This work aims to introduce the concept of the second-order Weak Fuzzy Complex-Differential Equations (WFC-DEs) for the first time. A special isomorphic transformation function could write a WFC-DE as two related Differential Equations (DEs) of the second order concerning their real variables. We study the second-order constant-coefficient linear homogeneous DE in a Weak Fuzzy Complex (WFC) variable with WFC constant coefficients. Also, we study the simple form of this type of WFC-DE with real constant coefficients. Thus, to find the general solution, we use the characteristic equation of the second-order DE. Also, we get a particular solution for the second-order constant-coefficient linear homogeneous Weak Fuzzy Complex-Initial Value Problem (WFC-IVP). To enhance understanding, we provide illustrative examples for each problem discussed.
functions, Weak fuzzy complex differential equation, Initial value problem.
Throughout history, the expansion of number sets has frequently arisen from the necessity to address particular mathematical challenges. Scholars have shown considerable interest in extending the set of real numbers R, with these newly formulated sets being characterized as two-dimensional generalizations of the real number system such as:
Complex numbers: {u + v i; u, v ∈ R; i = −1}, dual numbers: {u + vi; u, v ∈ R; i = 0}, Neutrosophic numbers: {u + vi; u, v ∈ R; i = i}, and split-complex numbers: {u + v i; u, v ∈ R; i = 1}.
Fuzzy numbers emerged as an extension of fuzzy set theory, which Lotfi Zadeh
In 2023, the WFC set FJ = {u + v J; u, v ∈ R; J = t ∈ ]0, 1[} was defined in
However, WFC numbers and their primary arithmetic operations were programmed by Python
Numerous physical phenomena are represented through functions and their derivatives. The first derivative, for instance, signifies the slope or velocity, while the second derivative conveys curvature or acceleration. DEs play a crucial role in various aspects of contemporary life, finding applications in different fields such as biology, medicine, control systems
That leads us to pursue an essential area of inquiry that remains underexplored about the DEs in FJ WFC-DEs. Our paper will introduce the second-order WFC-DEs. In Section 2, we will mention some main concepts about WFC numbers, a special isomorphism transformation function, the quadratic WFC equation, WFC functions, and Weak Fuzzy Complex-Ordinary Differential Equations (WFC-ODEs).
In Section 3, we will define the second-order ordinary DEs, introducing the type of linear homogeneous equation with constant coefficients from FJ. Also, we will take the simple case of them with real constant coefficients. We will find the roots of their characteristic equation and construct their general solution with suitable examples. Then, we will give the notion of Weak Fuzzy Complex-Initial Value Problems (WFC-IVPs) of the related second-order DEs in Section 4 and find their particular solutions.
This section presents important definitions to explain key elements within our analytical framework
Definition 1. The set of WFC numbers was defined as follows:
FJ = {x0 + x1 J ; x0 , x1 ∈ R , J = t ∈ ]0, 1[ }, where ‘J’ is the WFC operator ( J ∉ R ).
Definition 2. Let φ be the transformation function from FJ to R × R, which we define as follows:
φ: FJ ↦ R × R,
( 0 + 1 ) = ( 0 + 1 (−√ ), 0 + 1(+√ )) = ( 0 − 1 √ , 0 + 1√ ),
where J = t ∈ ]0, 1[ ⇒ J = ±√t, and x0, x1, y0, y1 ∈ R (This map is an isomorphism).
Definition 3. Let φ: FJ ↦ R × R such that φ(X) = (a, b), the inverse function of φ is defined as follows:
φ : R × R ↦ FJ,
φ (a, b) = [a + b] + J[b − a].
2 2√t Definition 4. Let X = x0 + x1 J, Y = y0 + y1 J∈ FJ, we say that X ≤ Y, if and only if:
x0 − x1√t ≤ y0 − y1 √t, x0 + x1 √t ≤ y0 + y1 √t.
Definition 5. Let A = a0 + a1 J, B = b0 + b1 J∈ FJ, we define the interval [A, B] if and only if A ≤ B, according to the definition of the partial order relation (≤).
I. If A ≰ B, then [A, B] = ϕ.
II. We can understand [A B] as follows:
[A, B] = {C = c0 + c1 J ∈ FJ; A ≤ C ≤ B}.
Theorem 1. Let aX + bX + c = 0 be a quadratic WFC equation, where a = a0 + a1 J, b = b0 + b1 J and c = c0 + c1 J ∈ FJ, then it is equivalent to:
A0X0 + B0X0 + C0 = 0, A1X1 + B1X1 + C1 = 0,
a = φ (A0, A1) = φ (a0 − a1 √t, a0 + a1 √t), A0 = a0 − a1 √t ∈ R, A1 = a0 + a1 √t ∈ R, B = φ (B0, B1) = φ (b0 − b1 √t, b0 + b1 √t), B0 = b0 − b1 √t ∈ R, B1 = b0 + b1 √t ∈ R, c = φ (C0, C1) = φ (c0 − c1 √t, c0 + c1 √t), C0 = c0 − c1 √t ∈ R, C1 = c0 + c1 √t ∈ R.
