On the Foundation of Solving a First-Order Weak Fuzzy Complex Initial Value Problem (WFC-IVP)RazoukLamaDepartment of Mathematics, Tartous University, Tartous, Syria.
The aim of this paper is to introduce for the first time the basic concept of the first-order Weak Fuzzy Complex-Initial Value Problems (WFC-IVPs). We find, using a special isomorphic transformation function, that solving a WFC-IVP is equivalent to solving two classical real IVPs with respect to their own real variables. Thus, we study “the existence and uniqueness”, “the stability”, and “the well-posedness” associated with the WFC-IVPs in terms of definitions, lemmas, and theorems. Then, we get the approximate solutions of a WFC-IVP by stable and convergent numerical methods. One of the most famous and simple methods to solve IVPs is Euler’s method, which is discussed for well-posed WFC-IVPs. However, we focus on a stable linear model WFC-IVP with real coefficients and real initial values, and we further investigate the properties of the results. Additionally, we present an example with tables and diagrams of its numerical solutions and absolute errors by Python to clarify how Euler’s algorithm works.
Weak fuzzy complex setWeak fuzzy complex functionsInitial value problemEuler’s methodIntroduction
Over centuries, the extensions of number sets have been created out of necessity often to solve specific problems. Researchers have been very interested in expanding the set of real numbers, where the developed sets have been defined as two-dimensional generalizations of real numbers 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}.
Recently, in a similar way to build sets of Complex, Dual, Neutrosophic, and Split-Complex numbers, the Weak Fuzzy Complex (WFC) set {u + v J; u, v ∈ R; J = t ∈ [0, 1]} was born in 2023 [1]. It is a new generalization of classical real numbers set with fuzzy operators J ∉ R. Hence, they introduced the solutions of linear and quadratic WFC equations [1], the WFC vector space [2], the inner products defined over this vector space [3], and linear systems by applying WFC matrices [4].
The linear Diophantine equation in two WFC variables was studied in [5]. Also, the non-linear WFC Diophantine equations have been solved in [6], [7] with the concepts of the WFC and anti-WFC Pythagoras triples and Pythagoras quadruples. Additionally, Abobala [8] defined some main notions in number theory, like division, ordering, and units in the set of WFC integers.
One important application of the WFC numbers is ‘A-Curves’ which display the geometrical characterization of the solutions for some vectorial equations defined by Euclidean norms [9]. Also, many new semi-module isomorphisms gave a full classification of A-Curves related to the WFC ring [10].
In computer [11], Python introduced WFC numbers and their main arithmetic operations.
In addition, Razouk et al. [12] created a special isomorphism transformation function that works between the WFC set and the Real number set, and they defined the foundation of WFC functions. Thus, using this transformation function, Edduweh et al. [13] provided the basic concepts of WFC Ordinary Differential Equations (WFC-ODEs) and found the solutions of some types of first-order first-degree separable, exact, and linear WFC-ODEs.
While Hatamleh [14] studied the novel topological space generated by WFC intervals based on the partial ordered ring structure of WFC numbers.
Otherwise, we know that an Initial Value Problems (IVPs) play an essential role in modeling World phenomena. Thus, the IVPs have predictive capability which is vital in many applications, including weather forecasting [15], financial modeling [16], and control systems [17].
That motivates us to study an important research gap about the IVPs for WFC-ODEs. In this work, we consider a numerical method for solving a stable model of first-order first-degree WFC-IVP.
Basic notions about WFC elements are mentioned in Section 2, where we introduce the Mean Value Theorem in FJ. Accordingly, we define the standard form of first-order WFC-IVP in Section 3, with the related fundamental issues: existence, uniqueness, stability, and well-posedness. Then, we show the methodology of the numerical techniques in Section 4 to obtain approximate solutions to the well-posed first-order WFC-IVP. In Section 5, we address Taylor series Methods with the numerical errors that can be formed. We explain “Euler’s method” which is considered the basis of methods for solving the IVP, and infer its properties for our WFC-IVP. A linear stable model problem will be considered in Section 6, where we apply Euler’s algorithm to it and prove the similar results between the original IVP in FJ and its two related IVPs in R. After that, a simple example appears with tables, and diagrams of solutions and errors obtained using Python language (the instructions executed by Python are included in Appendix) to illustrate the validity of our approach.
