Applications of fuzzy Kashuri-Fundo transforms

Introduction

The theory of fuzzy differential equations has gained a lot of interest lately and has been expanding quickly. Numerous writers conducted extensive research on it (see [1][2][3][4] ). Other academics have created some numerical approaches and algorithms to deal with the solutions of this type of problem (see [5][6][7][8] ). A new approach to solving fuzzy linear differential equations was put out by Allahviranloo et al. in [7] . It was constructed using the equivalent integral forms of the original problems and assumed highly generalized differentiability. However, their major finding was formal and unproven, and their approach was restricted to solving only fuzzy linear differential equations with crisp constant coefficients. We devised and expanded this operator approach in 2015 (see [9] ) to solve several first-order fuzzy linear differential equations with variable coefficients, motivated by their work. Additionally, we provided the solution to the general formula along with the required proofs. Prior to 2021, Alshibley [9] employed the fuzzy Aboodh transform in an analytic solution approach for a few first-order fuzzy differential equations under the highly generalized differentiability. The authors' primary finding was the relationship between a fuzzy function's first derivative and its fuzzy Kashuri-Fundo transformations. They provided two numerical examples, both of which are first-order fuzzy differential equations, to demonstrate the effectiveness of the approach. This work aims to solve second-order fuzzy differential equations under strongly generalized differentiability by extending their main result and developing their method by establishing the relationship between a function's second derivative and its fuzzy Kashuri-Fundo transforms.

Fundamental Preliminaries

The following fundamental concepts and theorems pertaining to our work in this field are included for completeness's sake. [10] The mapping Ƀ: Ɽ → [0, 1] is said to be fuzzy number if it fulfills the following:

Definition

Let 𝜛 be the set of all fuzzy number on Ɽ. The Ɣ -level set of a fuzzy number Ƀ ∈ 𝜛, 0 ≤ Ɣ ≤ 1 denoted by [Ƀ] Ɣ is defined as

Zadeh's extension principle states that the operation of addition on 𝜛 is given by

and a fuzzy number's scalar multiplication is provided by

The following characteristics are widely acknowledged to be true at all levels: [11] A pair that is sorted parametrically is a fuzzy number, (Ƀ, Ƀ ) of functions Ƀ(Ɣ), Ƀ(Ɣ), Ɣ ∈ [0,1], which fulfills:

Definition

1. Ƀ(Ɣ) is a continuous function with a right function of 0 and a left function of (0,1] that is non-decreasing.

2. Ƀ(Ɣ) is a bounded, non-increasing function with 0 continuous right and (0,1] continuous left.

For arbitrary Ƀ = (Ƀ(Ɣ), Ƀ(Ɣ)), Ƀ = (Ƀ (Ɣ), Ƀ (Ɣ)), 𝟎 ≤ Ɣ ≤ 𝟏 and 𝛒 > 𝟎 we define: . If ρ = 1 then ρ⨀Ƀ = -Ƀ [11] Let Ƀ and 𝛺 are fuzzy numbers, the Hausdorff distance between two fuzzy numbers is provided by: [11] Let Ɏ: Ɽ ⟶ 𝜛 be a function with fuzzy values. Assuming a random fixed point χ 0 ∈ Ɽ and ϵ > 0 there exists δ > 0 such that |χ -χ 0 | < δ ⟹ 𝛤(Ɏ(χ), Ɏ(χ 0 )) < ϵ, Ƀ is said to be a continuous fuzzy-valued function.

Definition

Definition

Definition [12]

A mapping Ɏ: Ɽ × 𝜛 ⟶ 𝜛 is referred to as continuous at one point (𝜏 0 , χ 0 ) ∈ Ɽ × 𝜛 supplied for any fixed Ɣ 0 ∈ [0,1] and arbitrary 𝜀 > 0 there exists an 𝛿(𝜀, Ɣ) such that 𝛤([Ɏ(𝜏, χ)] Ɣ , [Ɏ(𝜏 0 , χ 0 )] Ɣ ) < 𝜀 whenever |𝜏 -𝜏 0 | < 𝛿(𝜀, Ɣ) and 𝛤([χ] Ɣ , [χ 0 ] Ɣ < 𝛿(𝜀, Ɣ) for all 𝜏 ∈ Ɽ, χ ∈ 𝜛 2.5.1 Theorem [13] Assume Ɏ(𝜒) 2.6 Definition [12] Assume ψ, λ ∈ 𝜛.There is β ∈ 𝜛 such that ψ = λ⨁β then β is known the H-differential of ψ and λ and it is represented by ψ ⊝ λ. In this paper, the sign " ⊝ " always stands for H-difference, and also note that