Definition 6. Let f: FJ ↦ FJ be a WFC function in one variable, where φ(f(X)) = (f1(x0 − x1 √t ), f2( x0 + x1√t )); f1, f2: R ↦ R.
Then we say:
I. f is continuous on FJ if and only if f1 and f2 are continuous on R.
II. f is differentiable on FJ if and only if f1 and f2 are differentiable on R, concerning their variables.
III. f is integrable on FJ if only f1 and f2 are integrable on R.
Definition 7. Let f: FJ ↦ FJ be a differentiable/integrable function on FJ. We define I. f′(X) = φ (f′1(x0 − x1 √t ), f′2( x0 + x1√t )).
II. ∫ f(X). dX = φ (∫ f1. d(x0 − x1 √t), ∫ f2. d( x0 + x1√t )).
Definition 8. Let Y be a dependent variable, and Y = f(X) represents an unknown WFC function where the independent variable X = x0 + x1 J is a WFC variable. We define the WFC DE of n-th order (the highest derivative appearing in the DE) as the following:
ℱ(X, Y, Y′, Y′′, . . . , Y ) = 0, (1)
(n) d Y −1 −1
where Y = , Y = y0 + y1J = φ (y0 − y1 √t, y0 + y1 √t) = f(X) = φ (f1(x0 − x1 √t ), f2( x0 +
dXn
x1√t )) and f1, f2: R ↦ R.
Definition 9. A function Y = f(X) = φ (f1(x0 − x1 √t ), f2( x0 + x1√t )) is called the general solution of Eq. (1) on I ⊆ FJ if Y is n-times differentiable WFC function on I and satisfies the equation on I, where f1, f2: R ↦ R, I = φ (I1 × I2) ⊆ FJ and X = φ (x0 − x1 √t, x0 + x1 √t), x0 − x1 √t ∈ I1 ⊆ R, x0 + x1 √t ∈ I2 ⊆ R. We aim to study the second-order DEs that play a pivotal role in modeling systems where the relationship between acceleration and displacement is crucial, such as in mechanics and wave propagation.
We will now propose the definition of the Second-order WFC-DEs. Throughout the discussion, X = φ (x0 − x1 √t, x0 + x1 √t) ∈ FJ, x0 − x1 √t ∈ I1 ⊆ R, x0 + x1 √t ∈ I2 ⊆ R, I = φ (I1 × I2) ⊆ FJ, Y0 = y0 − y1 √t ∈ R, Y1 = y0 + y1 √t ∈ R, Y = φ (Y0, Y1) ∈ FJ.
Definition 10. The WFC-DE of the second order is written as follows:
ℱ(X, Y, Y , Y′′) = 0, (2)
′ ′ ′′ ′′
φ ℱ1(x0 − x1√t, y0 − y1√t, y − y √t, y − y √t) = 0, (3 − 1)
⇔ { (3)
′ ′ ′′ ′′
ℱ2(x0 + x1√t, y0 + y1√t, y + y √t, y + y √t) = 0, (3 − 2) where ℱ = φ (ℱ1, ℱ2), y0 − y1 √t = f1(x0 − x1 √t), y0 + y1 √t = f2(x0 + x1 √t).
dY −1
Y′ = = f′(X) = φ (f′1(x0 − x1 √t ), f′2( x0 + x1√t )),
dX d Y −1
Y′′ = = f′′(X) = φ (f′′1(x0 − x1 √t ), f′′2( x0 + x1√t )),
dX2 d −1 d d d −1 d d
and = φ ( , ) , = φ ( , ).
dX d(x −x t) d(x +x t) dX2 d(x −x t)2 d(x +x t)2 0 1 √ 0 1 √ 0 1 √ 0 1 √
Definition 11. Let Y0 = y0 − y1 √t = f1(x0 − x1 √t) and Y1 = y0 + y1 √t = f2(x0 + x1 √t) are the general solutions to Eq. (3-1) on I1 and Eq. (3-2) on I2, respectively, then
Y = f(X) = φ (Y0, Y1),
is the general solution of Eq. (2) on I ⊆ FJ.
In other words, Y = f(X) is the general solution of Eq. (2) on I ⊆ FJ if f(X) is a twice differentiable WFC function on I.
One of the simplest types of second-order ordinary DEs is the linear homogeneous equation with constant coefficients. We will concentrate on this type, which has many applications in science and engineering.
Definition 12. The general form of a second-order linear homogeneous WFC ordinary DE with constant coefficients is written as
aY + bY + cY = 0, (4)
where a, b, c, Y, a = a0 + a1 J, b = b0 + b1 J and c = c0 + c1 J ∈ FJ.
Remark 1. Using φ, we find that Eq. (4) is equivalent to two second-order linear homogeneous ordinary DEs with constant coefficients in R φ A Y ′′ + B Y ′ + C Y = 0, (5 − 1)
0 0 0 0 0 0
Eq. (4) ⇔ { (5)
A1Y1′′ + B1Y1′ + C1Y1 = 0, (5 − 2) where,
a = a0 + a1 J = φ (A0, A1), b = b0 + b1 J = φ (B0, B1) and c = c0 + c1 J = φ (C0, C1) ∈ FJ,
A0 = a0 − a1 √t ∈ R, A1 = a0 + a1 √t ∈ R, B0 = b0 − b1 √t ∈ R, B1 = b0 + b1 √t ∈ R, C0 = c0 − c1 √t ∈ R, C1 = c0 + c1 √t ∈ R.
