The notions of fuzzy differential equations and fuzzy integral equations with fuzzy control have attracted researchers. Different definitions of fuzzy derivatives and fuzzy integrals have been established and extended to fuzzy calculus. For the existence and uniqueness of the solution of these equations, sufficient conditions were provided, and different numerical algorithms were designed for the approximate solution. The fuzzy integral equation theory is new, and plays a key role in various fields of engineering, applied mathematics, physics, biological sciences, optimal control theory, mathematical economics, etc. Modeling any physical problem by using integral equations with exact parameters is not an easy task; in real-world problems, it is almost impossible. Thus, it is nearly impossible to deal with exact parameters in real-life problems. Therefore, many researchers have worked on these models to investigate the solutions of fuzzy integral equations. To overcome this problem, a reliable approach is to use the fuzzy concept. Elementary idea of “fuzzy set” by Zadeh [1, 2] in his publications of fuzzy set theory. In 1982, Dubois and Prade [3] presented arithmetic operations on fuzzy calculus. The study of fuzzy integral and differential equations has been rapidly advancing in recent years [4–8]. In recent years, research on fuzzy differential equations and fuzzy integral equations from both theoretical and numerical points of view have been developed. Analytical solutions of some fuzzy integro-differential equations (FIDEs) using the analytical method, Laplace homotopy perturbation method (HPM), were produced [9]. Hamoud and Ghadle [10] provided a solution for fuzzy Volterra integro-differential equation (IDE) via modified Adomian decomposition method. Hooshangian [11] proposed a solution for the fuzzy Volterra IDE of the n-th order with a nonlinear fuzzy kernel and converted it into a nonlinear integral equation by using the generalized Hakuhara derivative. Therefore, finding an efficient and accurate algorithm for investigating fuzzy integral equation has recently garnered significant attention among researchers. The existence and uniqueness of the solution of the nonlinear FIDE and impulsive semi-linear FIDEs with nonlocal conditions and forcing the term with memory is well addressed in [12, 13]. Recently, Padmapriya et al. [14] investigated the numerical solutions of fuzzy fractional delay differential equations using the proposed novel technique. The HPM was first proposed by He [15, 16], who also approximated the answers for several differential equations. This method is a merge of homotopy defined in topology with the traditional perturbation method. The author has practiced HPM on a wide range of applications in computing functional equations [17]. The Sumudu transform was first introduced byWatagula in [18]. The homotopy perturbation Sumudu transform method (HPSTM) is the coupling of the Sumudu transform and HPM. The proposed method has advantages such as a simple structure, ease of programming, and high accuracy. Motivated by the aforementioned work, in this study, we use a HPSTM to solve the nonlinear FIDE. The remainder of this paper is organized as follows. In Section 2, we introduce some preliminaries and basic definitions. In Section 3, we present the proposed algorithm. To demonstrate the effectiveness of the proposed method, we present some numerical results on several tests in Section 4. Finally, some concluding remarks are presented in Section 5.
The Sumudu transform of any set B is defined as
The Sumudu transform method [18] is written as follows:
The general form of the FIDE is as follows:
y(0, β) = (a_{0}, b_{0}), y′(0, β) = (a_{1}, b_{1}), y″(0, β) = (a_{2}, b_{2})′, ..., y^{m}^{−1}(0, β) = (a_{m}_{−1}, b_{m}_{−1}), where (a_{j}, b_{j}) with j = 0, 1, 2, 3, ...m − 1 are known to remain constant. y^{m}(t, β) is the m^{th} order derivative of the fuzzy function and is already given, β is a fuzzy parameter with values between [0, 1](0 ≤ β ≤ 1), λ is known to be a constant parameter, and K(t, s) is the kernel of this FIDE and depends on variable t and s. a(t) and b(t) are known to be the limits of this FIDE. If these limits are constant, the IDE will be called a fuzzy-Fredholm IDE, and if one of these limits is variable, then this equation will be said fuzzy-Volterra IDE:
In parametric form
In addition, 0≤β≤1 and F(y(s, β))=(F(y(s, β)), F̄(y(s, β))), g(t, β) = (g(t, β), ḡ(t, β)), with kernel
To illustrate the basic representation of the HPSTM on the FIDE in general form, the basic fuzzy condition is defined in
And the initial approximation is reserved as
Substituting
Applying Sumudu transform on both sides of
Applying the differential property of Sumudu transform on both sides of
The inverse transform on each side of
Assuming the solution of
By replacing the solution of
and so on. Finally, the solution of FIDE-2 is given as
Consider the nonlinear fuzzy Fredholm IDE of the second kind as [19]
with initial condition y(0, β) = (0, 0), λ = 1, 0 ≤ s ≤ 1, 0 ≤ β ≤ 1,
The exact solution of
The homotopy is
According to the procedure described above, we have
Consequently, we have
and so on. The series solution (Figure 1) is
Consider the nonlinear fuzzy Volterra IDE of the second kind as [9]
where y(0, β) = (0, 0), 0 ≤ s ≤ 1, 0 ≤ α ≤ 1, λ = 1, K(t, s) = 1, g(t, β) = (β, (7 − β)). According to the procedure described above, we have
and so on. The solution (Figure 2) is given as
Consider the nonlinear fuzzy Volterra IDE of 2nd second-order as [11]
with initial values y(0, β) = (β − 1, 1 − β), y′(0, β) = (β, 2 − β), where g(t, β) = (−β^{2}t^{3}, −(2 − β)^{2}t^{3}), 0 ≤ s ≤ t, 0 ≤ β ≤ t, λ = 1, K(t, s) = 1.
The exact solution is given by y(0, β) = (3tβ, 3t (2 − β)) . The homotopy as
According to the procedure described above, we have
The solution (Figure 3) in powers of p is represented as
Consider the nonlinear IDE of the second kind [11]
with the initial condition y(0, β) = (0, 0), where 0 ≤ t, s ≤ 1, 0 ≤ β ≤ 1, λ = 1,
The solution components are
and so on. The solution (Figure 4) is represented by utilizing the above iterations as
Finally, consider the nonlinear fuzzy VIDE of the first order as [11]
with initial condition y(0, β) = (0, 0), where 0 ≤ t, s ≤ 1, 0 ≤ β ≤ 1, λ = 1, K(t, s) = 1, and
The exact solution of
We construct the homotopy as
According to the procedure described above, we have
Comparing the powers of parameter p in
and so on. Utilizing the above results and we get series solution (Figure 5) as
In this study, we investigated the HPSTM in order to solve non-linear FIDEs based on the parametric form of fuzzy numbers. The solutions obtained by these fuzzy differential equations are considered to be controllers in applications. This method proved its effectiveness and reliability in solving uncertain types of equations by providing the best approximate solutions. The numerical outcomes obtained using the proposed technique are comparable to the exact solutions. Thus, the work can be extended to higher-order mixed-type FIDEs in more variables.
The authors have no conflict of interest.
Comparison of approximate solution (
Comparison of approximate solution (
Comparison of approximate solution (
Comparison of approximate solution (
Comparison of approximate solution (
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