Design of an operational matrix method based on Haar wavelets and evolutionary algorithm for time-fractional advection–diffusion equations

Najeeb Alam Khan; Mumtaz Ali; Muhammad Ayaz; Nadeem Alam Khan

doi:10.1515/eng-2025-0142

Artikel Open Access

Design of an operational matrix method based on Haar wavelets and evolutionary algorithm for time-fractional advection–diffusion equations

Najeeb Alam Khan , Mumtaz Ali , Muhammad Ayaz und Nadeem Alam Khan

Veröffentlicht/Copyright: 10. November 2025

Veröffentlicht von

Veröffentlichen auch Sie bei De Gruyter Brill

Manuskript einreichen Informationen für Autor*innen

Aus der Zeitschrift Open Engineering Band 15 Heft 1

Abstract

In this study, we introduce an operational matrix (OM) method based on Haar wavelets (HWs) to approximate the solution of time-fractional advection–diffusion equations (TFADEs). These problems involve fractional derivatives in the Atangana–Baleanu Caputo sense. The method employs an OM and a truncated HW series to transform TFADEs into objective functions. Differential evolution optimization was then applied to minimize these objective functions to determine the unknown Haar coefficients. The applicability and efficiency of the proposed method are demonstrated using illustrative examples. The accuracy of the proposed method was verified by comparing its results with those of exact solutions. Furthermore, the performance measures included the mean absolute deviation, root mean square error, Nash–Sutcliffe efficiency, and maximum absolute error, confirming the efficacy and precision of the proposed method.

Keywords: advection–diffusion equation; Atangana–Baleanu–Caputo derivative; operational matrix method; Haar wavelets, fractional calculus, evolutionary algorithm

1 Introduction

The concept of fractional differential equations (FDEs) has gained significant attention in modern mathematics, extending classical calculus through fractional calculus (FC), which generalizes derivatives and integrals to non-integer orders. Originating from the correspondence between Leibniz and L’Hopital, FC has evolved rapidly since the early 1900s, proving indispensable across scientific disciplines for providing more accurate models than integer-order counterparts [1,2,3]. Key fractional operators, such as Riemann–Liouville and Caputo [4], are pivotal for describing localized dynamics, but their traditional kernels fail to capture nonlocal behavior. In addition, Caputo and Fabrizio introduced the novel concept of fractional differentiation in 2015 [5], which was later examined by Losada and Nieto, who highlighted its non-singular and exponential behavior [6]. Subsequently, Atangana and Baleanu proposed a new class of fractional derivatives based on the Mittag–Leffler (ML) function [7]. With their nonlocal and non-singular kernels, these derivatives are excellent for modelling dynamic systems that have memory effects [8,9].

The advection–diffusion equation, a partial differential equation (PDE), characterizes several physical processes, including the transfer of energy, mass, and heat. Time-fractional advection–diffusion equations (TFADEs) offer an excellent description of energy transfer during the advection and diffusion processes. In recent years, several numerical algorithms have been developed to address the TFADEs. For instance, Tajadodi [10] used Bernstein polynomials to fix TFADEs involving the Atangana–Baleanu–Caputo (ABC) derivative, and Yadav et al. [11] applied the finite difference method to solve TFADEs involving ABC derivatives.

Among the various types of wavelets, Haar wavelets (HWs) are practical, orthonormal wavelets with compact support, characterized by their rectangular shape. HWs can be integrated analytically multiple times; however, their discontinuous nature means that derivatives are undefined at points of discontinuity, making them challenging to apply directly to differential equations. Two main approaches can be used to address this limitation. The first approach involves regularizing HWs by interpolating splines, as introduced by Cattani [12]. However, this approach adds complexity and reduces the simplicity of the HWs. The second approach, proposed by Chen and Hsiao [13], uses an integral method. This approach expresses the highest derivative in DE as an HW series, whereas lower-order derivatives are obtained through integration. This collocation-based approach forms the foundation of the HW approach for addressing differential equations. In recent years, numerical solutions of FDEs using HWs have become a prominent research focus [14,15,16,17,18,19,20,21].