7 Definition [14] A function Ɏ: (𝓅, 𝓆): ⟶ 𝜛 and 𝜒 0 ∈ (𝓅, 𝓆). We say that Ɏ is strongly generalized differentiable at 𝜒 0 If such an element exists (𝜒 0 ) ∈ 𝜛, such that:

i. ∀ 𝛯 > 0 that is adequately little, there are Ɏ(𝜒 0 + 𝛯) ⊖ Ɏ(𝜒 0 ), Ɏ(𝜒 0 ) ⊖ Ɏ(𝜒 0 -𝛯),

Where lim [15] Let Ɏ(𝜒): Ɽ ⟶ 𝜛be a function and represents Ɏ(𝜒) = ((Ɏ(𝜒, Ɣ),Ɏ(𝜒, Ɣ)) in every instance for Ɣ ∈ [0,1]. Then:

Theorem

2. If Ɏ(𝜒) is differentiable form ii, then (Ɏ(𝜒, Ɣ) and Ɏ(𝜒, Ɣ) are differentiable functions and Ɏ ′ (𝜒) = ( Ɏ ′ (𝜒, Ɣ), Ɏ ′ (𝜒, Ɣ)).

Definition [16]

A function Ɏ: (𝓅, 𝓆): ⟶ 𝜛. We're saying that a mapping Ɏ is strongly generalized differentiable at 𝜒 0 if an element Ɏ (2) (𝜒 0 ) ∈ 𝜛 exists, such that:

i. ∀ 𝛯 > 0 that is sufficiently small, there exist Ɏ ′ (𝜒 0 + 𝛯) ⊖ Ɏ ′ (𝜒 0 ), Ɏ ′ (𝜒 0 ) ⊖ Ɏ ′ (𝜒 0 -𝛯)

Or ii. ∀ 𝛯 > 0 that is sufficiently small, there exist Ɏ ′ (𝜒 0 ) ⊖ Ɏ ′ (𝜒 0 + 𝛯), Ɏ ′ (𝜒 0 -𝛯) ⊖ Ɏ ′ (𝜒 0 )

Or iii. ∀ 𝛯 > 0 that is sufficiently small, there exist Ɏ ′ (𝜒 0 + 𝛯) ⊖ Ɏ ′ (𝜒 0 ), Ɏ ′ (𝜒 0 -𝛯) ⊖ Ɏ ′ (𝜒 0 )

Or iv. ∀ 𝛯 > 0 that is sufficiently small, there exist Ɏ ′ (𝜒 0 ) ⊖ Ɏ ′ (𝜒 0 + 𝛯), Ɏ ′ (𝜒 0 ) ⊖ Ɏ ′ (𝜒 0 -𝛯)

Ɏ ′ (𝜒 0 )⊖Ɏ ′ (t𝜒 0 -𝛯) 𝛯 = Ɏ (2) (𝜒 0 ). [17] Let Ɏ(𝜒) and Ɏ ′ (𝜒) are two differentiable fuzzy-valued functions. Denoted that Ɣcut is fuzzy valued function representation Ɏ(𝜒) with Ɏ(𝜒) = [Ɏ(𝜒, Ɣ), Ɏ(𝜒, Ɣ), then:

Theorem

1. Let Ɏ(𝜒) and Ɏ ′ (𝜒) are (i)-differentiable, or let Ɏ(𝜒) and Ɏ ′ (𝜒) are (ii)-differentiable then Ɏ(𝜒, Ɣ)and Ɏ(𝜒, Ɣ) have derivatives in first and second order such that

2. Let Ɏ(𝜒) be (i)-differentiable and Ɏ ′ (𝜒) be (ii)-differentiable, or let Ɏ(𝜒) be (ii)-differentiable and Ɏ ′ (𝜒) be (i) differentiable then Ɏ(𝜒, Ɣ)and Ɏ(𝜒, Ɣ) have derivatives in first and second order such that Ɏ (2) (𝜒) = [Ɏ (2) (𝜒, Ɣ), Ɏ (2) (𝜒, Ɣ)].

Fuzzy Kashuri-Fundo Transform

The fuzzy start and boundary value issues related to fuzzy differential equations are resolved by the fuzzy Kashuri-Fundo transform method. Fuzzy Kashuri-Fundo transformations simplify a fuzzy differential equation by turning it into an algebraic problem. Operational calculus, a crucial branch of practical mathematics, is the transition from calculus operations to algebraic operations on transformations. Artion Kashuri and Akli Fundo. [18] The Kashuri-Fundo Transform is an integral transform that was proposed by Kashuri and Fundo. The function defines this transform Ɏ(𝜒) as:

Kashuri-Fundo Integral Transform

, Where 𝜒 ≥ 0, -ϻ 1 ≤ 𝓇 ≤ ϻ 1 and in the function Ɏ, the variable 𝓇 is used as a factor to the variable 𝜒

Definition

Let Ɏ(𝜒) be a fuzzy-valued continuous function. Suppose that

is being called fuzzy Kashuri-Fundo transform and is known as

Using the definition of classical Kashuri-Fundo transform, to get:

Theorem

Let Ɏ(𝜒), ¥(𝜒) be continuous fuzzy-valued functions, ℳ 1 and ℳ 2 are constants, then (1).