Preliminaries
The subsequent definitions will help clarify the various elements at play in our analysis, thereby facilitating a deeper understanding of the results we present for solving IVPs in a WFC variable [10], [18].
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).
To simplify, we transform from elements in FJ to real elements using a special transformation function, and then go back to FJ by its inverse [12].
Definition 2. Let φ be the transformation function from FJ to R × R, which we define as follows:
φ: FJ ↦ R × R, φ(x0 + x1 J) = (x0 + x1 (−√t), x0 + x1(+√t)) = (x0 − x1 √t, x0 + x1√t), 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}.
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, f2 are continuous on R.
II. f is differentiable on FJ if and only if f1, f2 are differentiable on R, with respect to their own variables. III. f is integrable on FJ if and only if f1, 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 )).
Remark 1. The Lipschitz condition equation is equivalent to the two Lipschitz conditions (due to Definition
|ℱ1(X0, Z0) − ℱ1(X0, S0)| ≤ L0|Z0 − S0|, |ℱ2(X1, Z1) − ℱ2(X1, S1)| ≤ L1|Z1 − S1|, where (X0, Z0) and (X0, S0) are in D1 = {(X0, Y0); X0 ∈ I1 = [A0, B0] ⊆ R, −∞ < Y0 < +∞}, (X1, Z1) and (X1, S1) are in D2 = {(X1, Y1); X1 ∈ I2 = [A1, B1] ⊆ R, −∞ < Y1 < +∞}, and L0, L1 > 0 are Lipschitz constants for ℱ1, ℱ2, respectively.
Theorem 1 (The Mean Value Theorem on ). If f is a WFC continuous function on an interval [A, B] ⊆ FJ and differentiable on (A, B), then a WFC number C in (A, B) exists with f(B) − f(A)
′(C)
f = , B − A where Y = φ (Y0, Y1), Y0 = y0 − y1√t, Y1 = y0 + y1√t, X = φ (X0, X1), X0 = x0 − x1 √t, X1 = x0 + x1 √t, 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.
Proof: f is a WFC function, then we have φ(f(X)) = (f1(x0 − x1 √t ), f2( x0 + x1√t )); f1, f2: R ↦ R, when f is continuous on [A, B] ⊆ FJ, then f1, f2 are continuous on [A0, B0], [A1, B1] ⊆ R, respectively.
Also, f is differentiable on (A, B) ⊆ FJ, then f1, f2 are differentiable on (A0, B0), (A1, B1) ⊆ R, with respect to their own variables.
Depending on the mean value theorem in R [9] numbers C1, C2 in (A0, B0), (A1, B1) exist with:
f1(B0) − f1(A0) ′(C ) f2(B1) − f2(A1) f1′(C1) = and f = .
The Initial-Value Problem for a first-Order Differential Equation
in
The IVP for first-order differential equations serves as a crucial tool in understanding dynamic systems across various disciplines. We first establish key definitions that will serve as the foundation for our subsequent discussions.
Definition 12. The WFC- IVP (WFC-Cauchy problem) consists of finding the solution of a WFC-ODE given suitable initial conditions. The standard form of the first-order WFC-ODE that has been adopted is Y′ = ℱ(X, Y), dY = ℱ(X, Y)dX,
{ { (1)
Y(A) = C (the initial condition), Y(A) = C, where a WFC function ℱ(X, Y(X)) defined on a set D, and a fixed WFC value (A, C) ∈ D be given. A function Y(X) is sought that is differentiable in an interval I with A ∈ I, and D = {(X, Y); X ∈ I; I = [A, B] ⊆ FJ, −∞ < Y0, Y1 < +∞} ⊆ FJ × FJ; D = φ (D1, D2), D1 = {(X0, Y0); X0 ∈ I1 = [A0, B0] ⊆ R, −∞ < Y0 < +∞}, D2 = {(X1, Y1); X1 ∈ I2 = [A1, B1] ⊆ R, −∞ < Y1 < +∞} X = φ (X0, X1), X0 = x0 − x1 √t, X1 = x0 + x1 √t, Y = φ (Y0, Y1), Y0 = y0 − y1 √t, Y1 = y0 + y1 √t, 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.
