International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(1): 23-47
Published online March 25, 2022
https://doi.org/10.5391/IJFIS.2022.22.1.23
© The Korean Institute of Intelligent Systems
Moa’ath N. Oqielat^{1}, Ahmad El-Ajou^{1}, Zeyad Al-Zhour^{2 }, Tareq Eriqat^{1}, and Mohammed Al-Smadi^{3};^{4}
^{1}Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Salt, Jordan
^{2}Department of Basic Engineering Sciences, College of Engineering, Imam Abdulrahman Bin Faisal University, Dammam, Saudi Arabia
^{3}Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun, Jordan
^{4}Nonlinear Dynamics Research Centre (NDRC), Ajman University, Ajman, UAE
Correspondence to :
Zeyad Al-Zhour (zeyad1968@yahoo.com)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this paper, we present in detail the power series solutions to fuzzy quadratic Riccati differential equations (QRDEs) along with a suitable fuzzy constant through an interactive derivative, more specifically, the Hukuhara-strongly generalized differentiability (H-SGD) based on our new technique. This technique is called the Laplace residual power series (LRPS) method, and it mainly depends on a new expansion and the combination of the Laplace transform technique with the residual power series method. To validate the accuracy of our proposed algorithm, numerous examples were examined numerically and graphically, and we compared the results of the optimal homotopy asymptotic (OHA), multiagent neural network (MNN), and fourth-order Runge-Kutta (RK-4) methods with the LRPS method at γ = 1.
Keywords: Fuzzy-valued function, Strongly generalized differentiability, Quadratic Riccati differential equation, Laplace residual power series method, Laplace and inverse transforms
Fuzzy set theory is an important topic in the study and modeling of many real-life problems related to numerous physical phenomena and dynamical processes, such as astronomy, quantum mechanics, chromodynamics, quantum optics, electronic mechanisms, electronic-controlled systems, the modeling of anomalous hydraulic diffusion systems, and population dynamics models [1–14]. Moreover, several problems in applied mathematics that are associated with biology, medicine, physics, and technology can be modeled using fuzzy differential equations [1–6,15–17].
Fuzzy set theory has been explored since the 1920s; however, the term fuzzy derivative was introduced by Chang and Zadeh [15]. Subsequently, Dubois and Prade [16] proposed the fuzzy differential calculus concept. Many researchers have recently presented other results and notations for fuzzy mapping [18–21]. Moreover, Bede and his colleagues [22,23] defined strongly generalized differentiability (SGD) as a fuzzy-valued function (FVF). In general, it is not straightforward to obtain the exact solution for these types of problems because of the difficulties involved; therefore, reliable numerical techniques are needed, including a reliable numerical algorithm for handling fuzzy integral equations of the second kind in the Hilbert spaces [4], a residual power series (RPS) for solving uncertain Riccati differential equations
[24], and a fuzzy Picard method for solving fuzzy quadratic Riccati and fuzzy Painlevl equations [25]. Other examples include a novel extended approach for obtaining numerical solutions to fuzzy conformable fractional differential equations [22], a reproducing kernel algorithm for solving second-order two-point fuzzy boundary value problems and second-order fuzzy Volterra integro-differential equations [3,18,26], a Runge-Kutta method applied to the SI epidemiological model [27], a homotopy analysis method (HAM) for solving fuzzy initial value problems (FIVPs) [28], and bidimensional IVP with interactive fuzzy numbers [14]. Additional solutions are a logistic model of population growth through an interactive derivative [29], an HIV viral dynamics model for individuals under antiretroviral treatment [17], a numerical approach in the Hilbert space to solve a fuzzy Atangana-Baleanu fractional hybrid system [30], a novel technique for solving fuzzy fractional delay differential equations [31], and a homotopy perturbation Sumudu transform method to find the analytical fuzzy solution of nonlinear fuzzy integro-differential equations [32].
The existence, uniqueness, and controllability of fuzzy solutions for such nonlinear fuzzy (fractional) differential equations have been studied and discussed in [33–39]. Kwun et al. [33–36] and Lee et al. [37] respectively studied the existence and uniqueness of fuzzy (periodic) solutions for the nonlinear fuzzy integro-differential equations on
Quadratic Riccati differential equations (QRDEs) form a precise nonlinear model for defining a certain type of physical scheme with applications in the diffusion process, optimal control, optical networks, and artificial intelligence, and usually have the following general form [40–43] of
subject to
Many numerical and analytical techniques have been employed to obtain QRDEs solutions, including the HAM [40], homotopy perpetuation method (HPM) [41], differential transformation method [42], and Adomian decomposition method [43].
Several authors [24,26,28,40–45] have reformulated the QRDE as a fuzzy differential equation along with suitable fuzzy constants under the SGD concept, the fuzzy form of which is as follows [24]:
subject to the fuzzy initial condition (FIC):
where
The main objective of the research described in this study is to extend the application of the LRPS method and provide a power series solution to the FIVP in
The LRPS method is a new method that combines the Laplace transform technique with the RPS method, which is used to solve the integral and differential equations [7–13]. The Laplace transform method involves transforming the equation from the space within it into a new space, presenting its solution in the new space, and then returning the solution to the original space and thus obtaining the solution to the equation. However, it is sometimes difficult to offer a solution to an equation in a new space. The RPS method assumes a power series solution to the equation and provides an easy technique for determining the series coefficients. The LRPS method uses the two methods together to provide a series solution to the equation in the new space and uses the principle of the RPS method to determine the series coefficients. However, through a new approach, it is easier and faster than the user in the original technique. In fact, our proposed method is an analytical and efficient method that can be easily applied to solve different types of fuzzy problems.
