1. Introduction
Physical models of real-world phenomena typically have considerable uncertainty and ambiguity, which can be attributed to a variety of causes. For the purpose of examining these issues, Zadeh introduced the fuzzy set theory in 1965 [
1]. There are many disciplines in fuzzy sets, such as fuzzy topology, fuzzy arithmetic, fuzzy algebraic structures, fuzzy differential calculus, fuzzy geometry, fuzzy relational calculus, fuzzy databases, and fuzzy decision-making [
2].
Many researchers attempted to apply the fuzzy set theory in the context of decision-making. A method of
X fuzzy mathematics, also known as the analytical hierarchy process (AHP), was developed by Wang and Khadidos [
3] and was applied in the basketball scoring strategy. The results showed that the proposed method is conceivable and credible in the scoring and analysis of basketball strategies. In [
4], the authors used the fuzzy mathematics evaluation method to measure basketball players’ skill levels. The suggested approach can be used to assess basketball skill development and instruction, and it also serves as a general framework for enhancing players’ basketball skills. In 2021, a study was conducted on the usage of the eye movement method with the help of fractional order fuzzy differential equation (FDE) to maximise the effect of artistic images in artistic image migration [
5]. The impact of the fuzzy mathematics simulation model on the growth of the martial arts industry in Wushu has been studied by Sun, Lv, Khadidos, and Kharabsheh [
6]. According to the report, the martial arts market is still in its early stages of development, with a significant market potential and high benefits expectations. Based on the mathematical model of the fuzzy comprehensive evaluation of the differential equations, a study on the audio-visual family restoration of children with mental problems was undertaken [
7].
In calculus, the study of the FDE is a topic that researchers are interested in these days [
8]. For example, it produced a mathematical model for issues pertaining to hydraulic systems and population dynamics [
9,
10]. In 1972, Chang and Zadeh [
11] pioneered the idea of fuzzy derivatives. The extension principle was then defined and developed by Dubois and Prade [
12]. Next, Puri and Ralescu [
13] applied the Hukuhara differentiability (H-differentiability) to the concept of fuzzy functions. Following that, Kaleva [
14] and Seikkala [
15] established some theorems for FDE using H-differentiability. Moreover, Friedman, Ma, and Kandel [
16] presented the Friedman–Ming–Kandel derivative, a new approach to fuzzy derivatives, and compared it to previous approaches. Other than that, a historical survey of the FDE is carried out by Mazandarani and Xiu [
17].
Research is now being carried out to find a solution to the fuzzy initial value problem (FIVP) and fuzzy boundary value problem (FBVP), both of which can be utilized to find solutions to real-world issues. However, due to the difficulty in finding their analytical solution, not all FIVP and FBVP could be solved precisely [
18]. Therefore, reliable and efficient numerical techniques may be needed to handle the corresponding FDE.
The FDE has been studied and solved numerically using a variety of methods. Ma, Friedman, and Kandel [
19] developed the Euler method to determine the pointwise convergence of approximate solutions to exact solutions. Meanwhile, Tapaswini, and Chakraverty [
20] proposed a novel method, the improved Euler method (IEM). The authors considered two methods to solve both FIVP and FBVP in this study: max-min IEM and average IEM. The IEM performs better than the established approach and is discovered to be in good agreement. Moreover, Georgieva [
21] presented the double fuzzy Sumudu transform to solve the partial Volterra fuzzy integro-differential equation with the convolution kernel under H-differentiability. In [
22], You, Cheng, and Ma studied the stability of the fuzzy Euler method related to the FDE. The stabilities involved are the asymptotical stability, the mean square (MS) stability, the exponential stability and the
A stability. The numerical examples were identified by comparing the effects of three fuzzy Euler schemes on the asymptotical and MS stabilities and examining the impact of
in a semi-implicit fuzzy Euler scheme on the MS stability.
