Solution of Second Order Fuzzy Differential Equation by Fuzzy

Abstract:

In this paper, we introduce a new method for solving a numerical second-order fuzzy differential equation using the fuzzy Sumudu transforms method. Here, we present the solution procedure with an initial condition as a fuzzy number and finally provide numerical examples to demonstrate the efficiency of the proposed methods.

Keywords:

Second Order Differential Equation, Fuzzy Differential Equation, Fuzzy Sumudu Transform

1. Introduction

The study of real-life problems can be modeled as mathematical equations, such as differential equations. Fuzzy differential equations provide a suitable approach for mathematical modeling of real-world problems in which uncertainties or vagueness are pervasive.

In 1987, Chang and Zadeh introduced the term "fuzzy differential equations" [1].

One of the most significant applications of integral transform methods is to solve ordinary differential equations. The Sumudu transform is particularly useful for solving differential equations and was introduced by Watugala[2] for solving engineering control problems. R. Goetschel, Jr. and W. Voxman used elementary calculus in fuzzy set theory [3], and O. Kaleva analyzed fuzzy differential equations[4,7].

Fuzzy initial value problems of the first order were solved by S. Seikkala [5,6], and S. Abbasbandy and T. Allahviranloo[10]. J.J. Buckley, T. Feurihg, and E. Eslami introduced fuzzy set theory with fuzzy differential equations [8,9]. The generalization of the differentiability of fuzzy numbers was used by Bede, B, Gal, SG [11]. Many authors worked on fuzzy differential equations of the first order linear equation using generalized fuzzy differential concepts [12-17].

Then, numerical solutions of Second-Order Fuzzy Differential Equations were obtained using the Improved Runge-Kutta Nystrom Method by F. Rabiei, F.

Ismail et al [20]. Nth order fuzzy differential equations were solved using the STHWS method by S. Sekar and S. Senthilkumar[21]. Abdul Rahman, N.A.; Ahmad used the Sumudu transform to solve partial and fractional differential equations [28,29,30]. The solution of solving fuzzy linear Volterra integro-differential equations using the fuzzy Sumudu transform was presented by A.Rajkumar, C.Jesuraj, and A.Mohammed Shapique[31]. In general, the Sumudu transform is considered a powerful tool for solving differential equations due to its unique properties, which ease the process of finding solutions. Finally, in this paper, we utilize the fuzzy Sumudu transform to solve second-order fuzzy differential equations with initial values represented as fuzzy numbers.

This paper is arranged as follows:

• Section 2: Basic concepts, definitions, and theorems.
• Section 3: Formulas of fuzzy derivatives of order two and fuzzy Sumudu transforms.
• Section 4: Solution procedure with examples of fuzzy initial value problems of order two.
• Section 5: Conclusion and future work.

2. Preliminaries

Fuzzy Sets Definitions

In this section, we review the basic definitions of fuzzy sets and fuzzy numbers.

2.1 Definition: Fuzzy Set

A fuzzy set is characterized by a membership function mapping of a domain space (i.e., a mapping between the universe of discourse U to the unit interval [0,1]) given by:

_[0,1]^μ_A(x)

Here, ^[0,1]_U is called the degree of membership function of the fuzzy set A.

2.2 Definition: Normal Fuzzy Set

A fuzzy set A of the universe of discourse U is called a normal fuzzy set if there exists at least one x _U such that ^μ_A(x) = 1.

2.3 Definition: Height of a Fuzzy Set

The largest membership grade obtained by any element in the fuzzy set, denoted as ^h_A, is given by:

^h_A = sup {^μ_A(x) | x _U}

2.4 Definition: Convex Fuzzy Set

A fuzzy set A is said to be convex if and only if for any x₁, x_{2 U}, the membership function of A satisfies the condition:

_[0,1]^{μ_A(λx₁ + (1-λ)x₂)} ≥ min {^μ_A(x₁), ^μ_A(x₂)}

for all λ in the interval [0, 1].