In this study, we examined the subsequent TFADE containing ABC derivatives

(1) D τ α 0 A B C ψ ( ξ , τ ) = η ∂ 2 ψ ( ξ , τ ) ∂ ξ 2 − λ ∂ ψ ( ξ , τ ) ∂ ξ + ϒ ( ξ , τ ) , α ∈ ( 0 , 1 ] , ξ ∈ [ a , b ] and τ ∈ [ τ 0 , T ] ,

with initial condition (IC)

(2) ψ ( ξ , τ 0 ) = f ( ξ ) ,

and boundary conditions (BCs)

(3) ψ ( a , τ ) = g 0 ( τ ) , ψ ( b , τ ) = g 1 ( τ ) ,

where ϒ ( ξ , τ ) denotes the source term, η > 0 is the diffusivity coefficient, and λ is the average velocity component.

This study was motivated by a recent breakthrough in numerical analysis of TFADEs. Given the widespread applications of such models, the objective of this study is to approximate TFADEs using the Haar wavelet operational matrix method (HWOMM). While previous studies have employed HWs to solve TFADEs, none have utilized the HWOMM in conjunction with the ABC derivative. We developed an operational matrix (OM) of AB integration based on the HWs. HW series and OM approaches are employed to transform TFADEs into objective functions, which are then minimized by differential evolution (DE) optimization for unknown HW coefficients. The efficiency and practicality of the proposed method are illustrated by solving a numerical examples. A comparison of the numerical outcomes with the analytical results reveals that the proposed method provides accurate results. The proposed method for TFADEs is innovative and, to the best of our knowledge, has not been explored previously.

2 Preliminary tools

We begin with the definition of the AB fractional derivative. This derivative is a nonlocal fractional derivative characterized by a non-singular kernel, and it is associated with a wide range of applications.

Definition 2.1

[7] Let μ ∈ T 1 ( 0 , L ) , L > 0 , then the AB derivative of the μ ( t ) in RL sense is given as

(4) D t α 0 A B R μ ( t ) = B ( α ) ( 1 − α ) d d t ∫ 0 t μ ( τ ) E α − α 1 − α ( t − τ ) α d τ .

Definition 2.2

[7] Let μ ∈ T 1 ( 0 , L ) , L > 0 , then the AB derivative in the Caputo sense of μ ( t ) given as

(5) D t α 0 A B C μ ( t ) = B ( α ) ( 1 − α ) d d t ∫ 0 t μ ′ ( τ ) E α − α 1 − α ( t − τ ) α d τ ,

where E α ( t ) = ∑ n = 0 ∞ t n Γ ( α n + 1 ) is an ML function, and B ( α ) = 1 − α + α Γ ( α ) is a normalization function.

Definition 2.3

[7] The fractional integral of a function μ ( t ) is given as

(6) I t α 0 A B μ ( t ) = ( 1 − α ) B ( α ) μ ( t ) + ( α ) B ( α ) J t α 0 μ ( t ) ,

where J t α 0 μ ( t ) = 1 Γ ( α ) ∫ 0 t ( t − τ ) α − 1 μ ( τ ) d τ .

Lemma 2.1

[22] For 0 < α ≤ 1 , we have that

( I t α 0 A B D t α 0 A B C μ ) ( t ) = μ ( t ) − μ ( 0 ) .

3 HWs and function approximation

This section discusses HWs and the associated OM of the AB fractional integration.

3.1 HWs

The HW family for the interval τ ∈ L 2 ( [ 0 , 1 ) ) is defined as [23]

(7) ℏ i ( τ ) = 1, τ ∈ [ β 1 , β 2 ) , − 1, τ ∈ [ β 2 , β 3 ) , 0 , elsewhere .

Here, β 1 = κ m , β 2 = κ + 0.5 m , and β 3 = κ + 1 m is introduced. The integer m = 2 j , j = 0 , 1 , 2 , … , J , indicates the level of the wavelet, whereas κ = 0 , 1 , 2 , .. . , m − 1 acts as a translation parameter. The maximum resolution level is J and the index i in Equation (7) is determined as i = m + κ + 1 . For the lowest values, m = 1 , κ = 0 is i = 2 . The highest possible value of i is 2 M , where M = 2 J .

The scaling function for the HW family is as follows:

(8) ℏ 1 ( τ ) = 1 , τ ∈ [ 0 , 1 ) , 0 , elsewhere .

3.2 Approximation of square integrable function

Any function f ( ξ , τ ) defined over the interval L 2 ( [ 0 , 1 ) × [ 0 , 1 ) ) can be expanded in the HW series as follows:

(9) f ( ξ , τ ) = ∑ i = 1 ∞ ∑ j = 1 ∞ ω i j ℏ i ( ξ ) ℏ j ( τ ) ,

where the HW coefficients ω i j , s are resolved by the inner product 〈 ℏ i ( ξ ) , 〈 f , ℏ j ( τ ) 〉〉 .

