Computer Modeling in Engineering & Sciences |
DOI: 10.32604/cmes.2021.014460
ARTICLE
Solution and Analysis of the Fuzzy Volterra Integral Equations via Homotopy Analysis Method
1School of Quantitative Sciences, Universiti Utara Malaysia, Kedah, Sintok, 06010, Malaysia
2Department of Mathematics, Faculty of Science and Technology, Irbid National University, Irbid, 2600, Jordan
3Department of Mathematics, Faculty of Science, Yarmouk University, Irbid, 21163, Jordan
*Corresponding Author: Ali. F. Jameel. Email: alifareed@uum.edu.my
Received: 29 September 2020; Accepted: 25 February 2021
Abstract: Homotopy Analysis Method (HAM) is semi-analytic method to solve the linear and nonlinear mathematical models which can be used to obtain the approximate solution. The HAM includes an auxiliary parameter, which is an efficient way to examine and analyze the accuracy of linear and nonlinear problems. The main aim of this work is to explore the approximate solutions of fuzzy Volterra integral equations (both linear and nonlinear) with a separable kernel via HAM. This method provides a reliable way to ensure the convergence of the approximation series. A new general form of HAM is presented and analyzed in the fuzzy domain. A qualitative convergence analysis based on the graphical method of a fuzzy HAM is discussed. The solutions sought by the proposed method show that the HAM is easy to implement and computationally quite attractive. Some solutions of fuzzy second kind Volterra integral equations are solved as numerical examples to show the potential of the method. The results also show that HAM provides an easy way to control and modify the convergence area in order to obtain accurate solutions.
Keywords: Homotopy analysis method; convergence control parameter; fuzzy Volterra integral equations
Integral equations have been used to model problems in a number of fields [1–4]. In real-world problems, inaccuracy, uncertainty and lack of information exist and are discussed both theoretically and numerically. The way to address this lack of information is to model uncertainty as fuzziness [5]. It is therefore possible to refer to fuzzy integral equations rather than to use deterministic models in the crisp domain. In order to study and solve many of the problems in applied mathematics, integral equations in fuzzy form are important, particularly for physics, for medical modelling [6]. In many applications, certain problem parameters are typically defined by a fuzzy number rather than a crisp number, and it is therefore important to establish mathematical models and numerical procedures for the proper handling of fuzzy integral equations. Numerical approaches to fuzzy integral equations have inspired many research works in the last decade due to their use in scientific phenomena [7–11]. The existence and uniqueness of a second kind fuzzy Volterra equation solution was introduced in [12]. Among the approximate methods for fuzzy Volterra integral equations, there are the Differential Transform Method (DTM) to obtain analytical solution of linear fuzzy Volterra integral equations of second kind [13], Homotopy Perturbation Method (HPM) for solving linear and nonlinear fuzzy Volterra integral equations of second kind [14,15] and Variational Iteration Method (VIM) together with Taylor method to solve linear fuzzy Volterra integral equations of second kind [16]. The main application of fuzzy integral equations is biomathematical modelling. For example, a model based on fuzzy integral equations [16] was proposed to study the dynamics of diseases transmitted through direct contact between susceptible and infected individuals. Solving mathematical problems with approximation methods usually lead to approaches in series or polynomial functions which often have a better interpretability and this can contribute to pave the way to future processes and solutions of given problems, without the shortcoming of a suitable discretization. In the 90s, Liao introduced a new approximation approach called homotopy analysis method [17]. The approximate solution is obtained as an infinite series function that has been shown to converge to the exact solution in many mathematical problems involving engineering applications [18–22]. The homotopy, a fundamental concept in topology, is a landmark of the approximation methods, since it provides more flexibility in handling the equations and their solution [23]. The validity of HAM depends on homotopy topology, regardless of the physical parameters. It is worth recalling that the Adomian Decomposition Method (ADM) and the VIM are non-perturbation techniques, which do not depend on a series of physical parameters, but such non-perturbation techniques in some cases do not guarantee the convergence of the solution series [24]. The HAM also allows to select the proper base function without any constraint to approximate the solution of some nonlinear problems [25].