Definition 13. Suppose that ℱ1 , ℱ2 and ℱ = φ (ℱ1, ℱ2) are differentiable in I1 = [A0, B0], I2 = [A1, B1] and I = [A, B] ⊆ FJ with respect to their own variables X0, X1 and X, respectively. A function Y(X): I → FJ is called the solution to a WFC-IVP Eq. (1) and given as follows:
Y = φ (Y0(X0), Y1(X1)) = (Y0(X0) + Y1(X1)) + J(Y1(X1) − Y0(X0)), 2 2√t where Y0(X0) and Y1(X1) are the corresponding solutions of Eq. (2) and Eq. (3), respectively dY0 = ℱ1(X0, Y0)dX0
{ ; A0 ≤ X0 ≤ B0, (2)
Y0(A0) = C0 dY1 = ℱ2(X1, Y1)dX1
{ ; A1 ≤ X1 ≤ B1. (3)
Y1(A1) = C1 Proof: using the function φ for Eq. (1), we get
φ (Y0(A0), Y1(A1)) = φ (C0, C1), dY0 = ℱ1(X0, Y0)dX0 { ; A0 ≤ X0 ≤ B0, Y0(A0) = C0 dY1 = ℱ2(X1, Y1)dX1 { ; A1 ≤ X1 ≤ B1, { Y1(A1) = C1 So, we can say that finding the solution for the WFC-IVP Eq. (1) is equivalent to solve the two related Cauchy problems in R, Eq. (2) and Eq. (3) by φ, and the solution has the form
In advance of trying to solve a WFC-IVP, it is desirable to know whether a solution exists, and when it does, whether it is the only solution to the problem.
The Existence and Uniqueness
Theorem 2 (The Existence and Uniqueness theorem). Suppose ℱ(X, Y) is continuous on D = {(X, Y); X ∈ I; I = [A, B] ⊆ FJ, −∞ < Y0, Y1 < +∞}. If ℱ satisfies a Lipschitz condition on D in the variable Y, then the WFC-IVP Eq. (1) has a unique solution Y(X) for X ∈ I.
Proof: we know that solving WFC-IVP Eq. (1) is equivalent to solve Eq. (2) and Eq. (3) in R. And I. Since ℱ(X, Y) is continuous on D if and only if ℱ1, ℱ2 are continuous on D1, D2 ⊆ R.
II. Since ℱ(X, Y) satisfies a Lipschitz condition on D if and only if ℱ1, ℱ2 satisfy Lipschitz conditions on D1, D2. Then each IVP Eq. (2) and Eq. (3) has a unique solution Y0(X0), Y1(X1), respectively (The existence and uniqueness theorem in R [9], [19]).
It means that Y(X)-the solution of Eq. (1), is a unique structure of two unique solutions in R:
2 2√t Consequently, we can assume the following lemma.
Lemma 1. “The existence of the unique solution Y(X) ∈ FJ of Eq. (1)” is equivalent to “the existence of the two unique solutions Y0(X0) and Y1(X1) ∈ R of Eq. (2) and Eq. (3), respectively”
Example 1.
Y = 1 + Xsin(XY),
{ (4)
Y(0) = 0.
To show that there is a unique solution to the IVP Eq. (4) where 0 ≤ X ≤ 2 + 2J, holding X constant and applying the Mean Value Theorem to the function ℱ(X, Y) = 1 + Xsin(XY).
We find that when Z1 < Z2, a number ξ in the interval (Z1, Z2) exists with ℱ(X, Z1) − ℱ(X, Z2) ∂ = ℱ(X, ξ) = X cos(ξ X).