We will use the characteristic equation to solve Eq. (4).
Definition 13. The characteristic equation of the constant-coefficient second-order linear homogeneous WFC-DE Eq. (4) is defined as the quadratic WFC equation.
am + bm + c = 0, a, b, c ∈ FJ, (6)
which holds the following, It is equivalent to the corresponding characteristic equations
A0λ + B0λ + C0 = 0, (its discriminant ∆1> 0), (7 − 1)
{ (7)
A1μ + B1μ + C1 = 0, (its discriminant ∆2> 0). (7 − 2) It has 4 roots when its discriminant ∆= b − 4ac:
l1 = φ (λ1, μ1), l2 = φ (λ2, μ2), l3 = φ (λ1, μ2), l4 = φ (λ2, μ1).
Proof:
Depending on Remark 1 and Theorem 1, we know that the characteristic equations of second-order DEs Eq. (5-1), Eq. (5-2) have the forms Eq. (7-1), Eq. (7-2), respectively,
A0λ + B0λ + C0 = 0, A1μ + B1μ + C1 = 0,
−1 2 2 −1
φ (A0λ + B0λ + C0, A1μ + B1μ + C1) = φ (0,0) −1 2 2 −1 −1 −1
⇒ φ (A0λ , A1μ ) + φ (B0λ, B1μ) + φ (C0, C1) = φ (0,0)
−1 −1 2 2 −1 −1 −1 −1
⇒ φ (A0, A1)φ (λ , μ ) + φ (B0, B1)φ (λ, μ) + φ (C0, C1) = φ (0,0) ⇒ am + bm + c = 0, where,
m = φ (λ, μ), a, b, c ∈ FJ, a = φ (A0, A1) = φ (a0 − a1 √t, a0 + a1 √t), A0 = a0 − a1 √t ∈ R, A1 = a0 + a1 √t ∈ R, b = φ (B0, B1) = φ (b0 − b1 √t, b0 + b1 √t), B0 = b0 − b1 √t ∈ R, B1 = b0 + b1 √t ∈ R, c = φ (C0, C1) = φ (c0 − c1 √t, c0 + c1 √t), C0 = c0 − c1 √t ∈ R, C1 = c0 + c1 √t ∈ R.
Each characteristic equation has its discriminant,
A0λ + B0λ + C0 = 0, ∆1= B0 − 4A0C0, { ⇒ {
2 the discriminants 2
A1μ + B1μ + C1 = 0, ∆2= B1 − 4A1C1.
We will study the case when ∆1, ∆2> 0 ⟹ the corresponding solutions of the characteristic equations Eq. (7- 1), Eq. (7-2) are real and formed respectively as
−B0 − √∆1 −B0 + √∆1 λ1 = , λ2 = ∈ R, 2A0 2A0 −B1 − √∆2 −B1 + √∆2 μ1 = , μ2 = ∈ R, { 2A1 2A1 ⇒ the solutions of the characteristic equations Eq. (6)
l1 = φ (λ1, μ1), l2 = φ (λ2, μ2), l3 = φ (λ1, μ2), l4 = φ (λ2, μ1).
Theorem 2. Let aY + bY + cY = 0 is a second-order constant-coefficients linear homogeneous DE. And its characteristic equation is am + bm + c = 0. Then, the general solution of the WFC-ODE has the form
l1X l2X l3X l4X
Y = M1 e + M2e + M3e + M4e .
Proof:
we found that
aY + bY + c
A0Y0′′ + B0Y0′ + C0Y0 = 0, A0λ + B0λ + C0 = 0 (its discriminant ∆1), ⇔ { ⇒ {
the 2
A1Y1′′ + B1Y1′ + C1Y1 = 0, A1μ + B1μ + C1 = 0 (its discriminant ∆2).
characteristic equations
λ1 , λ2 ∈ R,
Since ∆1, ∆2> 0, the characteristic equations have the real different roots, {
μ1 , μ2 ∈ R.
Those roots directly influence the behavior of the solutions of the second-order corresponding Ordinary Differential Equations (ODEs) Eq. (5-1) and Eq. (5-2)
λ1X0 λ2X0
Y0 = He + Ie
⇒ { , H,I,G, and K, are arbitrary constants.
the general μ1X1 μ2X1
Y1 = Ge + Ke
solutions
Using φ , we get the general solution of WFC-DE Eq. (4), −1 −1 λ1X0 λ2X0 μ1X1 μ2X1
Y = φ (Y0, Y1) = φ (He + Ie , Ge + Ke )
λ1X0 λ2X0 μ1X1 μ2X1 μ1X1 μ2X1 λ1X0 λ2X0
= (He + Ie + Ge + Ke ) + J(Ge + Ke − He − Ie ).