2.5 Definition: Extension Principle

Let X be a Cartesian product of the universe X and A₁, A₂, ..., A_n be n fuzzy numbers. If f: X → Y is a mapping, then the extension principle defines the fuzzy set B in Y by:

^B(y) = sup { min {^μ_A1(x₁), ^μ_A2(x₂), ..., ^μ_An(x_n), ^{f(x₁, x₂, ..., x_n)}} | (x₁, x₂, ..., x_n) _X}

For n = 1, the above extension principle reduces to:

^B(y) = sup { min {^μ_A1(x), ^f(x)} | x _X}

2.6 Definition: Triangular Fuzzy Number

Let (A in F(mathbb{R})), (1 ≤ a₁ ≤ a₂ ≤ a₃) be called a triangular fuzzy number if its membership function is given by:

[μ_A(x) =
begin{cases}
0, & x < a₁ leq a₂ \
frac{{x - a₁}}{{a₂ - a₁}}, & a₁ leq x < a₂ \
frac{{a₃ - x}}{{a₃ - a₂}}, & a₂ ≤ x ≤ a₃ \
0, & x > a₃
end{cases}
]

2.7 Definition: Generalized Triangular Fuzzy Number

Let (A in F(mathbb{R})), (1 ≤ a₁ ≤ a₂ ≤ a₃) and (ω) be called a Generalized triangular fuzzy number if its membership function is given by:

[μ_A(x) =
begin{cases}
0, & x < a₁ leq a₂ \
frac{{x - a₁}}{{a₂ - a₁}}^ω, & a₁ leq x < a₂ \
frac{{a₃ - x}}{{a₃ - a₂}}^ω, & a₂ ≤ x ≤ a₃ \
0, & x > a₃
end{cases}
]

2.8 Definition: Fuzzy Ordinary Differential Equation

Consider an ordinary differential equation whose initial condition is described in fuzzy numbers:

[ frac{{d}}{{dt}} left( int_{0}^{1} f(t, y_t) , dt right) = y_t quad text{with} quad y_0 = (a, b, c; α, β) ]

Then we say the given differential equation is a fuzzy differential equation.

2.9 Definition: Strong and Weak Solution of Fuzzy Differential Equation [30]

Consider a first-order non-homogeneous linear fuzzy ordinary differential equation (frac{{dx}}{{dt}}(t) = f(t, x(t))), (x(t_0) = x') with (x') as a GTFN. Let (x(t)) and ({x_t^α}) be the solution and α-cuts of the above differential equation, respectively.

We say the solution (x(t)) is a strong solution if it satisfies the condition:

(∀ α ∈ mathbb{R}), ([x_t^α] ≤ x_t^α ≤ [overline{x}_t^α]) for all (t ∈ T).

Otherwise, it is a weak solution. In such cases, ({x_t^α}) can be expressed as:

[ {x_t^α} = min left{max left{[x_t^α], underline{x}_t^αright}, overline{x}_t^αright} ]

2.10 Definition: Fuzzy Derivative

Let (f: (a, b) to mathbb{E}) and ((a, b)) be a strongly generalized differential at ((a, b)) (Bede-Gal differential) if there exists an element ((x, E') in mathbb{E}') such that:

(i) (∀ h > 0), ((h to 0)), ((h neq 0)), (o) of (x), (h) of (x), (f(x, h)), (f(x, o)), (f(x, h_o)), ((h to 0)), (o) of (x), (h) of (f(x, h)), (f(x, o)), (f(x, h_o)), (f(x, h)), (o) of (h), ((h to 0)), (f(x, o)), (f(x, h_o)), ((h neq 0)) and the limits are given by:

[0 = lim_{h to 0} lim_{h_o to 0} lim_{h to 0} f(x, h) - f(x, o) - f(x, h_o) - f(x, h_o)]

or

(ii) (∀ h > 0), ((h to 0)), ((h neq 0)), (o) of (x), (h) of (x), (f(x, h)), (f(x, o)), (f(x, h_o)), ((h to 0)), (o) of (x), (h) of (f(x, h)), (f(x, o)), (f(x, h_o)), (f(x, h)), (o) of (h), ((h to 0)), (f(x, o)), (f(x, h_o)), ((h neq 0)) and the limits are given by:

[0 = lim_{h to 0} lim_{h_o to 0} lim_{h to 0} f(x, h) - f(x, o) - f(x, h_o) - f(x, h_o)]

or

(iii) (∀ h > 0), ((h to 0)), ((h neq 0)), (o) of (x), (h) of (x), (f(x, h)), (f(x, o)), (f(x, h_o)), ((h to 0)), (o) of (x), (h) of (f(x, h)), (f(x, o)), (f(x, h_o)), (f(x, h)), (o) of (h), ((h to 0)), (f(x, o)), (f(x, h_o)), ((h neq 0)) and the limits are given by:

[0 = lim_{h to 0} lim_{h_o to 0} lim_{h to 0} f(x, h) - f(x, o) - f(x, h_o) - f(x, h_o)]

or

(iv) (∀ h > 0), ((h to 0)), ((h neq 0)), (o) of (x), (h) of (x), (f(x, h)), (f(x, o)), (f(x, h_o)), ((h to 0)), (o) of (x), (h) of (f(x, h)), (f(x, o)), (f(x, h_o)), (f(x, h)), (o) of (h), ((h to 0)), (f(x, o)), (f(x, h_o)), ((h neq 0)) and the limits are given by:

[0 = lim_{h to 0} lim_{h_o to 0} lim_{h to 0} f(x, h) - f(x, o) - f(x, h_o) - f(x, h_o)]

2.11 Definition

Let (f: (a, b) to mathbb{E}) be a function, and let (f(t)) and (f(t)^alpha) for (alpha in [0, 1]) be defined as ((1)) If (f) is (i) differentiable, then (f(t)) and (f(t)^alpha) are differentiable functions, and ((2)) If (f) is (ii) differentiable, then (f(t)) and (f(t)^alpha) are differentiable functions.

2.12 Theorem

Let (f: mathbb{R} to F(mathbb{R})) and (f(t)) be defined as ({f(t) = f(t)^alpha | 0,1 in alpha}). For any fixed (alpha in [0, 1]), assume that (f(t)) and (f(t)^alpha) are Riemann-Integrable on ([a, b]) where (b > a), and assume that:

[ int_{a}^{b} f(t) , dt = C quad text{and} quad int_{a}^{b} f(t)^alpha , dt = C^alpha quad text{where} quad C, C^alpha text{ are any two positive quantities}.]

Then, (f(t)) is improper fuzzy Riemann-Integrable on ([a,∞)), and the improper fuzzy Riemann-Integral is a fuzzy number. Thus:

[
int_{a}^{infty} f(t) , dt = (t) int_{a}^{infty} f(t)^alpha , dt = (t)
]

Proposition

If (1 leq 2), and (f(x)) and (f(x)) are a fuzzy-valued function and fuzzy Riemann-Integrable on ([a,∞)), then:

(1 leq 2 (f(x) oplus f(x))) is also fuzzy Riemann-Integrable on ([a,∞)).

[
1 leq 2 int_{a}^{infty} (f(x) oplus f(x)) , dx = 1 leq 2 int_{a}^{infty} (f(x) oplus f(x))
]

3. Fuzzy Sumudu Transform

3.1 Theorem

Let (f: mathbb{R} to F(mathbb{R})) be a continuous fuzzy valued function. Suppose that ((e^{talpha}f(t))) is an improper fuzzy Riemann-integrable on ([a, ∞)), then:

((talpha) int_{a}^{infty} e^{-talpha}f(t) , dt) is called the fuzzy Sumudu transform and it is denoted by:

[
left[ (talpha) int_{0}^{∞} e^{-talpha}f(t) , dt right]_{alpha}^{(tau)} = left[ (talpha) int_{0}^{∞} e^{-talpha}f(t) , dt right]_{(tau)}
]