Z1 − Z2 ∂Y Thus, |F(X, Z1) − ℱ(X, Z2)| = |Z1 − Z2||X cos(ξ X)| ≤ (2 + 2J) |Z1 − Z2|, and ℱ satisfies a Lipschitz condition in Y with a Lipschitz constant L = (2 + 2J) .
Moreover, ℱ(X, Y) is continuous when 0 ≤ X ≤ 2 + 2J and −∞ < Y0, Y1 < +∞ , so Theorem 2 implies that a unique solution exists to the IVP Eq. (4).
Now, we can move to the next important considerations.
Stability
In view of the stability analysis of the WFC-IVP Eq. (1), by perturbing both the initial value C and the function ℱ, we consider the perturbed problem:
dZ = ℱ(X, Z(X)) + δ(X),
{ dX (5)
Z(A) = C + δC, where δ(X) is a WFC continuous function for all X ∈ [A, B] and the constant δC ∈ FJ.
The sensitivity of the solution Z to the perturbations will be characterized in the following.
Definition 14. Let I = [A, B] ⊆ FJ be a bounded set. The WFC Cauchy problem Eq. (1) is stable in I if, for any perturbation (δC, δ(X)) satisfying |δC| < E, |δ(X)| < E, ∀X ∈ I .
With E > 0 suffciently small to guarantee that the solution to the perturbed problem Eq. (5) does exist, and ∃k > 0 constant depends in general on initial problem data A, C and ℱ then |Z(X) − Y(X)| < kE.
Lemma 2. The WFC-IVP Eq. (1) is a stable problem if its two corresponding IVPs Eq. (2) and Eq. (3) are stable.
Proof: the stability of Eq. (2) and Eq. (3) means that:
There exist constants ε0, ε1 > 0, and k0, k1 > 0 such that for any ε, ε′, with ε0 > ε > 0 and ε1 > ε′ > 0, whenever δ0(X0), δ1(X1) are continuous with |δ0(X0)| < ε for all X0 ∈ [A0, B0] and |δ1(X1)| < ε′ for all X1 ∈ [A1, B1], and when |δ00| < ε & |δ11| < ε′, the initial-value problems:
there exist a WFC constant Ec = φ (ε0, ε1) > 0 and k = φ (k0, k1) > 0 such that for any E = φ (ε, ε′), with Ec > E > 0, whenever δ(X) = φ (δ0(X0), δ1(X1)) is continuous with
−1 −1
|δ(X)| = φ (|δ0(X0)|, |δ1(X1)|) < E for all X ∈ [A, B], and when |δC| = φ (|δ00|, |δ11|) < E, the IVP (where δC = φ (δ00, δ11)) dZ = ℱ(X, Z) + δ(X), { dX Z(A) = C + δC, has a unique solution Z(X) = φ (Z0(X0) , Z1(X1)), that satisfies
−1 −1 ′
φ (|Z0(X0) − Y0(X0)|, |Z1(X1) − Y1(X1)|) < φ (k0ε, k1ε ) ⇒ |Z(X) − Y(X)| < kE, then, the WFC Cauchy problem Eq. (1) is stable.
Well-Posedness
It is significant to know whether small perturbations, in the statement of the problem introduce parallelly small changes in the solution, that the effects of errors are bounded when the problem is well-posed.
Definition 15. The WFC-IVP dY = ℱ(X, Y)dX { ; A ≤ X ≤ B Y(A) = C is said to be a well-posed problem if, I. Its unique solution exists (existence and uniqueness).
II. Its solution depends continuously on the initial condition (stability).
Lemma 3. The WFC-IVP Eq. (1) is a well-posed problem if its two corresponding IVPs Eq. (2) and Eq. (3) are well-posed.
Proof: the well-posedness of Eq. (2) and Eq. (3) intends [9]:
I. The existence of the two unique solutions Y0(X0) and Y1(X1) ∈ R of Eq. (2) and Eq. (3), respectively ⇒ the existence of the unique solution Y(X) ∈ FJ of Eq. (1) (by Lemma 1).