Two fuzzy numbers are considered non-interactive when the joint possibility distribution involving both is given by the minimum t-norm. Thus, any operation between two fuzzy numbers, aside from the one obtained by the minimum t-norm, must be considered to be interactive. Following this reasoning, when we use the Hukuhara difference (H-difference) for two fuzzy numbers, we assume that they are interactive. Thus, all derivatives obtained by a fuzzy arithmetic operation that differs from the standard operation must be considered interactive. In this study, we find analytical and numerical solutions for FIVPs along with a suitable fuzzy constant through an interactive derivative, more specifically, the Hukuhara-strongly generalized differentiability (H-SGD) based on our new LRPS technique. The solution to the FIVP obtained here using H-SGD is more suitable than the classical derivative because the latter uses an interactive derivative obtained by the H-difference, whereas the standard sum is used to compute from the non-interactive operation, that is, the sum operation is based on the minimum t-norm. This is because the arithmetical operations compatible with the H-difference are not known. Thus, it makes no sense to combine interactive and non-interactive H-SGD. Furthermore, with the H-SGD theory, it is possible to extend and generalize our analysis and results in this study to all fuzzy interactive arithmetic operations, and not only to the H-difference operation in future studies. In addition, in the present paper, we introduce a new expansion to Taylor’s formula, which is required for creating the LRPS method and formulating the FIVP. Moreover, several examples are solved and examined numerically and graphically using this new technique, and the results of the optimal homotopy asymptotic (OHA), multiagent neural network (MNN), and fourth-order Runge-Kutta (RK-4) methods are compared with those of the LRPS method at
This paper consists of five sections. In Section 2, we study some basic concepts of fuzzy theory, establish a new expansion to Taylor’s formula, and formulate the FIVP. In Section 3, the LRPS algorithm is created and successfully tested to obtain analytical and numerical solutions of the nonlinear FIVP. In Section 4, we test the accuracy of our proposed algorithm (LRPS) by solving three important examples numerically and graphically, and comparing the results of the OHA, MNN, and RK-4 methods with the LRPS method at
This section reviews some theories and concepts necessary for our study and is divided into three subsections: the first relates to fuzzy calculus, the second relates to a new expansion needed to construct the LRPS solution, and we formulate the FIVP in third.
In this subsection, we study the most important definitions and notations related to fuzzy analysis and fuzzy calculus [3,4,12, 15,16,18–23,48–63] used throughout this paper. The provided ℝ_{ℱ} denotes the set of all the fuzzy numbers.
Letting
(i)
(ii)
(iii)
(iv)
Let
Therefore, if
in addition, if
In addition, if
Note that a fuzzy set
Some conditions must be satisfied by two functions:
Suppose that
(i)
(ii)
(iii)
(iv)
Subsequently,
Let
(i) standard sum,
(ii) standard differences,
(iii) scalar multiplication,
(iv) multiplication,
(v) quotient,
(vi) equality, the two fuzzy numbers
Letting
This definition was modified by Bede and Stefanini [23] to the generalized H-difference (gH-difference), as follows:
with
Very recently, Wasques et al. [17,27,29,59] generalized the Hukuhara and H-differences and used interactivities for all arithmetic operations and not only for the difference operation. We summarize this work as follows.
In general, a fuzzy subset
and
Note that the fuzzy subset
Let
for all
for all
Here,
for all
Otherwise, the fuzzy numbers are said to be
The Pompeiu-Hausdorff distance
where
and gives the Hausdorff distance between [
It is provided that (ℝ_{ℱ},
(i)
(ii)
(iii)
Note that the metric space
Let [
which is called the
The complete metric structure on ℝ_{ℱ} is given by the Hausdorff distance mapping
Let
Let
Let
(i) the H-differences
or all
(ii) The H-differences
for all
(i) If
(ii) The limits in
Let
(i) if
and (ii) if
Let
A strongly measurable and limited integrable fuzzy function is known as an integrable function. The fuzzy integral of an FVF
for all
Similarly, the fuzzy integral of an FVF
for all
Let
(i) If
and
(ii) If
and
In this subsection, we present a new form of Taylor’s formula, which we will mainly use in our proposed method for solving FIVPs.
Let
Then
We assume that
For the other aspect as well, multiplying
It is clear that
If we multiply
Now, we can see the patterns and find the general formula for
The inverse Laplace transform of the expansion in Theorem 2.5 has the following form:
which is the classical Taylor’s formula.
The following theorem explains and determines the conditions for convergence of the new form of Taylor’s formula, which is presented in Theorem 2.5, as mentioned above.
Let
First, suppose that ℒ[
From the definition of the reminder
It follows from
Thus, by reformulating
In this subsection, we state the conditions and assumptions required to formulate the FIVP, as in
with the FIC
Let
Finally, to construct the LRPS solution for the target equation in a
If
with FICs,
We then have the following procedures:
(i) Construct the LRPS solution for the system in
(ii) Ensure that the solutions [
(iii) Then, [
If
with FICs,
In addition, we have the following procedures:
(i) Construct the LRPS solution for systems
(ii) Ensure that the solutions [
(iii) Then, [
In this section, we present the (1)-solution for the FIVP, as in
Using the properties of the Laplace transform and the initial conditions in
where
Next, the LRPS technique represents the solution as an infinite series, and the series solution to
Because
and
this leads to
Therefore, the expansions in
Consequently, the
Prior to applying the LRPS technique to determine the values of the coefficients
and the
It is clear that
Because
for
For the other aspects as well, to determine the value of the first unknown coefficients
Now, multiplying both sides of
Computing the limits of both sides of
Considering
Similarly, to determine the values of the second unknown coefficients
By multiplying both sides of
Using the fact that
By solving the resultant algebraic system in
For
Therefore, the third LRPS estimation can also be given.
By continuing the same process until random coefficients order
subsequently
Finally, the exact or approximate series solution obtained must be transformed into the original space by using the inverse Laplace transform to obtain the series solution to the original FIVP, as in
To validate the accuracy of our proposed algorithm, numerous examples are examined numerically in this section. For calculations and results, Maple 2018 is used to perform the computations.