Over the last few years, the Runge–Kutta (RK) method has received a lot of attention from researchers, and many researchers have proposed some RK modifications. By presenting new third and fourth-order numerical methods, Goeken and Johnson [
23] proposed a higher derivative approximation of RK to solve the FDE. The RK of third and fourth-order methods were then introduced by Wu [
24] by reducing the function of evaluations. Alternatively, Jayakumar, Maheskumar, and Kanagarajan [
25] are interested in using the RK of the fifth-order method (RK5) to solve the FDE, and this method is compared with the IEM. It is evident that the RK5 performed better than the IEM because it has ordered five while the IEM only has ordered two. In addition, the fuzzy reduced Runge–Kutta of the fourth-order method (FRRK4) has been proposed to solve the hybrid FDE [
26]. The hybrid FIVP is solved using the FRRK4 and, compared with the analytical solution, the RK of the fourth-order method (RK4), and the Euler method. The findings indicated that the FRRK4 method’s solution is quite close to an analytical solution. Moreover, the proposed method has undergone fewer function evaluations, demonstrating that it is more effective than the traditional RK method. Apart from that, Ramli, Ahmad, Din, and Salleh [
27] proposed a developed Runge–Kutta of the fourth-order method (DRK4) to improve the solution of FDE. The authors presented the new approach and compared the results to the RK4. The discovery demonstrates that the proposed method is close to the analytical solution.
Likewise, Kanagarajan and Suresh [
28] proposed the fuzzy Runge–Kutta of the fourth-order method (FRK4) to solve the FDE under a generalized differentiability. The proposed method provides a superior solution when compared to the Euler method. Ahmadian, Salahshour, Chan, and Baleanu [
29] introduced an extended Runge–Kutta of the fourth-order method (ERK4), which the authors used to solve the autonomous and non-autonomous fuzzy systems. The method is compared to the RK4, showing that the ERK4 produces better results because it requires only three function evaluations, whereas the RK4 requires four function evaluations. In 2019, Rajkumar and Rubanraj [
30] introduced a Runge–Kutta of the seventh-order method (RK7) to solve the FIVP. When this proposed method was compared to Euler’s and the RK4 methods, it was discovered that the RK7 provided a better solution. Majid, Rabiei, Hamid, and Ismail [
31] proposed a new method, namely the fuzzy general linear method (FGL), to solve the fuzzy Volterra-integro differential equation (FVIDE). This method is combined with Simpson’s method and Lagrange’s interpolation polynomial, and the numerical examples are performed. To test the efficiency of the proposed method, the solutions of the FVIDE using the FGL are compared with the Homotopy perturbation method and the Runge–Kutta of the third-order method (RK3). It demonstrated that the proposed method provides an accurate result with an analytical solution.
Based on the earlier investigations, the proposed numerical methods to solve the FDE were discovered to be extensions of the traditional numerical methods. The researchers compared them to other numerical methods to demonstrate their efficacy. According to our observations, more researchers paid attention to the non-autonomous FDE than the autonomous FDE. In light of the current trend in this area, we concentrated on the creation of a novel numerical technique termed fuzzy Runge–Kutta Cash–Karp of the fourth-order method (FRKCK4) to solve both autonomous and non-autonomous FDEs. The major goal is to provide more precise instructions for creating sophisticated algorithms. To the best of our knowledge, this is the first time that the traditional Runge–Kutta Cash–Karp of the fourth-order method (RKCK4) is presented in a fuzzy setting. The goal of this paper is to solve the autonomous and non-autonomous FDEs using the FRKCK4. This paper is set up as follows:
Section 2 goes over the definitions, and
Section 3 goes over the methods, which consist of the characterization theorem for the FDEs and the traditional RKCK4. Furthermore, the proposed model and numerical simulations are discussed in
Section 4 and
Section 5. Tables and graphs will be used to present a brief summary of our findings. Meanwhile,
Section 6 presents the advantages of the proposed model. We present our findings’ conclusions at the end of this paper, and future work is suggested.
5. Numerical Simulations
This section provides numerical simulations to demonstrate the capability of the proposed method. The proposed method is then compared to the analytical solutions, as well as the FRK4 for the (i) and (ii)–differentiability. In order to calculate the error, the absolute values of the analytical and approximate solutions are subtracted. The following sections discuss the examples of autonomous and non-autonomous fuzzy differential equations. The idea of Example 1 below is based on Example 5.1 in [
25]. Some modification is made by changing the value of
and
Example 1. Let the autonomous FDE be given as follows:
Using the (i)–differentiability, we translate Equation (14) into the ODE system.
In order to solve Equation (15), the following is proposed:
The analytical solution for the (i)–differentiability is:
Using the (ii)–differentiability, we translate Equation (14) into the ODE system.
In order to solve Equation (16), the following is proposed:
The analytical solution for the (ii)–differentiability is:
By implementing the proposed method, we tested Example 1 using the step size
,
and the result was adjusted to 12 decimal places. The comparisons between the analytical and approximate solutions at
for the (i)–differentiability are quantified and tabulated in
Table 1, while the errors
are listed in
Table 2. For the (ii)–differentiability, the comparisons between the analytical and approximate solutions at
are tabulated in
Table 3, while the errors are listed in
Table 4.