3.2 Theorem

If (f: mathbb{R} to F(mathbb{R})) be a continuous fuzzy valued function and if (left[ (talpha) int_{0}^{∞} e^{-talpha}f(t) , dt right]_{alpha}^{(tau)}), then:

(left[ (0) int_{h}^{∞} F_u f(t) , dt right]_{u>0}^{(tau)} = left[ (0) int_{h}^{∞} F_u f(t) , dt right]_{u>0}^{(tau)}) if (f) is (i) differentiable, (u>0) and:

(left[ (h) int_{h}^{∞} F_u f(t) , dt right]_{u>0}^{(tau)} = left[ (h) int_{h}^{∞} F_u f(t) , dt right]_{u>0}^{(tau)}) if (f) is (ii) differentiable, (u>0).

Proof: Case (i)

Let’s assume (f) is (i) Differentiable, then:

[
(talpha) left[ int_{0}^{∞} e^{-talpha}f(t) , dt right]_{alpha}^{(tau)} = (talpha) left[ int_{0}^{∞} e^{-talpha}f(t) , dt right]_{(tau)}
]

[
left[ int_{0}^{∞} e^{-talpha}f(t) , dt right]_{(tau)}' = left[ (talpha) int_{0}^{∞} e^{-talpha}f(t) , dt right]_{alpha}^{(tau)}'
]

Hence:

[
left[ (0) int_{h}^{∞} F_u f(t) , dt right]_{u>0}^{(tau)}' = left[ (0) int_{h}^{∞} F_u f(t) , dt right]_{u>0}^{(tau)}'
]

Case (ii)

Assume (f) is (ii) Differentiable, then:

[
(talpha) left[ int_{0}^{∞} e^{-talpha}f(t) , dt right]_{alpha}^{(tau)} = (talpha) left[ int_{0}^{∞} e^{-talpha}f(t) , dt right]_{(tau)}
]

[
left[ int_{0}^{∞} e^{-talpha}f(t) , dt right]_{(tau)}' = left[ (talpha) int_{0}^{∞} e^{-talpha}f(t) , dt right]_{alpha}^{(tau)}'
]

Hence:

[
left[ (h) int_{h}^{∞} F_u f(t) , dt right]_{u>0}^{(tau)}' = left[ (h) int_{h}^{∞} F_u f(t) , dt right]_{u>0}^{(tau)}'
]

3.3 Theorem

If (f: mathbb{R} to F(mathbb{R})) be a continuous fuzzy valued function and if (left[ (talpha) int_{0}^{infty} e^{-talpha}f(t) , dt right]_{alpha}^{(tau)}), then:

[
1 - frac{1}{1 + atau u}S e^{f(t)} , dt = frac{atau u}{(1 + atau u)^2}
]

Proof: by definition 2.11

[
left[ int_{0}^{infty} e^{-talpha}f(t) , dt right]_{(tau)} = left[ int_{0}^{infty} e^{-atau u t}f(t) , dt right]_{(tau)}
]

[
frac{1}{tau} left[ int_{0}^{infty} e^{-atau u t}f(t) , dt right]_{(tau)}' = frac{1}{tau} left[ int_{0}^{infty} e^{-atau u t}f(t) , dt right]_{(tau)}'
]

Hence:

[
1 - frac{1}{1 + atau u}S e^{f(t)} , dt = frac{atau u}{(1 + atau u)^2}
]

3.4 Theorem

If (f: mathbb{R} to F(mathbb{R})) be a continuous fuzzy valued function and if (left[ (talpha) int_{0}^{infty} e^{-talpha}f(t) , dt right]_{alpha}^{(tau)}), then:

[
left( t - S e^{f(t)} , dt right) u = F_u
]

Proof: Assume the function (g) is (i) Differentiable, then:

Let:

[
(talpha) int_{0}^{infty} e^{-talpha}f(t) , dt = (talpha) int_{0}^{infty} e^{-talpha}g(t) , dt
]