Consider the following FIVP:
with FIC,
For
Using Definition 2.10, the FIVP, as in
subject to
Applying the LRPS method as below, taking the Laplace transform of both sides of
and
where
Now, we assume that an infinite series solution of the algebraic system in
In addition, the
Depending on the initial data,
The
To determine the first unknown coefficients,
Now, by remultiplying both sides of
Using the fact that
Similarly, to determine the values of the second unknown coefficients
Now, re-multiplying
According to the fact:
Solving
Hence, the second-approximate LRPS solution to
The
and
The LRPS solution to
By applying the inverse Laplace transform to
As a special case, when
Table 1 shows the numerical results of the exact and approximate solutions using the LRPS method for Example 4.1 at
Table 2 shows a numerical comparison between the tenth-LRPS solution and the OHA method [2], the MNN method [3], and the RK-4 method [27]. We concluded that the results obtained using the LRPS method were similar to those obtained using other methods.
Figure 1 shows the upper and lower bounds of the triangular fuzzy LRPS solution at
Consider the following quadratic FIVP:
with FIC,
For
According to Definition 2.10, the FIVPs, as in
subject to
According to the procedure of the LRPS algorithm presented in Section 3, the Laplace transform of
where
Depending on the initial conditions in
and the
By utilizing lim
and
We can express the LRPS solution of
Applying the inverse Laplace transform to
As a special case, when
Tables 3 and 4 present the numerical results of the fuzzy solution upper and lower bounds of the system in
Figure 3 shows the upper and lower bounds of the triangular fuzzy LRPS solution at
Consider the following FIVP:
with FIC,
For
The FIVP, as in
subject to
Similar to the steps used in previous examples, the Laplace transform of
According to the initial data in
The
By utilizing the following facts:
and
We can express the LRPS solution of
Applying the inverse Laplace transform to
As a special case, when
Table 5 presents the error analysis of the proposed algorithm for the system in
Figure 5 shows the upper and lower bounds of the triangular fuzzy LRPS solution at
Many numerical and analytical methods have been applied to solve fuzzy differential equations, some of which have advantages over others. Some of them are accurate and effective; however, they require mathematical operations that can be difficult, long, or sometimes failed. Others are simple and fast; however, they may not provide precise solutions. Our new method, called LRPS, is characterized by its accuracy, speed, and simplicity in finding exact and accurate approximate solutions of linear and nonlinear fuzzy differential equations. In this study, we succeeded in providing accurate approximate and sometimes exact solutions to FIVPs using the newly proposed technique. The proposed expansion of our study enabled us to obtain a serial solution for equations in the Laplace transform space. The LRPS method provides an easy and fast technique for determining the proposed series coefficients as a solution to the equation. Unlike the traditional RPS method, which specifies series coefficients, the derivative must be calculated each time, whereas LRPS only requires the concept of the limit at infinity in determining series coefficients, which distinguishes this method. It is worth noting that the LRPS method can be applied to solve other types of fuzzy ordinary and partial differential equations for an integer or fractional order, such as fuzzy Poisson, KdV, Schrödinger, and apple equations. Furthermore, based on the H-SGD theory, it is possible to extend and generalize our analysis and results to all fuzzy interactive arithmetic operations mentioned in Section 2 in future studies.
No potential conflict of interest relevant to this article was reported.
Triangular fuzzy solution plots for the eighth-LRPS solutions of Example 4.1. Solution plots for (a)
The 3-dim plot for Example 4.1. Blue and yellow are the lower and upper bounds, respectively, of the tenth-LRPS fuzzy solution.
Triangular fuzzy solution plots for the eighth-LRPS solutions of Example 4.2. Solution plots for (a)
3-dim plot for Example 4.2. Blue and yellow are the lower and upper bounds, respectively, of the tenth-LRPS fuzzy solution.
Triangular fuzzy solution plots for the eighth-LRPS solutions of Example 4.3. Solution plots for (a)
3-dim plot for Example 4.3. Blue and yellow are lower and upper bounds, respectively, of tenth-LRPS fuzzy solution.
Table 1. Numerical results of the LRPS solutions for Example 4.1 at
Exact solution | Approximate solution | Absolute error | Relative error | |
---|---|---|---|---|
0.1 | 0.110295196916 | 0.110295196917 | 1.52656 × 10^{−16} | 1.38406 × 10^{−15} |
0.2 | 0.241976799621 | 0.241976799621 | 5.55112 × 10^{−17} | 2.29407 × 10^{−16} |
0.3 | 0.395104848660 | 0.395104848660 | 2.77556 × 10^{−16} | 7.02486 × 10^{−16} |
0.4 | 0.567812166293 | 0.567812166293 | 3.33067 × 10^{−16} | 5.86579 × 10^{−16} |
0.5 | 0.756014393431 | 0.756014393431 | 3.33067 × 10^{−16} | 4.40556 × 10^{−11} |
0.6 | 0.953566216472 | 0.953566216472 | 1.11022 × 10^{−16} | 1.16429 × 10^{−16} |
0.7 | 1.152948966979 | 1.152948966980 | 5.32907 × 10^{−14} | 4.62212 × 10^{−14} |
0.8 | 1.346363655368 | 1.346363655422 | 5.39830 × 10^{−11} | 4.00954 × 10^{−11} |
0.9 | 1.526911313280 | 1.526911336846 | 2.35658 × 10^{−8} | 1.54336 × 10^{−8}) |
Table 2. Approximate solutions of Example 4.1 by various methods at
Exact | 10th LRPS | OHA | MNN | RK-4 | |
---|---|---|---|---|---|
0.1 | 0.110295 | 0.110295 | 0.110328 | 0.110295 | 0.100000 |
0.2 | 0.241977 | 0.241977 | 0.242273 | 0.241976 | 0.219000 |
0.3 | 0.395105 | 0.395105 | 0.396175 | 0.395089 | 0.358004 |
0.4 | 0.567812 | 0.567812 | 0.570231 | 0.567660 | 0.516788 |
0.5 | 0.756014 | 0.756014 | 0.75955 | 0.755134 | 0.693439 |
0.6 | 0.953566 | 0.953566 | 0.955049 | 0.949964 | 0.884041 |
0.7 | 1.152949 | 1.152949 | 1.142444 | 1.141423 | 1.082696 |
0.8 | 1.346364 | 1.346358 | 1.300569 | 1.315723 | 1.282012 |
0.9 | 1.526911 | 1.526814 | 1.400444 | 1.456545 | 1.474059 |
Table 3. Numerical results of lower bound LRPS-solution of Example 4.2.