II. The IVPs Eq. (2) and Eq. (3) are stable ⇒ the WFC-IVP Eq. (1) is stable (by Lemma 2).
III. which means that the WFC-IVP Eq. (1) is well-posed.
Theorem 3 (Well-Posed WFC-IVP). Suppose D = {(X, Y); X ∈ I; I = [A, B] ⊆ FJ, −∞ < Y0, Y1 < +∞}. If ℱ(X, Y) is continuous, and satisfies a Lipschitz condition on the set D in the variable Y, then the WFC-IVP Eq. (1) is a well-posed.
Proof: since ℱ(X, Y) is continuous, and satisfies a Lipschitz condition on the set D, then I. ℱ1, ℱ2 are continuous on D1, D2 ⊆ R.
II. ℱ1, ℱ2 satisfy Lipschitz conditions on D1, D2 in their own variables Y0(X0), Y1(X1), respectively.
Then each IVP Eq. (2) and Eq. (3) is a well-posed (well-posed problem theorem [9]).
So, Eq. (1) is a WFC-IVP well-posed problem (Lemma 3).
Unless the original problem Eq. (1) is well-posed, there is a little reason to anticipate that the numerical solution to a perturbed problem Eq. (5) will provide accurate approximations to the original problem. Since it is not always easy to find an explicit solution, we will use numerical methods which can be applied to any ODE with an initial condition to get a unique solution.
Solving the First-Order Weak Fuzzy Complex-Initial Value Problems Numerically
We will combine the numerical solutions of the two IVPs Eq. (2) and Eq. (3) in R to get the numerical solution of WFC-IVP Eq. (1) dY = ℱ(X, Y)dX { ; A ≤ X ≤ B.
Y(A) = C
Numerical Methodology to Solve a Weak Fuzzy Complex-Initial Value Problem
The object of numerical methods is to obtain approximations Z[i] and S[i] to well-posed first-order IVPs Eq. (2) and Eq. (3):
dY0 = ℱ1(X0, Y0)dX0, Y0(A0) = C0.
dY1 = ℱ2(X1, Y1)dX1, Y1(A1) = C1.
At N + 1 equally spaced numbers* in the intervals [A0, B0], [A1, B1], respectively, Z[i] and S[i]; i = 0,1,2, ⋯ , N.
*In our study, the two discrete sets have the same number of nodes. That, we will carefully choose the WFC numbers A < B (borders of X) to give us the same value of step sizes h0 = h1 = h.
Taking these nodes in the form, X0[i] = A0 + ih0, X1[i] = A1 + ih1; i = 0,1,2, ⋯ , N,
|B0−A0| |B1−A1|
where h0 = (the step size for the first IVP Eq. (2)) and h1 = (the second IVP Eq. (3)).
−1 −1
Using φ , we get approximate solutions to the well-posed first-order WFC-IVP Eq. (1), Wi = φ (Z[i], S[i]) at a discrete set of nodes, X[i] = φ (X0[i], X1[i]); i = 0,1,2, ⋯ , N.
Numerical Stability and Convergence
The reason that stability considerations are influential is that in each step after the first step of a numerical technique we begin again with a new IVP, where the initial condition is the approximate solution value computed in the previous step.
Definition 16. A method is stable when the results depend continuously on the initial data, i.e., for small changes in the initial conditions elicit only small changes in the solution; otherwise, it is unstable. Some methods are stable only for certain choices of initial data, and are called conditionally stable.
Definition 17. For a WFC-IVP Eq. (1), we say that a numerical method is absolutely stable (zero-stable) if, for h fixed, the approximate solutions Wi remains bounded, i.e., |Wi| → 0 when Xi → +∞.
Lemma 4. If a numerical method is absolutely stable for each IVPs Eq. (2) and Eq. (3), then it is an absolutely stable method for the WFC-IVP Eq. (1).