2 2√t We can write this structure as y1or y2,
1 λ X μ X λ X μ X 1 μ X λ X
y = ((He 1 0 + Ge 1 1) + (Ie 2 0 + Ke 2 1)) + J((Ge 1 1 − He 1 0) +
2 2√t μ2X1 λ2X0
(Ke − Ie ))
−1 λ1X0 μ1X1 −1 λ2X0 μ2X1
= φ (He , Ge ) + φ (Ie , Ke )
−1 −1 λ1X0 μ1X1 −1 −1 λ2X0 μ2X1
= φ (H, G)φ (e , e ) + φ (I, K)φ (e , e )
−1 l1X −1 l2X
= φ (H, G) e + φ (I, K) e ,
or
1 λ X μ X λ X μ X 1 μ X λ X
y = ((He 1 0 + Ke 2 1 ) + (Ie 2 0 + Ge 1 1)) + J((Ge 1 1 − Ie 2 0) +
2 2√t μ2X1 λ1X0
(Ke − He ))
−1 λ1X0 μ2X1 −1 λ2X0 μ1X1
= φ (He , Ke ) + φ (Ie , Ge )
−1 −1 λ1X0 μ2X1 −1 −1 λ2X0 μ1X1
= φ (H, K)φ (e , e ) + φ (I, G)φ (e , e )
−1 l3X −1 l4X
= φ (H, K) e + φ (I, G) e .
The linearity of equation Eq. (4) means that if we have two solutions y1 and y2, then any linear combination of y1 and y2 is also a solution of this equation, i.e.
Y(X) = p1y1(X) + p2y2(X)
−1 l1X −1 l2X −1 l3X −1 l4X
= p1(φ (H, G) e + φ (I, K) e ) + p2(φ (H, K) e + φ (I, G) e )
l1X l2X l3X l4X
= M1 e + M2e + M3e + M4e ,
−1 −1 −1
is a solution for any choice of constants p1, p2, M1 = p1φ (H, G), M2 = p1φ (I, K), M3 = p2φ (H, K), M4 = p2φ (I, G).
Now, let’s verify that the solution Y(X) satisfies Eq. (4).
l1X l2X l3X l4X l1X l2X l3X
a(M1 e + M2e + M3e + M4e ) + b(M1 e + M2e + M3e +
l4X l1X l2X l3X l4X
M4e ) + c(M1 e + M2e + M3e + M4e )
l1X l2X l1X l2X l1X l2X
= [a(M1 e + M2e ) + b(M1 e + M2e ) + c(M1 e + M2e )]
l3X l4X l3X l4X l3X l4X
+ [a(M3e + M4e ) + b(M3e + M4e ) + c(M3e + M4e )]
−1 l1X −1 l2X −1 l1X
= p1[a( φ (H, G) e + φ (I, K) e ) + b( φ (H, G) e +
−1 l2X −1 l1X −1 l2X
φ (I, K) e ) + c( φ (H, G) e + φ (I, K) e )]
−1 l3X −1 l4X −1 l3X −1 l4X
+p2[a(φ (H, K) e + φ (I, G) e ) + b(φ (H, K) e + φ (I, G) e ) +
−1 l3X −1 l4X
c(φ (H, K) e + φ (I, G) e )]
′′ ′ ′′ ′
= p1[ay1 + by1 + cy1] + p2[ay2 + by2 + cy2] = p1 (0) + p2(0) (because y1 and y2 are solutions of Eq. (4))
= 0,
−1 −1 −1 −1 −1
where m = φ (λ, μ), l1 = φ (λ1, μ1), l2 = φ (λ2, μ2), l3 = φ (λ1, μ2), l4 = φ (λ2, μ1) ∈ FJ.
a = a0 + a1 J = φ (A0, A1) = φ (a0 − a1 √t, a0 + a1 √t), A0 = a0 − a1 √t ∈ R, A1 = a0 + a1 √t ∈ R, b = b0 + b1 J = φ (B0, B1) = φ (b0 − b1 √t, b0 + b1 √t), B0 = b0 − b1 √t ∈ R, B1 = b0 + b1 √t ∈ R,
c = c0 + c1 J = φ (C0, C1) = φ (c0 − c1 √t, c0 + c1 √t), C0 = c0 − c1 √t ∈ R, C1 = c0 + c1 √t ∈ R.
′′ ′ 45
Example 1. (3 + 2J) Y + J Y − (15 + J)Y = 0.
Its characteristic equation is
(3 + 2J)m + Jm − (15 + J) = 0.
The two related DEs of the WFC-DE are
′′ ′ ′′ ′
(3 − 2√t)Y0 − √t Y0 − (15 − √t ) Y0 = 0 2Y0 − Y0 − Y0 = 0, { ⇒ { 45 for 1 105
′′ ′ ′′ ′
(3 + 2√t)Y + √t Y − (15 + √t ) Y = 0 1 4Y + Y − Y = 0,
1 1 1 2 1 1 1
2 J =t= 2 4 2λ − λ − = 0 (its discriminant ∆1= > 0) ⇒ { ⇒
the characteristic 2 1 105 1681
4μ + μ − = 0 (its discriminant ∆2= > 0)
equations 2 4 4
λ1 = − , λ2 = ∈ R, μ1 = − , μ2 = ∈ R,
− X0 X0,
Y = He 4 + Ie2
the general − X X ,
Y1 = Ge + Ke
solutions
H, I, G, K are arbitrary constants.