The difference between HAM and other approximation methods is the auxiliary convergence control parameter, denoted by h, that can optimize and rate the convergence of the method per order of solution [26]. In HAM the selections of proper initial approximation, operator and auxiliary function with the optimal value of h allows to solve the deformation equations and develop a solution series [20].
To control the error of HAM solution, there is a convergence-control parameter, whose value, if properly selected, can lead to an accurate convergent series or faster convergence [27]. Van Gorder et al. [28] discussed the application of HAM for nonlinear ordinary differential equation and the effectiveness of a suitable choice of initial approximation, auxiliary linear operator, auxiliary function and convergence control parameter. There are several methods to obtain the best value of the convergence-control parameter such as control of residual errors, minimization of error functional and optimal selection of the homotopy of auxiliary function which were introduced and suggested for the approximate solution of semi-linear elliptic equation in [29]. Moreover, an algorithm was proposed in [30] to optimize the solution of singular and integral equation of first kind via HAM by computing the optimal auxiliary control parameter value.
HAM has been used to solve several types of problems in crisp and fuzzy domains, e.g., fluid flow and heat transfer problems [31], fractional differential equations involving biological models [32], system of nonlinear ordinary differential equations describing HIV infection models [33], Abel’s integral equations of the first kind [34], fuzzy boundary value problem [35] and fuzzy delay differential equation [36].
The present work deals with the approximate solution of fuzzy Volterra integral equations of second kind, since to the best knowledge of the authors no study has been carried out by formulating the general concept of HAM from the crisp domain to the fuzzy domain for solving such class problems.
The paper is structured as follows. The Volterra integral equations of the second kind with the defuzzification details are recalled in the next section. A new description of the fuzzy HAM general formula is presented in Section 3, where a convergence analysis is also outlined. In Section 4, some test problems are considered and the numerical results discussed. Finally, there is a short conclusion that includes a summary of this work. Note that some of the basic fuzzy definitions, remarks and concepts not described in this paper are well-known. Notions of fuzzy level sets, fuzzy numbers and their operations, fuzzy functions, fuzzy Zadeh extension theory and integral of fuzzy functions can be easily retrieved from the literature, e.g., [9,37–41].
2 Fuzzy Volterra Integral Equation
The general fuzzy version of the standard second kind Volterra integral equation [9] is defined below:
where
Eq. (1) follows the properties of the standard second kind Volterra integral equation in crisp domain by means of defuzzification according to [9]. Hence, Eq. (1) with the fuzzy parametric forms are given as follows:
where
with
The sufficient conditions for the existence of a unique solution to Eq. (2) are given and proved in [11].
The crisp form of HAM was introduced in [42]. To describe the dynamic of the HAM under homotopy theory in the fuzzy domain, we start off with:
where
where
while if p = 1, one gets
Thus, by imposing
It is
From [13], if p = 0 and p = 1, the homotopy equations becomes
As p changes from 0 to 1, the fuzzy solution
where
The auxiliary linear operator L, the initial guess
which is one of the solutions of the given equation to be solved by HAM. Notice that if all the values of
which represents the homotopy perturbation method (HPM), implying that HPM is a special case of HAM [43]. From Eq. (10) the governing equations can be deduced from the zero-order deformation Eq. (12) by defining the vectors:
By deriving with respect to p both sides of Eq. (10) m times, at = 0, and after that dividing them by m!, we obtain the mth-order deformation equation
where
4 Fuzzy HAM for Fuzzy Volterra Integral Equations
In this section, the HAM solution of fuzzy Volterra integral equations is described in some steps.
Construct the zeroth-order deformation for Eq. (1) for all
Set the values of p = 0 and p = 1, implying
From Eq. (20), it follows that the fuzzy initial guess
where
From the mth-order deformation equation
where
Then the fuzzy HAM series solution can be written in the following form:
The convergence of Eq. (26) depends on selecting a suitable value of
It is worth noticing that since the defuzzification leads to a system of crisp equations, the theoretical achievements on the convergence in [30] can be adapted.