Exact solution | Approximate solution | Absolute error | Relative error | |
---|---|---|---|---|
0.1 | 0.34676399505 | 0.346763995059 | 1.11022 × 10^{−16} | 3.20167 × 10^{−16} |
0.2 | 0.51389727584 | 0.513897275841 | 1.11022 × 10^{−16} | 2.16040 × 10^{−16} |
0.3 | 0.69799087251 | 0.697990872514 | 2.22045 × 10^{−16} | 3.18120 × 10^{−16} |
0.4 | 0.89347216024 | 0.893472160244 | 3.33067 × 10^{−16} | 3.72778 × 10^{−16} |
0.5 | 1.09313499306 | 1.093134993060 | 2.22045 × 10^{−16} | 2.03126 × 10^{−16} |
0.6 | 1.28913717194 | 1.289137171947 | 2.22045 × 10^{−16} | 1.72243 × 10^{−16} |
0.7 | 1.47419437696 | 1.474194376963 | 1.05871 × 10^{−12} | 7.18161 × 10^{−13} |
0.8 | 1.64260136255 | 1.642601633577 | 1.02182 × 10^{−9} | 6.22073 × 10^{−10} |
0.9 | 1.79079791833 | 1.790798347506 | 4.29173 × 10^{−7} | 2.39655 × 10^{−7} |
Table 4. Numerical results of upper bound LRPS-solution of Example 4.2.
Exact solution | Approximate solution | Absolute error | Relative error | |
---|---|---|---|---|
0.1 | 0.34676399505 | 0.346763995059 | 1.66533 × 10^{−16} | 4.80250 × 10^{−16} |
0.2 | 0.51389727584 | 0.513897275841 | 1.11022 × 10^{−16} | 2.16040 × 10^{−16} |
0.3 | 0.69799087251 | 0.697990872514 | 2.22045 × 10^{−16} | 3.18120 × 10^{−16} |
0.4 | 0.89347216024 | 0.893472160244 | 3.33067 × 10^{−16} | 3.72778 × 10^{−16} |
0.5 | 1.09313499306 | 1.093134993060 | 2.22045 × 10^{−16} | 2.03126 × 10^{−16} |
0.6 | 1.28913717194 | 1.289137171947 | 2.22045 × 10^{−16} | 1.72243 × 10^{−16} |
0.7 | 1.47419437696 | 1.474194376963 | 1.05871 × 10^{−12} | 7.18161 × 10^{−13} |
0.8 | 1.64260136255 | 1.642601633577 | 1.02182 × 10^{−9} | 6.22073 × 10^{−10} |
0.9 | 1.79079791833 | 1.790798347506 | 4.29173 × 10^{−7} | 2.39655 × 10^{−7} |
Table 5. Numerical results of the LRPS solutions of Example 4.3 at
Exact solution | Approximate solution | Absolute error | Relative error | |
---|---|---|---|---|
0.16 | 0.158648504297 | 0.158648504297 | 5.55112 × 10^{−17} | 3.49900 × 10^{−16} |
0.32 | 0309506921213 | 0.309506921213 | 5.55112 × 10^{−17} | 1.79354 × 10^{−16} |
0.48 | 0.446243610249 | 0.446243610249 | 5.55112 × 10^{−17} | 1.24397 × 10^{−16} |
0.64 | 0.564899552846 | 0.564899552846 | 0.00000 | 0.00000 |
0.80 | 0.664036770268 | 0.664036770268 | 3.33067 × 10^{−16} | 5.01579 × 10^{−16} |
0.96 | 0.744276867362 | 0.744276867358 | 4.29645 × 10^{−12} | 5.77265 × 10^{−12} |
Table 6. Numerical comparison of Example 4.3 at
Exact | 10th-LRPS | OHAM | MNN | RK-4 | |
---|---|---|---|---|---|
0.1 | 0.099668 | 0.099668 | 0.099668 | 0.099668 | 0.100000 |
0.2 | 0.197375 | 0.197375 | 0.197376 | 0.197375 | 0.199000 |
0.3 | 0.291313 | 0.291313 | 0.291315 | 0.291313 | 0.295040 |
0.4 | 0.379949 | 0.379949 | 0.379949 | 0.379949 | 0.386335 |
0.5 | 0.462117 | 0.462121 | 0.462092 | 0.462121 | 0.471410 |
0.6 | 0.537050 | 0.537078 | 0.536910 | 0.537077 | 0.549187 |
0.7 | 0.604368 | 0.604514 | 0.603815 | 0.604513 | 0.619026 |
0.8 | 0.664037 | 0.664641 | 0.662245 | 0.664640 | 0.680707 |
0.9 | 0.716298 | 0.718392 | 0.711287 | 0.718390 | 0.734371 |
E-mail: moaathoqily@bau.edu.jo
E-mail: ajou44@bau.edu.jo
E-mail: zeyad1968@yahoo.com
E-mail: tareq.eriqat92@gmail.com
E-mail: mhm.smadi@yahoo.com
International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(1): 23-47
Published online March 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.1.23
Copyright © The Korean Institute of Intelligent Systems.