Equation (9) terminates finitely if the function f ( ξ , τ ) is approximated as a piecewise constant function within each interval then

(10) f ( ξ , τ ) ≈ ∑ i = 1 2 M ∑ j = 1 2 M ω i j ℏ i ( ξ ) ℏ j ( τ ) .

In order to obtain an approximate numerical solution for the function f ( ξ , τ ) , we employ the collocation points ( ξ i , τ j ) where i , j = 1 , 2 , … , 2 M . By discretizing Equation (10) at the collocation points, we obtain

(11) Ψ = H T C H ,

where C = [ ω i j ] 2 M × 2 M , Ψ = [ f ( ξ i , τ j ) ] 2 M × 2 M , and H = [ ℏ i ( ξ j ) ] 2 M × 2 M .

The HW matrix H of order 2 M , i.e.

H = ℏ 1 ( τ 1 ) ℏ 1 ( τ 2 ) ⋯ ℏ 1 ( τ 2 M ) ℏ 2 ( τ 1 ) ℏ 2 ( τ 2 ) ⋯ ℏ 2 ( τ 2 M ) ⋮ ⋮ ⋱ ⋮ ℏ 2 M ( τ 1 ) ℏ 2 M ( τ 2 ) ⋯ ℏ 2 M ( τ 2 M ) .

3.3 OM for fractional integration of HWs

Derive the OM for the AB fractional-order integration for HWs. We employ Equation (6) of the AB fractional integral operator I α .

I τ α 0 A B H = P η = [ I τ α 0 A B ℏ 1 ( τ ) , I τ α 0 A B ℏ 2 ( τ ) , .. . , I τ α 0 A B ℏ 2 M ( τ ) ] T , = 1 − α B ( α ) ℏ 1 ( τ ) + α B ( α ) J τ α 0 ℏ 1 ( τ ) , 1 − α B ( α ) ℏ 2 ( τ ) + α B ( α ) J τ α 0 ℏ 2 ( τ ) , … + 1 − α B ( α ) ℏ 2 M ( τ ) + α B ( α ) J τ α 0 ℏ 2 M ( τ ) T , = [ P ℏ 1 ( τ ) , P ℏ 2 ( τ ) , .. . , P ℏ 2 M ( τ ) ] T ,

where

(12) P ℏ 1 ( τ ) = 1 − α B ( α ) + τ α B ( τ ) Γ ( τ ) , τ ∈ [ 0 , 1 ) ,

and

(13) P ℏ i ( τ ) = 0 , τ ∈ [ 0 , β 1 ) , 1 − α B ( α ) + 1 B ( α ) Γ ( α ) τ − κ m α , τ ∈ [ β 1 , β 2 ) , 1 B ( α ) Γ ( α ) τ − κ m α − 2 τ − κ + 1 2 m α , τ ∈ [ β 2 , β 3 ) , 1 B ( α ) Γ ( α ) τ − κ m α − 2 τ − κ + 1 2 m α + τ − κ + 1 m α , τ ∈ [ β 3 , 1 ) .

For instance, if α = 0.5 , and J = 2 , we obtain

P 0.5 = 0.8196 0.9516 1.0425 1.1164 1.1803 1.2374 1.2895 1.3377 0.8196 0.9516 1.0425 1.1164 1.1803 − 0.0265 − 0.1562 − 0.2558 0.8196 0.9516 0.0425 − 0.1475 − 0.0851 − 0.0437 − 0.0285 − 0.0206 0 0 0 0 0.8196 0.9516 0.0425 − 0.1475 0.8196 − 0.0483 − 0.0411 − 0.0170 − 0.0099 − 0.0067 − 0.00499 − 0.0038 0 0 0.8196 − 0.0483 − 0.0411 − 0.0170 − 0.0099 − 0.0067 0 0 0 0 0.8196 − 0.0483 − 0.0411 − 0.0170 0 0 0 0 0 0 0.8196 − 0.0483 .