Thus
−1 −1 5 11 1 5 11 11 5 3
l1 = φ (λ1, μ1) = φ (− , − ) = (− − ) + J (− + ) = −2 − J,
4 4 2 4 4 4 4 2 −1 −1 3 5 1 3 5 5 3
l2 = φ (λ2, μ2) = φ ( , ) = ( + ) + J ( − ) = 2 + J,
2 2 2 2 2 2 2 −1 −1 5 5 1 5 5 5 5 5 15
l3 = φ (λ1, μ2) = φ (− , ) = (− + ) + J ( + ) = + J,
4 2 2 4 2 2 4 8 4 −1 −1 3 11 1 3 11 11 3 5 17
l4 = φ (λ2, μ1) = φ ( , − ) = ( − ) + J (− − ) = − − J.
2 4 2 2 4 4 2 8 4
Then, the general solution of WFC-DE
3 5 15 5 17
(−2− j)x (2+j)x ( + j)x (− − j)x
y = m e 2 + m e + m e 8 4 + m e 8 4 , where p1, p2, M1 = p1φ (H, G), M2 = p1φ (I, K), M3 = p2φ (H, K), M4 = p2φ (I, G) are arbitrary constants.
Now, we will study the simple form of this equation with a real constant.
The real constant-coefficient second-order linear homogeneous WFC-DE is adopted as
aY + bY + cY = 0, a, b, c ∈ R.
Its characteristic equation am + bm + c = 0, where its discriminant ∆= b − 4ac and its roots are l1, l2, l3, l4 ∈ FJ.
Since the coefficients of the quadratic WFC equation Eq. (6) are real a, b and c ∈ R, the second-order DEs Eq. (5-1), Eq. (5-2) have the following characteristic equations Eq. (8-1), and Eq. (8-2):
aY0′′ + bY0′ + cY0 = 0, aλ + bλ + c = 0, (8 − 1)
{ ⇒ { (8)
the characteristic 2
aY1′′ + bY1′ + cY1 = 0, aμ + bμ + c = 0. (8 − 2)
equations
Eq. (8-1), and Eq. (8-2) have the discriminants ∆1= ∆2= ∆= b − 4ac > 0.
Then, we will get the real roots of Eq. (8-1), and Eq. (8-2):
−b−√∆ −b+√∆
λ1 = μ1 = λ2 = μ2 = ,
2a 2a
and
−b+√∆
λ2 = μ2 = .
Example 2.
Y − 4Y + 3Y = 0. (9)
The characteristic equation of WFC-DE Eq. (9) is
m − 4m + 3 = 0. (10)
The two related DEs of Eq. (9) are
′′ 2
Y0 − 4Y0′ + 3Y0 = 0, λ − 4λ + 3 = 0 { ′′ ⇒ { 2 ⇒ ∆1= ∆2> 0 ⇒ Y − 4Y ′ + 3Y = 0, the characteristic μ − 4μ + 3 = 0
equations
λ1 = 3 , λ2 = 1 ∈ R, μ1 = 3 , μ2 = 1 ∈ R,
3X0 X0
Y0 = He + Ie ,
The general 3X1 X1
Y1 = Ge + Ke ,
solutions
H, I, G, K are arbitrary constants.
Then the roots of Eq. (10) are
−1 −1 1 1
l1 = φ (λ1, μ1) = φ (3,3) = (3 + 3) + J(3 − 3) = 3,
2 2√t −1 −1 1 1
l2 = φ (λ2, μ2) = φ (1,1) = (1 + 1) + J(1 − 1) = 1,
2 2√t −1 −1 1 1 1
l3 = φ (λ1, μ2) = φ (3,1) = (3 + 1) + J(1 − 3) = 2 − J,
2 2√t √t −1 −1 1 1 1
l4 = φ (λ2, μ1) = φ (3,1) = (1 + 3) + J(3 − 1) = 2 + J.
2 2√t √t
Then, the general solution of Eq. (9) (2− J)X (2+ J)X 3X X √t √t
Y = M1 e + M2e + M3e + M4e , where p1, p2, M1 = p1φ (H, G), M2 = p1φ (I, K), M3 = p2φ (H, K), M4 = p2φ (I, G) are arbitrary constants.
We know that arbitrary constants in any general solution of a DE are determined by specifying initial conditions. For a second-order DE in FJ, we should have two WFC initial conditions to determine the two arbitrary constants p1 and p2, which are equivalent to four initial conditions in R.
Definition 14. The second-order WFC-Initial Value Problem consists of finding the solution of a WFC-ODE given two initial conditions.
ℱ(X, Y, Y , Y′′) = 0,
{ , α ≤ X ≤ β, (11)
Y(α) = ν, Y (α) = ω, (the initial conditions) where a fixed WFC value (α, ν), (α, ω) ∈ FJ × FJ are given.