Moa’ath N. Oqielat^{1}, Ahmad El-Ajou^{1}, Zeyad Al-Zhour^{2 }, Tareq Eriqat^{1}, and Mohammed Al-Smadi^{3};^{4}
^{1}Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Salt, Jordan
^{2}Department of Basic Engineering Sciences, College of Engineering, Imam Abdulrahman Bin Faisal University, Dammam, Saudi Arabia
^{3}Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun, Jordan
^{4}Nonlinear Dynamics Research Centre (NDRC), Ajman University, Ajman, UAE
Correspondence to:Zeyad Al-Zhour (zeyad1968@yahoo.com)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this paper, we present in detail the power series solutions to fuzzy quadratic Riccati differential equations (QRDEs) along with a suitable fuzzy constant through an interactive derivative, more specifically, the Hukuhara-strongly generalized differentiability (H-SGD) based on our new technique. This technique is called the Laplace residual power series (LRPS) method, and it mainly depends on a new expansion and the combination of the Laplace transform technique with the residual power series method. To validate the accuracy of our proposed algorithm, numerous examples were examined numerically and graphically, and we compared the results of the optimal homotopy asymptotic (OHA), multiagent neural network (MNN), and fourth-order Runge-Kutta (RK-4) methods with the LRPS method at γ = 1.
Keywords: Fuzzy-valued function, Strongly generalized differentiability, Quadratic Riccati differential equation, Laplace residual power series method, Laplace and inverse transforms
Fuzzy set theory is an important topic in the study and modeling of many real-life problems related to numerous physical phenomena and dynamical processes, such as astronomy, quantum mechanics, chromodynamics, quantum optics, electronic mechanisms, electronic-controlled systems, the modeling of anomalous hydraulic diffusion systems, and population dynamics models [1–14]. Moreover, several problems in applied mathematics that are associated with biology, medicine, physics, and technology can be modeled using fuzzy differential equations [1–6,15–17].
Fuzzy set theory has been explored since the 1920s; however, the term fuzzy derivative was introduced by Chang and Zadeh [15]. Subsequently, Dubois and Prade [16] proposed the fuzzy differential calculus concept. Many researchers have recently presented other results and notations for fuzzy mapping [18–21]. Moreover, Bede and his colleagues [22,23] defined strongly generalized differentiability (SGD) as a fuzzy-valued function (FVF). In general, it is not straightforward to obtain the exact solution for these types of problems because of the difficulties involved; therefore, reliable numerical techniques are needed, including a reliable numerical algorithm for handling fuzzy integral equations of the second kind in the Hilbert spaces [4], a residual power series (RPS) for solving uncertain Riccati differential equations
[24], and a fuzzy Picard method for solving fuzzy quadratic Riccati and fuzzy Painlevl equations [25]. Other examples include a novel extended approach for obtaining numerical solutions to fuzzy conformable fractional differential equations [22], a reproducing kernel algorithm for solving second-order two-point fuzzy boundary value problems and second-order fuzzy Volterra integro-differential equations [3,18,26], a Runge-Kutta method applied to the SI epidemiological model [27], a homotopy analysis method (HAM) for solving fuzzy initial value problems (FIVPs) [28], and bidimensional IVP with interactive fuzzy numbers [14]. Additional solutions are a logistic model of population growth through an interactive derivative [29], an HIV viral dynamics model for individuals under antiretroviral treatment [17], a numerical approach in the Hilbert space to solve a fuzzy Atangana-Baleanu fractional hybrid system [30], a novel technique for solving fuzzy fractional delay differential equations [31], and a homotopy perturbation Sumudu transform method to find the analytical fuzzy solution of nonlinear fuzzy integro-differential equations [32].
The existence, uniqueness, and controllability of fuzzy solutions for such nonlinear fuzzy (fractional) differential equations have been studied and discussed in [33–39]. Kwun et al. [33–36] and Lee et al. [37] respectively studied the existence and uniqueness of fuzzy (periodic) solutions for the nonlinear fuzzy integro-differential equations on
Quadratic Riccati differential equations (QRDEs) form a precise nonlinear model for defining a certain type of physical scheme with applications in the diffusion process, optimal control, optical networks, and artificial intelligence, and usually have the following general form [40–43] of
subject to
Many numerical and analytical techniques have been employed to obtain QRDEs solutions, including the HAM [40], homotopy perpetuation method (HPM) [41], differential transformation method [42], and Adomian decomposition method [43].
Several authors [24,26,28,40–45] have reformulated the QRDE as a fuzzy differential equation along with suitable fuzzy constants under the SGD concept, the fuzzy form of which is as follows [24]:
subject to the fuzzy initial condition (FIC):
where
The main objective of the research described in this study is to extend the application of the LRPS method and provide a power series solution to the FIVP in
The LRPS method is a new method that combines the Laplace transform technique with the RPS method, which is used to solve the integral and differential equations [7–13]. The Laplace transform method involves transforming the equation from the space within it into a new space, presenting its solution in the new space, and then returning the solution to the original space and thus obtaining the solution to the equation. However, it is sometimes difficult to offer a solution to an equation in a new space. The RPS method assumes a power series solution to the equation and provides an easy technique for determining the series coefficients. The LRPS method uses the two methods together to provide a series solution to the equation in the new space and uses the principle of the RPS method to determine the series coefficients. However, through a new approach, it is easier and faster than the user in the original technique. In fact, our proposed method is an analytical and efficient method that can be easily applied to solve different types of fuzzy problems.