3.4 Procedure of implementation

This section illustrates the use of the OM method of HW to solve TFADEs with BCs. Let us consider TFADEs

(14) D τ α 0 A B C ψ ( ξ , τ ) = η ∂ 2 ψ ( ξ , τ ) ∂ ξ 2 − λ ∂ ψ ( ξ , τ ) ∂ ξ + ϒ ( ξ , τ ) , α ∈ ( 0 , 1 ] , ξ ∈ [ 0 , 1 ] and τ ∈ [ 0 , 1 ] ,

with IC

(15) ψ ( ξ , 0 ) = f ( ξ ) ,

and BCs

(16) ψ ( a , τ ) = g 0 ( τ ) , ψ ( b , τ ) = g 1 ( τ ) ,

where g 0 ( τ ) , g 1 ( τ ) and f ( ξ ) is a known function of L 2 ( [ 0 , 1 ) ) .

Let

(17) ∂ 2 + α ψ ( ξ , τ ) ∂ ξ 2 ∂ τ α = ∑ i = 1 2 M ∑ j = 1 2 M ω i j ℏ i ( ξ ) ℏ j ( τ ) ,

where ω i j ′ s are unknown Haar coefficients. By employing the AB fractional integral of the order α , we integrate of Equation (17) with respect to τ as

(18) ∂ 2 ψ ( ξ , τ ) ∂ ξ 2 = ∂ 2 ψ ( ξ , 0 ) ∂ 2 ξ + ∑ i = 1 2 M ∑ j = 1 2 M ω i j ℏ i ( ξ ) P j α ( τ ) .

By integrating Equation (18) with respect to ξ , twice, we obtain

(19) ∂ ψ ( ξ , τ ) ∂ ξ = ∂ ψ ( 0 , τ ) ∂ ξ + ∂ ψ ( ξ , 0 ) ∂ ξ − ∂ ψ ( 0 , 0 ) ∂ ξ + ∑ i = 1 2 M ∑ j = 1 2 M ω i j P i 1 ( ξ ) P j α ( τ ) .

Again integrating with respect to ξ , we obtain

(20) ψ ( ξ , τ ) = ψ ( 0 , τ ) + ξ ∂ ψ ( 0 , τ ) ∂ ξ − ∂ ψ ( 0 , 0 ) ∂ ξ + ψ ( ξ , 0 ) − ψ ( 0 , 0 ) + ∑ i = 1 2 M ∑ j = 1 2 M ω i j P i 2 ( ξ ) P j α ( τ ) .

Integrating Equation (18) with respect to ξ from 0 to 1 , using IC and BCs, we obtain

(21) ∂ ψ ( 0 , τ ) ∂ ξ − ∂ ψ ( 0 , 0 ) ∂ ξ = g 1 ( τ ) − g 0 ( τ ) + g 1 ( 0 ) − g 0 ( 0 ) − ∑ i = 1 2 M ∑ j = 1 2 M ω i j K i P j α ( τ ) .

Using Equation (20) in Equation (21), after simplification, we obtain

(22) ψ ( ξ , τ ) = ∑ i = 1 2 M ∑ j = 1 2 M ω i j P i 2 ( ξ ) P j α ( τ ) + g 0 ( τ ) − g 0 ( 0 ) + f ( ξ ) + ξ g 1 ( τ ) − g 0 ( τ ) + g 1 ( 0 ) − g 0 ( 0 ) − ∑ i = 1 2 M ∑ j = 1 2 M ω i j K i P j α ( τ ) .

By applying the AB derivative operator α times in Equation (22), with regard to τ , we obtain

(23) ∂ α ψ ( ξ , τ ) ∂ τ α = ∑ i = 1 2 M ∑ j = 1 2 M ω i j P i 2 ( ξ ) ℏ j ( τ ) − ξ ∑ i = 1 2 M ∑ j = 1 2 M ω i j K i ℏ j ( τ ) + ∂ α ∂ τ α ( g 0 ( τ ) + ξ ( g 1 ( τ ) − g 0 ( τ ) ) ) .

Using Equations (17), (18), and (23) in Equation (14), we obtain

(24) ∑ i = 1 2 M ∑ j = 1 2 M ω i j P i 2 ( ξ ) ℏ j ( τ ) − ξ ∑ i = 1 2 M ∑ j = 1 2 M ω i j K i ℏ j ( τ ) + ∂ α ∂ τ α ( g 0 ( τ ) + ξ ( g 1 ( τ ) − g 0 ( τ ) ) ) = η ∂ 2 ψ ( ξ , 0 ) ∂ ξ 2 + ∑ i = 1 2 M ∑ j = 1 2 M ω i j ℏ i ( ξ ) P j α ( τ ) − λ ∂ ψ ( 0 , τ ) ∂ ξ + g 1 ( τ ) − g 0 ( τ ) + g 1 ( 0 ) − g 0 ( 0 ) − ∑ i = 1 2 M ∑ j = 1 2 M ω i j K i P j α ( τ ) + ∑ i = 1 2 M ∑ j = 1 2 M ω i j P i 1 ( ξ ) P j α ( τ ) + ϒ ( ξ , τ ) .