(2). Suppose Ɏ(𝜒) = ( Ɏ(𝜒, Ɣ), Ɏ(𝜒, Ɣ) and ¥(𝜒) = ( ¥(𝜒, Ɣ), ¥(𝜒, Ɣ)

Fuzzy Kashuri-Fundo Transform for First -Order Fuzzy Differential Equation

The fuzzy Kashuri-Fundo transform of the derivative of a first order under generalized H-differentiability must be analyzed for high-order fuzzy differential equations.

Theorem

Let Ɏ(𝜒) is the primitive of Ɏ ′ (𝜒) on [0, ∞) and Ɏ(𝜒) is a fuzzy-valued function that is integrable, then:

Proof (1)

For a fixed, arbitrary 0 ≤ Ɣ ≤ 1,

Since Ɏ(𝝌) is differentiable of form i using Theorem (2.2)

Since Ɏ(𝝌) is differentiable of form ii using Theorem (2.2):

4.1 Example: Consider a fuzzy initial value problem:

Solution

Apply both sides' fuzzy Kashuri-Fundo transforms to get

Using upper and lower functions, to have

Using upper and lower functions, to have

Using inverse Kashuri-Fundo transform, Ɏ(𝜒, Ɣ) = (1 -Ɣ)e -𝜒 , Ɏ(𝜒, Ɣ) = (1 -Ɣ)e -𝜒

Fuzzy Kashuri-Fundo transform for second-order differential equation

In this part, fuzzy Ordinary differential equations with second order derivative are solved using the fuzzy Kashuri-Fundo transform.

Theorem

Let Ɏ(𝜒)e -s𝜒 , Ɏ ′ (𝜒)e -s𝜒 and Ɏ (2) (𝜒)e -s𝜒 are continuous and integrable Riemann functions on[0, ∞) so Ɏ(𝜒) is continuous fuzzy valued function. Thus:

Proof: (a) Ɏ(𝜒)and Ɏ ′ (𝜒) are (i)-differentiable

Since Ɏ(𝜒) and Ɏ ′ (𝜒) are (i)-differentiable, using Theorem (2.2):

Ɏ (2) (𝜒, Ɣ) = Ɏ (2) (𝜒, Ɣ), Ɏ (2) (𝜒, Ɣ) = Ɏ (2) (𝜒, Ɣ).

Since Ɏ(𝜒) is (i)-differentiable using Theorem (2.1), Ɏ ′ (0, Ɣ) = Ɏ ′ (0, Ɣ) and Ɏ ′ (0, Ɣ) = Ɏ ′ (0, Ɣ).

Since, Ɏ(𝜒) is (i)-differentiable and Ɏ ′ (𝜒) is (ii)-differentiable using Theorem (2.2):

Ɏ (2) (𝜒, Ɣ) = Ɏ (2) (𝜒, Ɣ), Ɏ (2) (t, Ɣ) = Ɏ (2) (𝜒, Ɣ).

Since Ɏ(𝜒) is (i)-differentiable using Theorem (2.1) getting, Ɏ ′ (0, Ɣ) = Ɏ ′ (0, Ɣ) and Ɏ ′ (0, Ɣ) = Ɏ ′ (0, Ɣ).

Since Ɏ(𝜒)is (ii)-differentiable and Ɏ ′ (𝜒) is (i)-differentiable using Theorem (2.2):

Ɏ (2) (𝜒, Ɣ) = Ɏ (2) (𝜒, Ɣ), Ɏ (2) (𝜒, Ɣ) = Ɏ (2) (𝜒, Ɣ).

Since Ɏ(𝜒) is (ii)-differentiable using Theorem (2.1) getting, Ɏ ′ (0, Ɣ) = Ɏ ′ (0, Ɣ) and Ɏ ′ (0, Ɣ) = Ɏ ′ (0, Ɣ). Since, Ɏ(𝜒) and Ɏ `(𝜒) are (ii)-differentiable using Theorem (2.2):

Ɏ (2) (𝜒, Ɣ) = Ɏ (2) (𝜒, Ɣ), Ɏ (2) (𝜒, Ɣ) = Ɏ (2) (𝜒, Ɣ).

Since Ɏ(𝜒) is (ii)-differentiable using Theorem (2.1) getting, Ɏ ′ (0, Ɣ) = Ɏ ′ (0, Ɣ) and Ɏ ′ (0, Ɣ) = Ɏ ′ (0, Ɣ).

₭[Ɏ (2) (𝜒, Ɣ) ] = (2) If Ɏ(𝜒)is (i)-differentiable but Ɏ ′ (t) is (ii)-differentiable,

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