Two fuzzy numbers are considered non-interactive when the joint possibility distribution involving both is given by the minimum t-norm. Thus, any operation between two fuzzy numbers, aside from the one obtained by the minimum t-norm, must be considered to be interactive. Following this reasoning, when we use the Hukuhara difference (H-difference) for two fuzzy numbers, we assume that they are interactive. Thus, all derivatives obtained by a fuzzy arithmetic operation that differs from the standard operation must be considered interactive. In this study, we find analytical and numerical solutions for FIVPs along with a suitable fuzzy constant through an interactive derivative, more specifically, the Hukuhara-strongly generalized differentiability (H-SGD) based on our new LRPS technique. The solution to the FIVP obtained here using H-SGD is more suitable than the classical derivative because the latter uses an interactive derivative obtained by the H-difference, whereas the standard sum is used to compute from the non-interactive operation, that is, the sum operation is based on the minimum t-norm. This is because the arithmetical operations compatible with the H-difference are not known. Thus, it makes no sense to combine interactive and non-interactive H-SGD. Furthermore, with the H-SGD theory, it is possible to extend and generalize our analysis and results in this study to all fuzzy interactive arithmetic operations, and not only to the H-difference operation in future studies. In addition, in the present paper, we introduce a new expansion to Taylor’s formula, which is required for creating the LRPS method and formulating the FIVP. Moreover, several examples are solved and examined numerically and graphically using this new technique, and the results of the optimal homotopy asymptotic (OHA), multiagent neural network (MNN), and fourth-order Runge-Kutta (RK-4) methods are compared with those of the LRPS method at
This paper consists of five sections. In Section 2, we study some basic concepts of fuzzy theory, establish a new expansion to Taylor’s formula, and formulate the FIVP. In Section 3, the LRPS algorithm is created and successfully tested to obtain analytical and numerical solutions of the nonlinear FIVP. In Section 4, we test the accuracy of our proposed algorithm (LRPS) by solving three important examples numerically and graphically, and comparing the results of the OHA, MNN, and RK-4 methods with the LRPS method at
This section reviews some theories and concepts necessary for our study and is divided into three subsections: the first relates to fuzzy calculus, the second relates to a new expansion needed to construct the LRPS solution, and we formulate the FIVP in third.
In this subsection, we study the most important definitions and notations related to fuzzy analysis and fuzzy calculus [3,4,12, 15,16,18–23,48–63] used throughout this paper. The provided ℝ_{ℱ} denotes the set of all the fuzzy numbers.
Letting
(i)
(ii)
(iii)
(iv)
Let
Therefore, if
in addition, if
In addition, if
Note that a fuzzy set
Some conditions must be satisfied by two functions:
Suppose that
(i)
(ii)
(iii)
(iv)
Subsequently,
Let
(i) standard sum,
(ii) standard differences,
(iii) scalar multiplication,
(iv) multiplication,
(v) quotient,
(vi) equality, the two fuzzy numbers
Letting
This definition was modified by Bede and Stefanini [23] to the generalized H-difference (gH-difference), as follows:
with
Very recently, Wasques et al. [17,27,29,59] generalized the Hukuhara and H-differences and used interactivities for all arithmetic operations and not only for the difference operation. We summarize this work as follows.
In general, a fuzzy subset
and
Note that the fuzzy subset
Let
for all
for all
Here,
for all
Otherwise, the fuzzy numbers are said to be
The Pompeiu-Hausdorff distance
where
and gives the Hausdorff distance between [
It is provided that (ℝ_{ℱ},
(i)
(ii)
(iii)
Note that the metric space
Let [
which is called the
The complete metric structure on ℝ_{ℱ} is given by the Hausdorff distance mapping
Let
Let
Let
(i) the H-differences
or all
(ii) The H-differences
for all
(i) If
(ii) The limits in
Let
(i) if
and (ii) if
Let
A strongly measurable and limited integrable fuzzy function is known as an integrable function. The fuzzy integral of an FVF
for all
Similarly, the fuzzy integral of an FVF
for all
Let
(i) If
and
(ii) If
and
In this subsection, we present a new form of Taylor’s formula, which we will mainly use in our proposed method for solving FIVPs.
Let
Then
We assume that
For the other aspect as well, multiplying
It is clear that
If we multiply
Now, we can see the patterns and find the general formula for
The inverse Laplace transform of the expansion in Theorem 2.5 has the following form:
which is the classical Taylor’s formula.
The following theorem explains and determines the conditions for convergence of the new form of Taylor’s formula, which is presented in Theorem 2.5, as mentioned above.
Let
First, suppose that ℒ[
From the definition of the reminder
It follows from
Thus, by reformulating
In this subsection, we state the conditions and assumptions required to formulate the FIVP, as in
with the FIC
Let
Finally, to construct the LRPS solution for the target equation in a
If
with FICs,
We then have the following procedures:
(i) Construct the LRPS solution for the system in
(ii) Ensure that the solutions [
(iii) Then, [
If
with FICs,
In addition, we have the following procedures:
(i) Construct the LRPS solution for systems
(ii) Ensure that the solutions [
(iii) Then, [
In this section, we present the (1)-solution for the FIVP, as in
Using the properties of the Laplace transform and the initial conditions in
where
Next, the LRPS technique represents the solution as an infinite series, and the series solution to
Because
and
this leads to
Therefore, the expansions in
Consequently, the
Prior to applying the LRPS technique to determine the values of the coefficients
and the
It is clear that
Because
for
For the other aspects as well, to determine the value of the first unknown coefficients
Now, multiplying both sides of
Computing the limits of both sides of
Considering
Similarly, to determine the values of the second unknown coefficients
By multiplying both sides of
Using the fact that
By solving the resultant algebraic system in
For
Therefore, the third LRPS estimation can also be given.
By continuing the same process until random coefficients order
subsequently
Finally, the exact or approximate series solution obtained must be transformed into the original space by using the inverse Laplace transform to obtain the series solution to the original FIVP, as in
To validate the accuracy of our proposed algorithm, numerous examples are examined numerically in this section. For calculations and results, Maple 2018 is used to perform the computations.