5 Dynamics of Fuzzy HAM Convergence
As mentioned before, the convergence of the approximate solution of Eq. (1) relies on the value of the parameter
then to use the least square method to optimize the values of
After that, the nonlinear equation coming from Eq. (29) in terms of
Finally, the equation is solved for
The HAM for seeking the approximate solution of Eq. (1) can be summarized in the following algorithm.
Step 1: Set the initial guess
Step 2: Set the value of
Step 3: Set number of terms, s.t.
Step 4: Set i = i + 1 and for i = 1 to
Step 5: Compute
Step 6: Set the fixed value of
Step 7: Define the residual form in Eq. (28) and use Eqs. (29)–(30) to find the best value of
Step 8: Replace again the optimal value of
In this section application examples are presented. For the remainder of this work,
Example 6.1: Consider the following linear fuzzy Volterra integral equation [13]:
where
The HAM formulation (see Section 5) for Eq. (33) is:
where the initial guess
From Fig. 2, the range of the valid values of
Fig. 3 shows the absolute errors defined in Eq. (31) of the fifth-order fuzzy HAM solution for the values
To be more precise regarding the convergence of HAM, if the values of
From Fig. 3, one can easily deduce that for values of
From Figs. 4, 5 one can notice that the fifth-order fuzzy HAM solutions of Eq. (31) are in the form of fuzzy numbers for any
Example 6.2: Consider the following linear fuzzy Volterra integral equation [16]:
The exact solution of Eq. (36) is given by
The HAM formulation of Eq. (36) is:
where the initial guess is assumed to be
The valid values of
From Fig. 7, one can see that the optimal value of the convergence control parameters in Tab. 6 is
In Tab. 9 there is a comparison of the mean errors by HAM, VIM and Taylor expansion method [16] (all of sixth-order) at
Clearly from Tab. 9 one can see that sixth-order HAM solution at the optimal value of the convergence control parameter
Example 6.3: Find the solution of the following nonlinear fuzzy Volterra integral equation:
where
The HAM formulation of Eq. (39) is:
where the initial guess
The
From Fig. 10 the valid values of
Tabs. 10–13 shows the fifth-order HAM approximate solution of Eq. (39) and accuracy for different values of
Example 6.4: Consider the following linear fuzzy Volterra integral equation [9]:
The exact solution for Eq. (42) is given by
The HAM formulation of Eq. (42) is:
where the initial guess is assumed to be
The valid values of
After testing the values of
In Tab. 15, there is a comparison of the absolute errors and the results by fifth-order HAM at
From Tab. 15 one can see that the fifth-order HAM solution at the optimal value of the convergence control parameter
This paper proposes HAM for solving fuzzy Volterra integral equation of the second kind with separable kernels. The fuzzy set theory was used to present a new formulation of HAM with application to the fuzzy Volterra integral equation of the second kind. The convergence of this approach was qualitatively discussed to find the optimal value of the convergence-control parameter. The examples presented show the potential of the method. Numerical results and graphs show that both linear and nonlinear fuzzy Volterra integral equations of the second type are well approximated by the method. Being a semi-analytical method, this approach has the advantage of lead to solutions in explicit form. The numerical experiments showed better performance of the method when compared against other approximation or numerical approaches such as VIM, the Taylor method and the Trapezoidal Quadrature Formula. Due to its accurate results, which do not violate the fuzzy sets theory solution, and a relatively low computational cost, HAM seems to be a reliable tool for solving fuzzy Volterra integral equation of the second kind. In future work, we will apply the approach to some problems in Biomathematics, such as cancer growth and epidemics.
Funding Statement: Dr. Ali Jameel and Noraziah Man are very grateful to the Ministry of Higher Education of Malaysia for providing them with the Fundamental Research Grant Scheme (FRGS) S/O No. 14188 that supported this research.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
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