Table 1 compared the analytical and numerical solutions for the (i)–differentiability produced by the FRKCK4 and FRK4 at time
. According to the table, the solution generated by the FRKCK4 is closer to the analytical solution than FRK4. This is obvious from error analysis, as shown in
Table 2.
Figure 1a–c presents the FRKCK4, FRK4, and analytical solutions, respectively, at the time intervals
. The figures make it quite evident that the fuzzy solutions are expanding as
increases. This is true under the property of the (i)–differentiability.
The analytical and numerical solutions for the (ii)–differentiability generated by the FRKCK4 and FRK4 at time
are presented in
Table 3. From the table, it indicates that the solution produced by the FRKCK4 is closer to the analytical solution than the FRK4. This can be seen clearly from the error analysis, as demonstrated in
Table 4. The FRKCK4, FRK4, and the analytical solutions at the time intervals
are shown in
Figure 2a–c, respectively. The figures clearly demonstrate that as
increases, the fuzzy solutions become contracted. This is true under the property of the (ii)–differentiability.
In the following example, we adopted Example 4.2 from [
29].
Example 2 [
29]. We consider the following non-autonomous FDE by implementing our proposed model:
Equation (17) above refers to an electrical circuit with an alternating current (AC) source that contains a resistor (R) and inductor (L). Let
, and
Therefore, Equation (17) may be written as follows:
Using the (i)–differentiability, we translate Equation (18) into the ODE system.
In order to solve Equation (19), the following is proposed:
The analytical solution for the (i)–differentiability is:
Using the (ii)–differentiability, we translate Equation (18) into the ODE system below
In order to solve Equation (20), the following is proposed:
The analytical solution for the (ii)–differentiability is:
By implementing the proposed method, we tested Example 2 using the step size
, and the result was adjusted to 12 decimal places. The comparisons between the analytical and approximate solutions at
for the (i)–differentiability is quantified and tabulated in
Table 5, while the errors are listed in
Table 6. For the (ii)–differentiability, the comparisons between the analytical and approximate solutions at
is tabulated in
Table 7, while the errors are listed in
Table 8.
A comparison of the analytical and numerical solutions for the (i)–differentiability obtained by the FRKCK4 and FRK4 at time
is shown in
Table 5. The table shows that the solution produced by the FRKCK4 is closer to the analytical solution than the FRK4. This is clear through error analysis, which is presented in
Table 6. The solutions of the FRKCK4, FRK4, and the analytical solutions at time intervals
are shown in
Figure 3a–c. The figures clearly show that the fuzzy solutions are expanding as
increases. This is true under the property of the (i)–differentiability.
Table 7 displays a comparison between the analytical and numerical solutions for the (ii)–differentiability obtained by the FRKCK4 and FRK4 at time
. The table demonstrates that the FRKCK4’s solution, as opposed to the FRK4, is closer to the analytical solution. The error analysis presented in
Table 8 makes this very evident.
Figure 4a–c displays the solutions of the FRKCK4 and FRK4 and the analytical solutions at time intervals
. It is clear from the figures that as
increases, the fuzzy solutions are contracting. This is true under the property of the (ii)–differentiability.
7. Conclusions
This study uses the characterization theorem to propose a traditional RKCK4 in a fuzzy setting. Under the interpretation of the (i) and (ii)–differentiability, the step-by-step procedures have been provided to solve the autonomous and non-autonomous fuzzy differential equations. Additionally, the theoretical convergence analysis is presented and proved mathematically.
To demonstrate the viability of the proposed method, numerical simulations of the autonomous and non-autonomous FDEs are then examined. The analytical solutions and the FRK4 were used to compare the results of the FRKCK4. The error analysis revealed that the FRKCK4 generates results that are more accurate than the FRK4. This is because the error produced by the FRKCK4 is smaller than the FRK4, when compared to the analytical solution.
Graphs are used to depict the study’s findings and demonstrate how the solutions to both FDEs behaved in the time domain . It is discovered that both solutions are expanding as increases under the (i)-differentiability. However, this is not the case under the (ii)-differentiability, which is contracting. This is consistent under the (i) and (ii)–differentiability interpretations.
In general, the FRKCK4 can solve FDEs with a good agreement. Moreover, for a better approximation, it can be further extended to higher-order degrees.