Proof: the numerical method is absolutely stable for each IVP Eq. (2) and Eq. (3) means, for h0 and h1 fixed, each Z[i] and S[i] remains bounded [20], i.e., |Z[i]| → 0 and |S[i]| → 0 when X0[i], X1[i] → +∞
−1 −1
⟹ |Wi| = φ (|Z[i]|, |S[i]|) → 0 when Xi = φ (X0i, X1i) → +∞.
Definition 18. A method is convergent in FJ if the approximation Wi approaches Yi -the exact value of the solution of the differential equation at Xi as the step size goes to zero, i.e., lim max |Yi − Wi| = 0.
h→0 0≤i≤N
Lemma 5. If a method is convergent in R for each IVPs Eq. (2) and Eq. (3), then it is convergent method for the WFC-IVP Eq. (1).
Proof: we know that the exact value of the solution of a WFC-IVP is Y[i] = φ (Y0i, Y1i) and the approximation is Wi = φ (Z[i], S[i]) at Xi When a method is convergent in R, the approximations Z[i], S[i] approach Y0i, Y1i the solutions to IVPs Eq. (2) and Eq. (3), respectively, as the steps size goes to zero ([9]). So that we can write
−1 −1
φ ( lim max |Y0i − Z[i]| , lim max | Y1i − S[i]| ) = φ (0,0)
h0→0 0≤i≤N h1→0 0≤i≤N
⟹ lim max φ (|Y0i − Z[i]|, |Y1i − S[i]| ) = 0
h→0 0≤i≤N
⟹ lim max |Yi − Wi| = 0
h→0 0≤i≤N
Euler’s Method to Solve the First-Order Weak Fuzzy Complex-Initial Value Problems
The most well-known and powerful numerical techniques used to solve IVPs are Taylor series expansions.
Taylor Series Expansions Methods
Definition 19. Assuming that our solution WFC function Y(X) has n + 1 continuous derivatives, is represented by its Taylor series
I. when only terms through Y (X) are included in the Taylor series, the method that results is called the Taylor series method of order (m).
II. the difference equation can be expressed as
(m−1)
Wi+1 = Wi + h F(Xi, Wi) + ℱ′(Xi, Wi) + ⋯ + ℱ (Xi, Wi); i = 0,1,2, ⋯ , N.
2! m!
III. the Taylor series method of order m = 1 is known as Euler’s method.
Errors will appear when applying numerical methods to find approximations [21].
Definition 20. At each step, if Wi is known, and Wi+1 is computed from the first few terms of the Taylor series, an error occurs because we have truncated the Taylor series. This error, then, is called the local truncation error, and defined by τ = Yi − Wi; i = 0,1,2, ⋯ , N, where Yi = Y[i] is the exact analytical solution in the step i, and Wi is the approximate solution in step i. Definition 21. The absolute error is the absolute value of the difference between exact value and approximate value, i.e., e[i] = |Yi − Wi|; i = 0,1,2, ⋯ , N.
Euler’s Algorithm in
In this paper, we will use Euler’s method [18] to solve WFC-IVPs of the form Eq. (1).
Definition 22. Given the WFC-IVP of the form, dY = ℱ(X, Y)dX { ; A ≤ X ≤ B, Y(A) = C where X -the WFC independent variable, we define Euler’s algorithm to find Wi -the approximate solutions for this problem at the node points X[i], as following, Input: endpoints A, B; integer N; initial condition C.
Step 1. Set h -the step size, for each i = 1,2, ⋯ , N; N ∈ ℕ, |B − A| h = , X[0] = A, W0 = C.
Step 2. For i = 1, 2, . . . , N run Steps 3 and 4:
Step 3. Set Wi+1 = Wi + h ℱ(X[i], Wi). (Compute Wi) - the difference equation-Step 4. X[i] = A + ih. (Compute X[i]) Step 5. Output: approximations Wi to Y[i] at the N + 1 values of X[i].
Absolute Error: e[i] = |Yi − Wi| where the exact (analytical) solution of a WFC-IVP is Yi = Y[i].