Using φ function for Eq. (11), we get
−1 ′ ′′ ′′
φ (ℱ1(X0, Y0, Y0 , Y0 ), ℱ2(X1, Y1, Y1′, Y1 )) = 0 φ (Y0(α0), Y1(α1)) = φ (ν0, ν1), φ (Y0′(α0), Y1′(α1)) = φ (ω0, ω1)
′ ′′
ℱ1(X0, Y0, Y0 , Y0 ) = 0, { ( ) α0 ≤ X0 ≤ β0, (12) ⇒ its solution is Y0(X0),
′ α0
Y0(α0) = ν0, Y = ω0, ℱ2(X1, Y1, Y1′, Y1 ) = 0, { ′(α ) α1 ≤ X1 ≤ β1, (13) ⇒ its solution is Y1(X1).
{ Y1(α1) = ν1, Y = ω1, Therefore, the particular solution of the second-order WFC-IVP Eq. (11) is a structure of the related solutions of the second-order initial-value problems Eq. (12) and Eq. (13),
Y = φ (Y0(X0), Y1(X1)), where
X = φ (X0, X1) ∈ [α, β] ⊆ FJ, X0 = x0 − x1 √t ∈ [α0, β0] ⊆ R, X1 = x0 + x1 √t ∈ [α1, β1] ⊆ R, Y = φ (Y0, Y1), Y0 = y0 − y1 √t, Y1 = y0 + y1 √t, α = φ (α0, α1), β = φ (β0, β1), ν = φ (ν0, ν1), ω = φ (ω0, ω1).
We get the particular solution when we substitute the initial conditions in the general solution.
Definition 15. The constant-coefficient second-order linear homogeneous WFC initial value problem (WFC-IVP) is formed as aY + bY + cY = 0,
{ ′ α ≤ X ≤ β, (12)
Y(α) = ν, Y (α) = ω, a = φ (A0, A1), A0 = a0 − a1√t, A1 = a0 + a1√t, b = φ (B0, B1), B0 = b0 − b1√t, B1 = b0 + b1√t, c = φ (C0, C1), C0 = c0 − c1√t, C1 = c0 + c1√t.
Remark 2. Using φ, we find that Eq. (12) is equivalent to two initial value problems of the same type in R, A0Y0′′ + B0Y0′ + C0Y0 = 0, { α0 ≤ X0 ≤ β0, φ Y0(α0) = ν0, Y0′(α0) = ω0,
Eq. (12) ⇒ ′′ ′ (13)
A1Y1 + B1Y1 + C1Y1 = 0, { ′(α ) α1 ≤ X1 ≤ β1, Y1(α1) = ν1, Y = ω1, where p1, p2, M1 = p1φ (H, G), M2 = p1φ (I, K), M3 = p2φ (H, K), M4 = p2φ (I, G) are determined constants.
Remark 3. Steps to get the particular solution of the second-order WFC-IVP Eq. (12):
1. Solving the second-order initial-value problems IVPs Eq. (13), we get the specific solutions.
3X0 X0
Y = H ̃e + Ĩe ,
3X1 X1,
Y = G ̃e + ̃ Ke H ̃, ̃ I, G ̃, K ̃ are determined constants.
2. Making a structure of the related particular solutions Y0 and Y1 of the first step.
−1 ̃ ̃ l1X −1 ̃ ̃ l2X −1 ̃ ̃ l3X −1 ̃ ̃ l4X
3. Y(X) = p1(φ (H, G) e + φ (I, K) e ) + p2(φ (H, K) e + φ (I, G) e ).
4. Substituting the initial conditions Y(α) = ν, Y (α) = ω to have two WFC linear equations.
5. Solving the two WFC linear equations to get values of p1, p2.
6. Substituting p1 and p2 in the structure of the second step.
In other words, to find the arbitrary constants p_1 〖,p〗_2,〖〖M_1=p_1 φ〗^(-1) (H,G),M_2=p_1 φ ^(-1) (I,K),M〗_3=p_2 φ^(-1) (H,K),M_4=p_2 φ^(-1) (I,G), we have to solve the six following equations: Y_0 (α_0 )=ν_0,Y_0^'(α_0 ) =ω_0 (which give the values of H and I ).
Y_1 (α_1 )=ν_1,Y_1'(α_1 )=ω_1 (which give the values of G and K ).
Then Y(α)=ν,Y^' (α)=ω (which give the values of p_1and p_2 ).
Example 3. Second-order linear Homogeneous WFC-IVP with WFC Constant Coefficients.