By minimizing the Equation (24) at the collocation points ( ξ i , τ j ) , i , j = 1 , 2 , … , 2 M , by DE optimization, we obtain the HW coefficients. By utilizing the HW coefficients in Equation (22), we obtain HW-based numerical solutions for Equation (14).

3.5 DE

This study uses the DE method to improve the process efficiency. The heuristic optimization method proposed by Price et al. [24] is exceptionally effective. DE is highly regarded by global optimizers because of its simplicity and robust population-based stochastic search methodology in continuous domains. The algorithm is characterized by three key control parameters: the population size NP, scaling factor Sf, and crossover constant CR. The DE optimization efficacy substantially affected by these factors. These factors significantly influence the performance of DE optimization. Thus, in [24], straightforward criteria were provided for selecting these values. In the DE algorithm, solutions can be quickly derived by providing the population set, an approximation solution, and the objective function. We employed built-in Mathematica 11.0 codes to minimize the object function via DE optimization.

3.6 Convergence of the HW bases

In this subsection, we suppose that ∂ ψ ( ξ , τ ) ∂ ξ is continuous and bounded on ( 0 , 1 ) × ( 0 , 1 ) , then

∃ K > 0 , ∀ ξ , τ ∈ ( 0 , 1 ) × ( 0 , 1 ) , ∂ ψ ( ξ , τ ) ∂ ξ ≤ K .

We assume that ψ 2 M ( ξ , τ ) is the following approximation of ψ ( ξ , τ ) :

ψ 2 M ( ξ , τ ) = ∑ i = 0 2 M ∑ j = 0 2 M ω i j ℏ i ( ξ ) ℏ j ( τ ) .

Then, we have

ψ ( ξ , τ ) − ψ 2 M ( ξ , τ ) = ∑ i = 2 M ∞ ∑ j = 2 M ∞ ω i j ℏ i ( ξ ) ℏ j ( τ ) ,

Theorem 1

Assuming that the ψ 2 M ( ξ , τ ) derived via HW serves as an approximation of ψ ( ξ , τ ) , subsequently, the error is bounded as follows:

(25) ∥ ψ ( ξ , τ ) − ψ 2 M ( ξ , τ ) ∥ E ≤ K 3 1 m 2 .

Proof

For complete proof, see [25].□

HWOMM is implemented by following the necessary steps:

Input: j , κ

Step 1: Define HW ℏ 1 ( τ ) , ℏ i ( τ ) in Equations (7) and (8)

Step 2: Construct the HW matrix H 2 M × 2 M

Step 3: Compute the AB fractional integral operator for HW P ℏ 1 ( τ ) and P ℏ i ( τ ) using Equations (12) and (13)

Step 4: Define Collocation points ( ξ i , τ j ) , where i , j = 1 , 2 , … , 2 M .

Step 5: Construct the objective functions based on the residual of the FDEs at the collocation points

Step 6: Minimize the objective functions in step 5 by using DE optimization

Output: Calculate the approximate results

3.7 Performance criteria

To investigate the accuracy and stability of a method by considering the values of various performance measures, such as MAD, RMSE, ENSE, and maximum absolute error L ∞ . The performance measures are mathematically defined as follows:

(26) MAD = 1 2 M ∑ i = 1 2 M ∣ ψ i − ψ ˜ i ∣ ,

where 2 M is the number of collocation points.

(27) RMSE = 1 2 M ∑ i = 1 2 M ( ψ i − ψ ˜ i ) 2 ,

(28) NSE = 1 − ∑ i = 1 2 M ( ψ i − ψ ˜ i ) 2 ∑ i = 1 2 M ψ i − 1 2 M ∑ i = 1 2 M ψ ˜ i 2 , ENSE = ∣ 1 − NSE ∣ ,

(29) L ∞ = max ( ∣ ψ i − ψ ˜ i ∣ ) , i = 1 , 2 , 3 , … 2 M .