Proof: we know that solving a WFC-IVP Eq. (1) is equivalent to solve the following IVPs in R dY0 = ℱ1(X0, Y0)dX0, Y0(A0) = C0, dY1 = ℱ2(X1, Y1)dX1, Y1(A1) = C1, where Y = φ (Y0, Y1), Y0 = y0 − y1√t, Y1 = y0 + y1√t, X = φ (X0, X1), X0 = x0 − x1 √t, X1 = x0 + x1 √t, 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.
Then, Euler’s Algorithm will be applied for each IVPs Eq. (2) and Eq. (3), as the following:
Eluer’s algorithm.
−1 1 −1 |B−A|
I. h = φ (h0 , h1) = φ (|B0 − A0| , |B1 − A1|) = .
II. Xi = X[i] = φ (X0i, X1i), where X0i = X0[i], X1i = X1[i],
Wi| Solving directly by Euler (the corresponding program is similar to previous programs).
Outputs of Euler’s Algorithm for IVPs (the corresponding program using Python is in the Appendix).
This table is shown in the PDF version of the article.
The results of approximations, and exact solutions for IVPs.
The errors for IVPs.
Outputs of Euler’s Algorithm for IVP.
This table is shown in the PDF version of the article.
Note: Under the conditions of our WFC-IVP test (where λ < 0 and λ, A, B, C ∈ R), we get the same real results in tables for the three problems, (Xi, W[i]) = (X0 , Z[i]) = (X1 , S[i]).
Conclusion
This paper included the main notions of the WFC-IVPs, and their related concepts of existence, uniqueness, stability, and well-posedness. We use a special isomorphic transformation function to construct the solution of a WFC-IVP from the solutions of the two related real IVPs analytically and numerically. Also, we introduced the definitions and lemmas of stable and convergent numerical methods to solve WFC-IVPs. Then, we used Euler’s method to solve a simple stable linear model IVP, which has real coefficients, and real initial values. After that, a simple example explains how Euler’s algorithm works to find approximations and absolute errors by Python.
However, we know that only two terms in the Taylor series are used in Euler’s method which causes a large truncation error, and the results cannot be computed with much accuracy. Thus, we aim to use higher-order Taylor series methods in the future.
Consent for Publication
The author confirms consent for the publication of this work.
Ethics Approval and Consent to Participate
The author confirms that this research did not involve human participants or animal subjects.
Funding
This research has received no external funding.
Data Availability
The data used and analyzed during the current study are available from the corresponding author upon reasonable request.
Conflicts of Interest
The author declare no conflict of interest regarding the publications of this paper.
Appendix
For results in Table 1: using Python Instructions to solve the first IVP (Left\blue Part of Table 1) Instructions to solve the first IVP (Right\Green Part of Table 1)
Figures
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Image from the PDF file; its caption could not be matched.
ReferencesHatip, A. (2023). An introduction to weak fuzzy complex numbers. Galoitica journal of mathematical structures and applications, 3(1), 8–13. https://doi.org/10.54216/GJMSA.030101Hatip, A. (2023). On the fuzzy weak complex vector spaces. Galoitica: Journal of mathematical structures and applications, 3(1), 29–32. https://doi.org/10.54216/gjmsa.030105Ali, R. (2023). On the weak fuzzy complex inner products on weak fuzzy complex vector spaces. Neoma journal of mathematics and computer science, 1. https://doi.org/10.5281/zenodo.7953682Alhasan, Y. A., Alfahal, A. M. A., Abdulfatah, R. A., Nordo, G., & Zahra, M. M. A. (2023). On some novel results about weak fuzzy complex matrices. International journal of neutrosophic science, 21(1), 