(3 + 2J) Y + J Y − (15 + J)Y = 0,
{ 2 0 ≤ X ≤ 1 + J. (14)
Y(0) = 0, Y (0) = 1 + J, This Initial Value Problem (IVP) is equivalent to the two following IVPs:
(3 − 2√t)Y0 − √t Y0 − (15 − √t ) Y0 = 0, { 2 0 ≤ X0 ≤ 1 − √t
′(0)
φ Y (0) = 0, Y = 1 − √t, Eq. (14) ⇒ (3 + 2√t)Y1 + √t Y1 − (15 + √t ) Y1 = 0, { 2 0 ≤ X1 ≤ 1 + √t { Y1(0) = 0, Y1(0) = 1 + √t, 2Y0 − Y0 − Y0 = 0, { 0 ≤ X0 ≤
⇒ ( ) 1 2 (15)
for Y (0) = 0, Y = ,
J =t= 4Y1 + Y1 − Y1 = 0,
⇒ { 0 ≤ X1 ≤ , (16)
for ′(0) 3 2 1 Y (0) = 0, Y = , J =t= 2 5 3 5 5 3 3
− X0 X0 ′ − X0 X0
Y = He 4 + Ie2 , Y = − He 4 + Ie2 , the general 11 5 11 11 5 5
− X1 X1 ′ − X1 X1
solutions Y = Ge 4 + Ke2 , Y = − Ge 4 + Ke2 .
(Example1) 4 2 After substituting the corresponding initial conditions to find the values of H, I, G, K:
0 = H + I, H ̃ = −2 5 2 3 11 − X0 X0 {1 5 3 ⇒ { ⇒ Y = e 4 + e2 , = − H + I, solving this system 2 the particular 11 11 2 4 2 ̃ I =
solution
11 for IVP Eq. (15) 0 = G + K, G ̃ = −6 11 6 5 21 − X1 X1 {3 11 5 ⇒ { ⇒ Y = e 4 + e2 , = − G + K, solving this system 6 the particular 21 21 2 4 2 K ̃ =
solution
{ 21 for IVP Eq. (16)
−1 −1 5 11 1 5 11 11 5 3
l1 = φ (λ1, μ1) = φ (− , − ) = (− − ) + J (− + ) = −2 − J,
4 4 2 4 4 4 4 2 −1 −1 3 5 1 3 5 5 3
l2 = φ (λ2, μ2) = φ ( , ) = ( + ) + J ( − ) = 2 + J,
2 2 2 2 2 2 2 −1 −1 5 5 1 5 5 5 5 5 15
l3 = φ (λ1, μ2) = φ (− , ) = (− + ) + J ( + ) = + J,
4 2 2 4 2 2 4 8 4 −1 −1 3 11 1 3 11 11 3 5 17
l4 = φ (λ2, μ1) = φ ( , − ) = ( − ) + J (− − ) = − − J,
2 4 2 2 4 4 2 8 4 −1 −2 −6 1 −2 6 −6 −2 18 8
( , ) = ( − ) + ( − ) = − − ,
11 21 2 11 21 21 11 77 77 −1 2 6 1 2 6 6 2 18 8
φ ( , ) = ( + ) + J ( − ) = + J,
11 21 2 11 21 21 11 77 77 −1 −2 6 1 −2 6 6 2 4 36
φ ( , ) = ( + ) + J ( + ) = + J,
11 21 2 11 21 21 11 77 77 −1 2 −6 1 2 −6 −6 2 4 36
φ ( , ) = ( + ) + J ( − ) = − − J.
11 21 2 11 21 21 11 77 77
Then, the general solution of Eq. (14) (from Example 1)
3 5 15 5 17
(−2− J)X (2+J)X ( + J)X (− − J)X
Y(X) = M e 2 + M e + M e 8 4 + M e 8 4
−1 l1X −1 l2X −1 l3X −1 l4X
= p (φ (H ̃, G ̃) e + φ (̃ I, K ̃) e ) + p (φ (H ̃, K ̃) e + φ (̃ I, G ̃) e )
−1 −2 −6 l X −1 2 6 l X −1 −2 6 l X
= p (φ ( , ) e 1 + φ ( , ) e 2 ) + p (φ ( , ) e 3 +
11 21 11 21 11 21 −1 2 −6 l X
φ ( , ) e 4 )
18 8 (−2− J)X 18 8 (2+J) X 4 36 ( + J)X
= p ((− − J) e 2 + ( + J) e ) + p (( + J) e 8 4 +
77 77 77 77 77 77 4 36 (− − J) X
(− − J) e 8 4 ), ′ 3 18 8 (−2− J)X 18 8 (2+J) X
Y (X) = p ((−2 − J) (− − J) e 2 + (2 + J) ( + J) e ) +
2 77 77 77 77 5 15 4 36 ( + J)X 5 17 4 36 (− − J) X
p (( + J) ( + J) e 8 4 + (− − J) (− − J) e 8 4 ).
8 4 77 77 8 4 77 77
Substituting the initial conditions: Y(0) = 0 in the general solution Y, and Y (0) = 1 + J in the first derivative of the general solution Y′, we get
18 8 18 8 4 36 4 36 0 = p1 ((− − J) + ( + J)) + p2 (( + J) + (− − J) ), 77 77 77 77 77 77 77 77 3 18 8 18 8 5 15 4 36 5 17 4 36 1 + J = p1 ((2 + J)( + J) + (2 + J) ( + J)) + p2 (( + J) ( + J) + ( + J)( + J) ) { 2 77 77 77 77 8 4 77 77 8 4 77 77
0 = 0p1 + 0p2 ⇒ { 18 8 5 4 36 10 32 .