4 Results and discussion

The numerical results of the two TFADEs are presented to validate the reliability, efficacy, and robustness of the proposed method.

Example 1

Consider TFADE [11]

(30) D τ α 0 A B C ψ ( ξ , τ ) = ∂ 2 ψ ( ξ , τ ) ∂ ξ 2 − ∂ ψ ( ξ , τ ) ∂ ξ + ϒ ( ξ , τ ) , ξ ∈ [ 0 , 1 ] , τ ∈ [ 0 , 1 ] ,

with IC

ψ ( ξ , 0 ) = 0 ,

and BCs

ψ ( 0 , τ ) = ψ ( 1 , τ ) = 0 ,

where source term ϒ ( ξ , τ ) = 2 B ( α ) 1 − α ξ ( ξ − 1 ) τ 2 E α , 3 − α 1 − α τ α − 2 τ 2 + ( 2 ξ − 1 ) τ 2 .

By applying the IC and BCs in Equation (24), Equation (30) becomes

(31) ∑ i = 1 2 M ∑ j = 1 2 M ω i j P i 2 ( ξ ) ℏ j ( τ ) − ξ ∑ i = 1 2 M ∑ j = 1 2 M ω i j K i ( τ ) ℏ j ( τ ) = ∑ i = 1 2 M ∑ j = 1 2 M ω i j ℏ i ( ξ ) P j α ( τ ) + ∑ i = 1 2 M ∑ j = 1 2 M ω i j K i ( τ ) P j α ( τ ) − ∑ i = 1 2 M ∑ j = 1 2 M ω i j P i 1 ( ξ ) P j α ( τ ) + ϒ ( ξ , τ ) .

Equation (31) can be written in matrix form as follows:

(32) ( P 2 ) T W H − X K T W H = H T W P α + K T W P α − ( P 1 ) T W P α + Λ ,

where P 2 = [ P i 2 ( ξ j ) ] 2 M × 2 M , P α = [ P j α ( τ i ) ] 2 M × 2 M , K = [ K i ( ξ j ) ] 2 M × 2 M , P 1 = [ P i 1 ( ξ j ) ] 2 M × 2 M ,

X = diag ( ξ 1 , ξ 2 , ξ 3 , .. . , ξ 2 M ) and Λ = [ ϒ ( ξ i , τ j ) ] 2 M × 2 M .

Equation (32) has unknown HW coefficients. The coefficients were established by formulating an objective function as follows:

(33) Ω = 1 4 M 2 ∑ i = 1 2 M ∑ j = 1 2 M ( ( P 2 ) T W H − X K T W H − H T W P α − K T W P α + ( P 1 ) T W P α − Λ ) 2 .

To obtain the unknown HW coefficients in Equation (34) using DE optimization. The exact solution of given problem is ψ ( ξ , τ ) = ξ ( ξ − 1 ) τ 2 .

Example 2

Consider TFADE [11]

(34) D τ α 0 A B C ψ ( ξ , τ ) = ∂ 2 ψ ( ξ , τ ) ∂ ξ 2 − ∂ ψ ( ξ , τ ) ∂ ξ + ϒ ( ξ , τ ) , ξ ∈ [ 0 , 1 ] , τ ∈ [ 0 , 1 ] ,

with IC

ψ ( ξ , 0 ) = 0 ,

and BCs

ψ ( 0 , τ ) = 0 = ψ ( 1 , τ ) ,

where the source term

ϒ ( ξ , τ ) = 120 B ( α ) 1 − α τ 5 sin ( π ξ ) E α , 6 − α 1 − α τ α + τ 5 π ( π sin ( π ξ ) + cos ( π ξ ) ) .

The exact solution to a given problem is ψ ( ξ , τ ) = τ 5 sin ( π ξ ) .