[
left( int_{0}^{infty} e^{-talpha}f(t) , dt right)' = left( talpha int_{0}^{infty} e^{-talpha}f(t) , dt right)'
]

Hence:

[
left( t - S e^{f(t)} , dt right) u = F_u
]

3.5 Theorem

3.5 Theorem

If (f: mathbb{R} to F(mathbb{R})) be a continuous fuzzy valued function and if ([F_u S f t]_0^1), then:

[
int_0^1 ttau S f t , dt = int_0^1 tau F_u , du
]

Proof: By definition of FST, ([tF_u S f t]_infty^0 = int_0^infty tF_u f ut e , dt)

(int_0^0 (tF_u f ut e - tF_u f ut e) , dt = int_0^infty (tF_u f ut e - tF_u f ut e) , dt)

Hence:

[
int_0^1 ttau S f t , dt = int_0^1 tau F_u , du
]

3.6 Theorem

If (y(t)) and (y'(t)) be(i)-differentiable. Then by theorem the above equation becomes

[ left[ S_y(t) right]_u^u + left[ S_y'(t) right]_u^u + left[ S_y''(t) right]_u^u = Theta_u^u ]

If (y(t)) is (i)-differentiable and (y'(t)) is (ii)-differentiable, then by theorem

[ left[ S_y(t) right]_u^u + left[ S_y'(t) right]_u^u + left[ S_y''(t) right]_u^u = -Theta_u^u ]

If (y'(t)) is (i)-differentiable and (y(t)) is (ii)-differentiable, then by theorem

[ left[ S_y(t) right]_u^u + left[ S_y'(t) right]_u^u + left[ S_y''(t) right]_u^u = -Theta_u^u ]

If (y'(t)) is (i)-differentiable and (y(t)) is (ii)-differentiable, then by theorem

[ left[ S_y(t) right]_u^u + left[ S_y'(t) right]_u^u + left[ S_y''(t) right]_u^u = Theta_u^u ]

4. Numerical example

Let us consider the following second-order fuzzy initial value problems using nanogonal fuzzy numbers:

[y''_t - y_t = 0, quad y₀ = 1, quad y'₀ = 1]

By using fuzzy Sumudu transforms, we have the following:

[ [Y''_s - Y_s] ast alpha ast alpha = [Y_s - Y₀] ast alpha ast alpha, quad [Y'_s - Y'₀] ast alpha ast alpha = [Y_s - Y₀] ast alpha ast alpha]

[ frac{Y''_s - Y_s}{alpha ast alpha} = frac{Y_s - Y₀}{alpha ast alpha}, quad frac{Y'_s - Y'₀}{alpha ast alpha} = frac{Y_s - Y₀}{alpha ast alpha}]

[ frac{Y''_s}{alpha ast alpha} - frac{Y_s}{alpha ast alpha} = frac{Y_s - Y₀}{alpha ast alpha}, quad frac{Y'_s}{alpha ast alpha} - frac{Y'₀}{alpha ast alpha} = frac{Y_s - Y₀}{alpha ast alpha}]

[ frac{Y''_s}{alpha ast alpha} - frac{Y_s}{alpha ast alpha} = frac{Y_s - 1}{alpha ast alpha}, quad frac{Y'_s}{alpha ast alpha} - frac{1}{alpha ast alpha} = frac{Y_s - 1}{alpha ast alpha}]

Here we are going to solve the above example by four different cases as follows:

Case 1: Let us consider (y_t) and (y'_t) be (i)-differentiable. Then by theorem, the above equation becomes:

[ [Y_s - 1] ast alpha ast alpha = left[ frac{Y'_s}{alpha ast alpha} - frac{1}{alpha ast alpha} right] ast alpha ast alpha]

Thus:

[ Y_s ast alpha ast alpha = left[ frac{Y'_s}{alpha ast alpha} - frac{1}{alpha ast alpha} right] ast alpha ast alpha]

[ Y_s = left[ frac{Y'_s}{alpha ast alpha} - frac{1}{alpha ast alpha} right] ast alpha ast alpha + 1]

Then we get the alpha-cut representation of the solution as:

[ (Y_s, Y_s) = (1 - 1 ast alpha ast alpha, 1 + 1 ast alpha ast alpha) ast e^{alpha ast t}]

Where (alpha) is a parameter.