Consider the following FIVP:
with FIC,
For
Using Definition 2.10, the FIVP, as in
subject to
Applying the LRPS method as below, taking the Laplace transform of both sides of
and
where
Now, we assume that an infinite series solution of the algebraic system in
In addition, the
Depending on the initial data,
The
To determine the first unknown coefficients,
Now, by remultiplying both sides of
Using the fact that
Similarly, to determine the values of the second unknown coefficients
Now, re-multiplying
According to the fact:
Solving
Hence, the second-approximate LRPS solution to
The
and
The LRPS solution to
By applying the inverse Laplace transform to
As a special case, when
Table 1 shows the numerical results of the exact and approximate solutions using the LRPS method for Example 4.1 at
Table 2 shows a numerical comparison between the tenth-LRPS solution and the OHA method [2], the MNN method [3], and the RK-4 method [27]. We concluded that the results obtained using the LRPS method were similar to those obtained using other methods.
Figure 1 shows the upper and lower bounds of the triangular fuzzy LRPS solution at
Consider the following quadratic FIVP:
with FIC,
For
According to Definition 2.10, the FIVPs, as in
subject to
According to the procedure of the LRPS algorithm presented in Section 3, the Laplace transform of
where
Depending on the initial conditions in
and the
By utilizing lim
and
We can express the LRPS solution of
Applying the inverse Laplace transform to
As a special case, when
Tables 3 and 4 present the numerical results of the fuzzy solution upper and lower bounds of the system in
Figure 3 shows the upper and lower bounds of the triangular fuzzy LRPS solution at
Consider the following FIVP:
with FIC,
For
The FIVP, as in
subject to
Similar to the steps used in previous examples, the Laplace transform of
According to the initial data in
The
By utilizing the following facts:
and
We can express the LRPS solution of
Applying the inverse Laplace transform to
As a special case, when
Table 5 presents the error analysis of the proposed algorithm for the system in
Figure 5 shows the upper and lower bounds of the triangular fuzzy LRPS solution at
Many numerical and analytical methods have been applied to solve fuzzy differential equations, some of which have advantages over others. Some of them are accurate and effective; however, they require mathematical operations that can be difficult, long, or sometimes failed. Others are simple and fast; however, they may not provide precise solutions. Our new method, called LRPS, is characterized by its accuracy, speed, and simplicity in finding exact and accurate approximate solutions of linear and nonlinear fuzzy differential equations. In this study, we succeeded in providing accurate approximate and sometimes exact solutions to FIVPs using the newly proposed technique. The proposed expansion of our study enabled us to obtain a serial solution for equations in the Laplace transform space. The LRPS method provides an easy and fast technique for determining the proposed series coefficients as a solution to the equation. Unlike the traditional RPS method, which specifies series coefficients, the derivative must be calculated each time, whereas LRPS only requires the concept of the limit at infinity in determining series coefficients, which distinguishes this method. It is worth noting that the LRPS method can be applied to solve other types of fuzzy ordinary and partial differential equations for an integer or fractional order, such as fuzzy Poisson, KdV, Schrödinger, and apple equations. Furthermore, based on the H-SGD theory, it is possible to extend and generalize our analysis and results to all fuzzy interactive arithmetic operations mentioned in Section 2 in future studies.
Triangular fuzzy solution plots for the eighth-LRPS solutions of Example 4.1. Solution plots for (a)
The 3-dim plot for Example 4.1. Blue and yellow are the lower and upper bounds, respectively, of the tenth-LRPS fuzzy solution.
Triangular fuzzy solution plots for the eighth-LRPS solutions of Example 4.2. Solution plots for (a)
3-dim plot for Example 4.2. Blue and yellow are the lower and upper bounds, respectively, of the tenth-LRPS fuzzy solution.
Triangular fuzzy solution plots for the eighth-LRPS solutions of Example 4.3. Solution plots for (a)
3-dim plot for Example 4.3. Blue and yellow are lower and upper bounds, respectively, of tenth-LRPS fuzzy solution.
Table 1 . Numerical results of the LRPS solutions for Example 4.1 at
Exact solution | Approximate solution | Absolute error | Relative error | |
---|---|---|---|---|
0.1 | 0.110295196916 | 0.110295196917 | 1.52656 × 10^{−16} | 1.38406 × 10^{−15} |
0.2 | 0.241976799621 | 0.241976799621 | 5.55112 × 10^{−17} | 2.29407 × 10^{−16} |
0.3 | 0.395104848660 | 0.395104848660 | 2.77556 × 10^{−16} | 7.02486 × 10^{−16} |
0.4 | 0.567812166293 | 0.567812166293 | 3.33067 × 10^{−16} | 5.86579 × 10^{−16} |
0.5 | 0.756014393431 | 0.756014393431 | 3.33067 × 10^{−16} | 4.40556 × 10^{−11} |
0.6 | 0.953566216472 | 0.953566216472 | 1.11022 × 10^{−16} | 1.16429 × 10^{−16} |
0.7 | 1.152948966979 | 1.152948966980 | 5.32907 × 10^{−14} | 4.62212 × 10^{−14} |
0.8 | 1.346363655368 | 1.346363655422 | 5.39830 × 10^{−11} | 4.00954 × 10^{−11} |
0.9 | 1.526911313280 | 1.526911336846 | 2.35658 × 10^{−8} | 1.54336 × 10^{−8}) |
Table 2 . Approximate solutions of Example 4.1 by various methods at
Exact | 10th LRPS | OHA | MNN | RK-4 | |
---|---|---|---|---|---|
0.1 | 0.110295 | 0.110295 | 0.110328 | 0.110295 | 0.100000 |
0.2 | 0.241977 | 0.241977 | 0.242273 | 0.241976 | 0.219000 |
0.3 | 0.395105 | 0.395105 | 0.396175 | 0.395089 | 0.358004 |
0.4 | 0.567812 | 0.567812 | 0.570231 | 0.567660 | 0.516788 |
0.5 | 0.756014 | 0.756014 | 0.75955 | 0.755134 | 0.693439 |
0.6 | 0.953566 | 0.953566 | 0.955049 | 0.949964 | 0.884041 |
0.7 | 1.152949 | 1.152949 | 1.142444 | 1.141423 | 1.082696 |
0.8 | 1.346364 | 1.346358 | 1.300569 | 1.315723 | 1.282012 |
0.9 | 1.526911 | 1.526814 | 1.400444 | 1.456545 | 1.474059 |
Table 3 . Numerical results of lower bound LRPS-solution of Example 4.2.
