have where .
Theorem 4 can be proven easily by following the procedure of Theorem 3. From Equation (42), we noted that as . Thus, the convergence of the -HW method is inferred.
5. Numerical Examples
We present several examples of how to get numerical solutions of -FRODEs using the -HW operational matrix approach.
Example 5. Consider the -FRODE
For and , the actual solution of Equation (43) is . We use the -HW technique to solve problem (43).
Let
Integrating Equation (44) with respect to and using the initial conditions, we have
Substituting (44) and (45) into (43), we have
Equation (46) has the following matrix form: where is the matrix representation of at the collocation points.
Solving the algebraic system given by Equation (47) for and substituting this value into Equation (45), we will have the required numerical solution. In Table 1, the max absolute error is given for and . Approximate solutions for and and different choices of the function are plotted in Figure 2. Also, actual and approximate results and the absolute error are given for , , and in Figure 2.
Example 6. Consider the composite -FRODE For .
The exact solution for the problem (48) is .
For the numerical solution, we employ the -HW technique.
Let
Integrating Equation (50) with respect to and utilizing the initial condition, we have Substituting Equations (50) and (51) in Equation (48), we get where . Equation (52) in matrix form is given as where is the matrix representation of .
Required approximate solutions can be obtained by using the value of from Equation (53) into Equation (51).
The absolute error is tabulated for and various choices of and in Table 2, which shows that the Maximum Absolute Error decreases by increasing the values of . Figure 3 represents approximate solutions for different choices of . Also, comparison of actual and approximate results and their absolute error is displayed in Figure 3.
Example 7. Consider the -FRODE where It is easy to verify that is the actual solution of Equation (54). For numerical approximation, we employ the -HW technique.
Let
Integrating Equation (55) with respect to and using the initial conditions, we have
Substituting (55) and (56) into (54), we have where
The matrix form of Equation (57) is where is the matrix representation of .
Required approximate solutions can be obtained by using the value of from Equation (58) in Equation (56). Table 3 shows that the Maximum Absolute Error decreases by increasing the values of . Approximate solutions are displayed in Figure 4 for various values of . Also, Figure 4 represents approximate and exact solutions and their max absolute error for , and .
Example 8. Consider the -FRODE For and , the exact solution of Equation (59) is .
For approximate solutions, we use the -HW technique.
Let
Integrating Equation (60) with respect to and using the initial conditions, we have
Substituting (60) and (61) into (59), we have
The matrix representation of Equation (62) is where is the matrix representation of at the collocation points.
Required approximate solutions can be obtained by using the value of from Equation (58) in Equation (61). Table 4 shows that the Maximum Absolute Error decreases by increasing the values of . Also the approximate solutions are displayed in Figure 5 for various values of .
6. Conclusion
This study introduces a numerical approach for solving a class of fractional differential equations with a -Caputo fractional derivative based on a novel type of operational matrix of fractional integration, namely, the -HW operational matrix. The convergence of the proposed method is demonstrated, and the numerical tests reported in Section 5 corroborate the efficacy of our approach.
Data Availability
The numerical data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.