Case 2: Let us consider (y_t) is (i)-differentiable and (y'_t) is (ii)-differentiable. Then by theorem, the above equations become:

[ [Y_s - 1] ast alpha ast alpha = left[ frac{Y''_s}{alpha ast alpha ast alpha ast alpha} - frac{1}{alpha ast alpha ast alpha ast alpha} right] ast alpha ast alpha ast alpha ast alpha]

Thus:

[ S Y_s ast alpha ast alpha = Sleft[ frac{Y''_s}{alpha ast alpha ast alpha ast alpha} - frac{1}{alpha ast alpha ast alpha ast alpha} right] ast alpha ast alpha ast alpha ast alpha]

[ S Y_s = Sleft[ frac{Y''_s}{alpha ast alpha ast alpha ast alpha} - frac{1}{alpha ast alpha ast alpha ast alpha} right] ast alpha ast alpha ast alpha ast alpha + 1]

Then we get the alpha-cut representation of solution is:

[ (Y_s, Y_s) = left(1 - 1 ast alpha ast alpha, 1 + 1 ast alpha ast alpharight) ast (sinh(alpha ast t), sin(alpha ast t))]

Case 3: Let us consider (y'_t) is (i)-differentiable and (y_t) is (ii)-differentiable. Then by theorem, the above equations become:

[ [Y_s - 1] ast alpha ast alpha = left[ frac{Y''_s}{alpha ast alpha ast alpha ast alpha} - frac{1}{alpha ast alpha ast alpha ast alpha} right] ast alpha ast alpha ast alpha ast alpha]

Thus:

[ S Y_s ast alpha ast alpha = Sleft[ frac{Y''_s}{alpha ast alpha ast alpha ast alpha} - frac{1}{alpha ast alpha ast alpha ast alpha} right] ast alpha ast alpha ast alpha ast alpha]

[ S Y_s = Sleft[ frac{Y''_s}{alpha ast alpha ast alpha ast alpha} - frac{1}{alpha ast alpha ast alpha ast alpha} right] ast alpha ast alpha ast alpha ast alpha - 1]

Then we get the alpha-cut representation of the solution is:

[ (Y_s, Y_s) = left(1 - 1 ast alpha ast alpha, 1 + 1 ast alpha ast alpharight) ast (sinh(alpha ast t), sin(alpha ast t))]

Case 4: Let us consider (y'_t) is (i)-differentiable and (y_t) is (ii)-differentiable. Then by theorem, the above equations become:

[ [Y_s - 1] ast alpha ast alpha = left[ frac{Y''_s}{alpha ast alpha ast alpha ast alpha} - frac{1}{alpha ast alpha ast alpha ast alpha} right] ast alpha ast alpha ast alpha ast alpha]

Thus:

[ S Y_s ast alpha ast alpha = Sleft[ frac{Y''_s}{alpha ast alpha ast alpha ast alpha} - frac{1}{alpha ast alpha ast alpha ast alpha} right] ast alpha ast alpha ast alpha ast alpha]

[ S Y_s = Sleft[ frac{Y''_s}{alpha ast alpha ast alpha ast alpha} - frac{1}{alpha ast alpha ast alpha ast alpha} right] ast alpha ast alpha ast alpha ast alpha]

Then we get the alpha-cut representation of the solution is:

[ (Y_s, Y_s) = (1, 1) ast (alpha ast t, alpha ast t) ast (e^{alpha ast t}, e^{alpha ast t})]

5. Conclusion

Using fuzzy sumudu transform method here we solved a second order fuzzy differential

equation with consider as initial condition is fuzzy numbers. We can solve higher order

fuzzy differential equation under strongly generalized H-differentiability. This method will

helpful for researcher and problems solvers.

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