134–140. https://doi.org/10.54216/IJNS.210112Abualhomos, M., Salameh, W. M. M., Bataineh, M., Al-Qadri, M. O., Alahmade, A., & Al-Husban, A. (2024). An effective algorithm for solving weak fuzzy complex diophantine equations in two variables. International journal of neutrosophic science, 23(4), 386–394. https://doi.org/10.54216/IJNS.230431Alfahal, A. M. A., Abobala, M., Alhasan, Y. A., & Abdulfatah, R. A. (2023). Generating weak fuzzy complex and anti weak fuzzy complex integer solutions for pythagoras diophantine equation X2 + Y2 = Z2. International journal of neutrosophic science, 22(2), 8–14. https://doi.org/10.54216/IJNS.220201Galarza, F. C., Flores, M. L., Rivero, D. P., & Abobala, M. (2023). On weak fuzzy complex pythagoras quadruples. International journal of neutrosophic science, 22(2), 108–113. https://doi.org/10.54216/IJNS.220209Abobala, M. (2024). On some novel results about weak fuzzy complex integers. Journal of neutrosophic and fuzzy systems, 8. https://doi.org/10.54216/JNFS.080202Burden, R.L. & Faires, J. D. (2010). Numerical Analysis. Richard Stratton. https://colab.research.google.com/drive/1QDCMJhtmlUVmCumB6DarPiRJz-6uiwdhShihadeh, A., Salameh, W. M. M., Bataineh, M., Al-Tarawneh, H., Alahmade, A., & Al-Husban, A. (2024). On the geometry of weak fuzzy complex numbers and applications to the classification of some a-curves. International journal of neutrosophic science, 23(4), 369–375. https://doi.org/10.54216/IJNS.230428Razouk, L., Mahmoud, S., & Ali, M. (2023). A computer program for the system of weak fuzzy complex numbers and their arithmetic operations using python. Galoitica: Journal of mathematical structures and applications, 8(1), 45–51. https://doi.org/10.54216/gjmsa.080104Razouk, L., Mahmoud, S., & Ali, M. (2024). On the foundations of weak fuzzy complex-real functions. Journal of fuzzy extension and applications, 5(1), 116–140. https://doi.org/10.22105/jfea.2024.435955.1369Edduweh, H., Heilat, A. S., Razouk, L., Khalil, S. A., Alsaraireh, A. A., & Al-Husban, A. (2025). On the weak fuzzy complex differential equations and some types of the 1st order 1st degree WFC-ODEs. International journal of neutrosophic science, 25(3), 450–468. https://doi.org/10.54216/IJNS.250338Hatamleh, R. (2025). On a novel topological space based on partially ordered ring of weak fuzzy complex numbers and its relation with the partially ordered neutrosophic ring of real numbers. Neutrosophic Sets and Systems, 78, 577-590.. https://doi.org/10.5281/zenodo.14457896Thomas Tomkins Warner. (2010). Numerical weather and climate prediction. Cambridge University Press. https://doi.org/10.1017/CBO9780511763243Hull, J. C., & Basu, Sh. (1988). Options, futures, and other derivatives. Pearson India. https://books.google.nl/books/about/Options_Futures_and_other_Derivatives.html?id=CREwDwAAQBA J&redir_esc=yKhalil, H. K. (2002). Nonlinear systems. Prentice Hall. https://books.google.nl/books/about/Nonlinear_Systems.html?id=v_BjPQAACAAJ&redir_esc=yAtkinson, K., Han, W., & Stewart, D. (2011). Numerical solution of ordinary differential equations. John Wiley & Sons, Inc. https://books.google.nl/books/about/Numerical_Solution_of_Ordinary_Different.html?id=SBvQ4ThK930 C&redir_esc=yGander, W., Gander, M., & Kwok, F. (2010). Scientific Computing. Springer. https://books.google.nl/books/about/Scientific_Computing_An_Introduction_usi.html?id=QX7HBAAAQB AJ&redir_esc=yQuarteroni, A., Sacco, R., and Saleri, F. (2006). Numerical mathematics. Springer. https://doi.org/10.1007/b98885Cheney, W., Kincaid, D. (2008). Numerical mathematics and computing. Bob Pirtle. chrome-extension://efaidnbmnnnibpcajpcglclefindmkaj/https://www.hlevkin.com/hlevkin/60numalgs/Pascal/Nu merical%20Mathematics%20and%20Computing.pdf