The HWOMM was applied to solve TFADEs, following the procedure outlined in Section 3.4. Figure 1(a)–(d) illustrate a robust agreement between the approximate and exact solutions for example 1 across various time instance τ = 0.15 , 0.45 , 0.75 and 0 .95 , fractional orders α = 0.1 , 0.25 and 0 .5 and collocation points 2 M = 8 , 1 6 and 32 . Figure 2 presents a 3D plot comparing the approximate and exact solution at different fractional orders α = 0.25 and 0 .5 and collocation points 2 M = 4 , 8 and 16 for example 1. Table 1 summarizes the performance metrics, including MAD, RMSE, ENSE, and maximum absolute error for different collocation points 2 M = 4 , 8 , 16, 32 and 64 in example 1. From figures and tables, it is evident that the accuracy improves with an increase in the number of the collocation points 2 M . Similarly, for example 2, Figure 3(a)–(d) shows a strong agreement between the approximate and exact outcomes at various times τ = 0.15 , 0.45 , 0.75 and 0 .95 , fractional order α = 0.1 , 0.25 and 0 .5 and 2 M = 8 , 1 6 and 32 phases. Figure 4 provides the 3D plot of approximate and exact outcomes at different phases of the fractional order α = 0.25 and 0 .5 and collocation point 2 M = 4 , 8 and 16 . In Table 2, the performance measures such as MAD, RMSE, ENSE, and maximum absolute error are tabulated for collocation points 2 M = 4 , 8 , 16, 32 and 64 for example 2. These figures and tables collectively demonstrate that the proposed method achieves a close agreement with the exact outcomes.

$Figure 1 Comparison of HWOMM and exact solutions at various phases of time for example 1. (a) 2 M = 32 2M=32 and α = 0.1 \alpha =0.1 , (b) 2 M = 8 2M=8 and α = 0.25 \alpha =0.25 , (c) 2 M = 16 2M=16 and α = 0.25 \alpha =0.25 , and (d) 2 M = 32 2M=32 and α = 0.5 \alpha =0.5 .$

Figure 1

Comparison of HWOMM and exact solutions at various phases of time for example 1. (a) 2 M = 32 and α = 0.1 , (b) 2 M = 8 and α = 0.25 , (c) 2 M = 16 and α = 0.25 , and (d) 2 M = 32 and α = 0.5 .

$Figure 2 3D plot for HWOMM and exact solutions for example 1. (a) Behavior of HWOMM solution at 2 M = 4 2M=4 where α = 0.25 \alpha =0.25 . (b) Behavior of HWOMM solution at 2 M = 8 2M=8 where α = 0.25 \alpha =0.25 . (c) Behavior of HWOMM solution at 2 M = 16 2M=16 where α = 0.5 \alpha =0.5 . (d) Behavior of exact solution.$

Figure 2

3D plot for HWOMM and exact solutions for example 1. (a) Behavior of HWOMM solution at 2 M = 4 where α = 0.25 . (b) Behavior of HWOMM solution at 2 M = 8 where α = 0.25 . (c) Behavior of HWOMM solution at 2 M = 16 where α = 0.5 . (d) Behavior of exact solution.

Table 1

MAD, RMSE, ENSE, L ∞ , and CPU time in seconds for example 1

α	2 M	MAD	RMSE	ENSE	L ∞	CPU time
0.25	4	2.3797 × 10⁻⁵	2.8839 × 10⁻⁵	4.2057 × 10⁻⁴	4.7654 × 10⁻⁵	0.034
	8	1.9567 × 10⁻⁵	2.3864 × 10⁻⁵	2.1942 × 10⁻⁴	4.1755 × 10⁻⁵	0.157
	16	1.8234 × 10⁻⁵	2.0925 × 10⁻⁵	1.5744 × 10⁻⁴	3.5012 × 10⁻⁵	0.375
	32	1.8451 × 10⁻⁵	2.0474 × 10⁻⁵	1.4978 × 10⁻⁴	2.9102 × 10⁻⁵	1.359
	64	1.9669 × 10⁻⁵	2.2240 × 10⁻⁵	1.7622 × 10⁻⁴	3.6559 × 10⁻⁵	10.187
0.5	4	6.8150 × 10⁻⁵	7.6467 × 10⁻⁵	2.9568 × 10⁻³	1.1676 × 10⁻⁴	0.047
	8	3.6716 × 10⁻⁵	4.6381 × 10⁻⁵	8.2881 × 10⁻⁴	7.8120 × 10⁻⁵	0.125
	16	2.7031 × 10⁻⁵	3.0439 × 10⁻⁵	3.3597 × 10⁻⁴	4.9898 × 10⁻⁵	0.485
	32	2.9893 × 10⁻⁵	3.4727 × 10⁻⁵	4.3091 × 10⁻⁴	5.9983 × 10⁻⁵	1.344
	64	3.7018 × 10⁻⁵	4.6639 × 10⁻⁵	7.7436 × 10⁻⁴	8.2687 × 10⁻⁵	8.141

$Figure 3 Comparison of HWOMM and exact solutions at various phases of time for example 2. (a) 2 M = 32 2M=32 and α = 0.1 \alpha =0.1 , (b) 2 M = 8 2M=8 and α = 0.25 \alpha =0.25 , (c) 2 M = 32 2M=32 and α = 0.25 \alpha =0.25 , (d) 2 M = 32 2M=32 and α = 0.5 \alpha =0.5 .$

Figure 3

Comparison of HWOMM and exact solutions at various phases of time for example 2. (a) 2 M = 32 and α = 0.1 , (b) 2 M = 8 and α = 0.25 , (c) 2 M = 32 and α = 0.25 , (d) 2 M = 32 and α = 0.5 .