1 + J = ( + J) (4 + J) p1 + ( + J) ( + J)p2 77 77 2 77 77 8 4
We find that those two equations are satisfied when
[1 + J − ( + J) ( + J) p2] p1 = = P(p2).
[( + J) (4 + J)] Then the solution of Eq. (15) is in terms of p2 for any p1:
18 8 (−2− J)X 18 8 (2+J) X 4
Y(X) = P(p ) ((− − J) e 2 + ( + J) e ) + p (( +
77 77 77 77 77 36 ( + J)X 4 36 (− − J) X
J) e 8 4 + (− − J) e 8 4 ).
Now, we will solve a simple form, ‘the real constant-coefficient second-order linear homogeneous WFC ordinary DE.' Example 4. Second-order linear Homogeneous WFC-IVP with Real Constant Coefficients.
Y − 4Y + 3Y = 0,
{ 0 ≤ X ≤ 10. (17)
Y(0) = 3, Y′(0) = 5, This IVP is equivalent to the two following IVPs:
Y0 − 4Y0′ + 3Y0 = 0, { 0 ≤ X0 ≤ 10, φ Y (0) = 3, Y ′(0) = 5, Eq. (17) ⇒ ′′ ′ Y1 − 4Y1 + 3Y1 = 0, { 0 ≤ X1 ≤ 10, { Y1(0) = 3, Y1′(0) = 5,
3X0 X0 3X0 X0
Y0 = He + Ie , Y0′ = 3He + Ie ,
the general 3X1 X1 ′ 3X1 X1
Y1 = Ge + Ke , Y1 = 3Ge + Ke .
solutions from Example 2
After substituting the corresponding initial conditions, 3 = H + I H ̃ = 1 3X X , { ⟹ { ⇒ Y = e 0 + 2e 0 5 = 3H + I ̃ I = 2 3 = G + K G ̃ = 1 3X X .
{ ⟹ { ⇒ Y = e 1 + 2e 1 5 = 3G + K K ̃ = 2 The general solution of Eq. (17), −1 ̃ ̃ l1X −1 ̃ ̃ l2X −1 ̃ ̃ l3X −1 ̃ ̃ l4X Y(X) = p1(φ (H, G) e + φ (I, K) e ) + p2(φ (H, K) e + φ (I, G) e )
−1 l1X −1 l2X −1 l3X −1 l4X
= p1(φ (1, 1) e + φ (2,2) e ) + p2(φ (1, 2) e + φ (2,1) e )
−1 1 1
φ (1,1) = (1 + 1) + J(1 − 1) = 1,
2 2√t
φ (2,2) = 2,
−1 1 1 3 1
φ (1,2) = (1 + 2) + J(2 − 1) = + J,
2 2√t 2 2√t −1 1 1 3 1
φ (2,1) = (2 + 1) + J(1 − 2) = − J,
2 2√t 2 2√t
l1 = φ (λ1, μ1) = 3, l2 = φ (λ2, μ2) = 1,
−1 1
l3 = φ (λ1, μ2) = 2 − J,
−1 1
l4 = φ (λ2, μ1) = 2 + J.
Then
3 1 (2− J)X 3 1 (2+ J)X
3X X √t √t
Y(X) = p1( e + 2 e ) + p2 [( + J) e + ( − J) e ], 2 2√t 2 2√t
′ 3X X 1 3 1 (2− J)X 1 3
Y (X) = p ( 3e + 2 e ) + p [(2 − J) ( + J) e √t + (2 + J) ( −
√t 2 2√t √t 2 1 (2+ J)X
J) e √t ].
Substituting the initial conditions: Y(0) = 1 in the general solution Y, and Y′(0) = 1 in the first derivative of the general solution Y′, we get
3 = p1( 1 + 2) + p2 ( + J + − J), 2 2√t 2 2√t 1 3 1 1 3 1 5 = p1( 3 + 2) + p2 [(2 − J) ( + J) + (2 + J) ( − J) ] , { √t 2 2√t √t 2 2√t
3 = 3p1 + 3p2 ⇒ { 1 3 1 1 3 1 5 = 5p1 + p2 [3 − J − J − + 3 − J + J − ] √t 2√t 2 √t 2√t 2 1 = p1 + p2 ⇒ { 2 5 = 5p1 + (5 − J) p2.
Solving these two equations
3X X
Y(X) = e + 2 e .
In this paper, we have defined the second-order WFC-DEs. We have focused on solving the second-order constant-coefficient linear homogeneous WFC DE using its characteristic equation, where we have found its general solution. We have solved the related second-order constant-coefficient linear homogeneous IVPs by adding initial conditions. We have explained all examples for this type of DEs and IVPs with WFC constant coefficients and real constant coefficients.
Conceptualization and Methodology, L. R., writing-reviewing and editing L. R., S. M. and M. A. All authors have read and agreed to the published version of the manuscript.
All authors have provided their consent for the publication of this manuscript.
This article does not involve studies with human participants or animals conducted by any authors.
The authors declare that no external funding or support was received for the research.
All data supporting the reported findings in this research paper are provided within the manuscript.
The authors declare no conflict of interest.