Exact solution | Approximate solution | Absolute error | Relative error | |
---|---|---|---|---|
0.1 | 0.34676399505 | 0.346763995059 | 1.11022 × 10^{−16} | 3.20167 × 10^{−16} |
0.2 | 0.51389727584 | 0.513897275841 | 1.11022 × 10^{−16} | 2.16040 × 10^{−16} |
0.3 | 0.69799087251 | 0.697990872514 | 2.22045 × 10^{−16} | 3.18120 × 10^{−16} |
0.4 | 0.89347216024 | 0.893472160244 | 3.33067 × 10^{−16} | 3.72778 × 10^{−16} |
0.5 | 1.09313499306 | 1.093134993060 | 2.22045 × 10^{−16} | 2.03126 × 10^{−16} |
0.6 | 1.28913717194 | 1.289137171947 | 2.22045 × 10^{−16} | 1.72243 × 10^{−16} |
0.7 | 1.47419437696 | 1.474194376963 | 1.05871 × 10^{−12} | 7.18161 × 10^{−13} |
0.8 | 1.64260136255 | 1.642601633577 | 1.02182 × 10^{−9} | 6.22073 × 10^{−10} |
0.9 | 1.79079791833 | 1.790798347506 | 4.29173 × 10^{−7} | 2.39655 × 10^{−7} |
Table 4 . Numerical results of upper bound LRPS-solution of Example 4.2.
Exact solution | Approximate solution | Absolute error | Relative error | |
---|---|---|---|---|
0.1 | 0.34676399505 | 0.346763995059 | 1.66533 × 10^{−16} | 4.80250 × 10^{−16} |
0.2 | 0.51389727584 | 0.513897275841 | 1.11022 × 10^{−16} | 2.16040 × 10^{−16} |
0.3 | 0.69799087251 | 0.697990872514 | 2.22045 × 10^{−16} | 3.18120 × 10^{−16} |
0.4 | 0.89347216024 | 0.893472160244 | 3.33067 × 10^{−16} | 3.72778 × 10^{−16} |
0.5 | 1.09313499306 | 1.093134993060 | 2.22045 × 10^{−16} | 2.03126 × 10^{−16} |
0.6 | 1.28913717194 | 1.289137171947 | 2.22045 × 10^{−16} | 1.72243 × 10^{−16} |
0.7 | 1.47419437696 | 1.474194376963 | 1.05871 × 10^{−12} | 7.18161 × 10^{−13} |
0.8 | 1.64260136255 | 1.642601633577 | 1.02182 × 10^{−9} | 6.22073 × 10^{−10} |
0.9 | 1.79079791833 | 1.790798347506 | 4.29173 × 10^{−7} | 2.39655 × 10^{−7} |
Table 5 . Numerical results of the LRPS solutions of Example 4.3 at
Exact solution | Approximate solution | Absolute error | Relative error | |
---|---|---|---|---|
0.16 | 0.158648504297 | 0.158648504297 | 5.55112 × 10^{−17} | 3.49900 × 10^{−16} |
0.32 | 0309506921213 | 0.309506921213 | 5.55112 × 10^{−17} | 1.79354 × 10^{−16} |
0.48 | 0.446243610249 | 0.446243610249 | 5.55112 × 10^{−17} | 1.24397 × 10^{−16} |
0.64 | 0.564899552846 | 0.564899552846 | 0.00000 | 0.00000 |
0.80 | 0.664036770268 | 0.664036770268 | 3.33067 × 10^{−16} | 5.01579 × 10^{−16} |
0.96 | 0.744276867362 | 0.744276867358 | 4.29645 × 10^{−12} | 5.77265 × 10^{−12} |
Table 6 . Numerical comparison of Example 4.3 at
Exact | 10th-LRPS | OHAM | MNN | RK-4 | |
---|---|---|---|---|---|
0.1 | 0.099668 | 0.099668 | 0.099668 | 0.099668 | 0.100000 |
0.2 | 0.197375 | 0.197375 | 0.197376 | 0.197375 | 0.199000 |
0.3 | 0.291313 | 0.291313 | 0.291315 | 0.291313 | 0.295040 |
0.4 | 0.379949 | 0.379949 | 0.379949 | 0.379949 | 0.386335 |
0.5 | 0.462117 | 0.462121 | 0.462092 | 0.462121 | 0.471410 |
0.6 | 0.537050 | 0.537078 | 0.536910 | 0.537077 | 0.549187 |
0.7 | 0.604368 | 0.604514 | 0.603815 | 0.604513 | 0.619026 |
0.8 | 0.664037 | 0.664641 | 0.662245 | 0.664640 | 0.680707 |
0.9 | 0.716298 | 0.718392 | 0.711287 | 0.718390 | 0.734371 |
Triangular fuzzy solution plots for the eighth-LRPS solutions of Example 4.1. Solution plots for (a)
The 3-dim plot for Example 4.1. Blue and yellow are the lower and upper bounds, respectively, of the tenth-LRPS fuzzy solution.
|@|~(^,^)~|@|Triangular fuzzy solution plots for the eighth-LRPS solutions of Example 4.2. Solution plots for (a)
3-dim plot for Example 4.2. Blue and yellow are the lower and upper bounds, respectively, of the tenth-LRPS fuzzy solution.
|@|~(^,^)~|@|Triangular fuzzy solution plots for the eighth-LRPS solutions of Example 4.3. Solution plots for (a)
3-dim plot for Example 4.3. Blue and yellow are lower and upper bounds, respectively, of tenth-LRPS fuzzy solution.