$Figure 4 3D plots for HWOMM and exact solutions for example 2. (a) Behavior of HWOMM solution at 2 M = 4 2M=4 where α = 0.25 \alpha =0.25 . (b) Behavior of HWOMM solution at 2 M = 8 2M=8 where α = 0.25 \alpha =0.25 . (c) Behavior of HWOMM solution at 2 M = 16 2M=16 where α = 0.5 \alpha =0.5 . (d) Behavior of exact solution.$

Figure 4

3D plots for HWOMM and exact solutions for example 2. (a) Behavior of HWOMM solution at 2 M = 4 where α = 0.25 . (b) Behavior of HWOMM solution at 2 M = 8 where α = 0.25 . (c) Behavior of HWOMM solution at 2 M = 16 where α = 0.5 . (d) Behavior of exact solution.

Table 2

MAD, RMSE, ENSE, L ∞ , and CPU time in seconds for example 2

α	2 M	MAD	RMSE	ENSE	L ∞	CPU time
0.25	4	1.0970 × 10⁻⁶	1.2389 × 10⁻⁶	3.6353 × 10⁻³	1.8924 × 10⁻⁶	0.031
	8	4.1118 × 10⁻⁷	4.9194 × 10⁻⁷	4.6909 × 10⁻⁴	7.8978 × 10⁻⁷	0.156
	16	3.9823 × 10⁻⁷	4.7827 × 10⁻⁷	4.2465 × 10⁻⁴	7.8014 × 10⁻⁷	0.375
	32	3.2379 × 10⁻⁷	4.0872 × 10⁻⁷	3.0692 × 10⁻⁴	6.9137 × 10⁻⁷	1.360
	64	2.5177 × 10⁻⁷	3.2809 × 10⁻⁷	1.9726 × 10⁻⁴	5.8103 × 10⁻⁷	8.282
0.5	4	6.8150 × 10⁻⁵	5.1974 × 10⁻⁷	6.3975 × 10⁻⁴	8.3544 × 10⁻⁷	0.047
	8	1.3076 × 10⁻⁶	1.4589 × 10⁻⁶	4.1255 × 10⁻³	2.1342 × 10⁻⁶	0.109
	16	4.5671 × 10⁻⁷	5.8863 × 10⁻⁷	6.4325 × 10⁻⁴	1.0074 × 10⁻⁶	0.359
	32	7.5519 × 10⁻⁷	9.1656 × 10⁻⁷	1.5434 × 10⁻³	1.4895 × 10⁻⁶	1.297
	64	5.5782 × 10⁻⁷	6.9163 × 10⁻⁷	8.7658 × 10⁻⁴	1.1580 × 10⁻⁶	8.062

5 Conclusion

In this study, we developed an OM method for AB fractional integration based on the HWs. The truncated HW series and OM of the AB fractional integration were used to transform TFADEs into objective functions. The unknown Haar coefficients were subsequently determined by minimizing these objective functions by DE optimization. Two examples were considered to validate the proposed method, and performance metrics such as MAD, RMSE, ENSE, and maximum absolute error L ∞ confirmed the high accuracy and consistent convergence of the method. The contributions of the aforementioned method is summarized as follows:

The designed OM method, which is based on HW, is well suited for solving TFADEs.
Fractional derivative and integration are in the sense of ABC.
The HWOMM transforms TFADEs into objective functions and then employs DE optimization to minimize the objective function for unknown HW coefficients.
The numerical results confirmed the accuracy and reliability of the HWOMM.
HWOMM’s high precision and consistent convergence can be further improved by increasing the collocation points 2 M .

Several problems related to this study should be addressed in future investigations. HWOMM has potential applications in addressing variable-order fractional, pantograph fractional, space-fractional, and higher-dimensional fractional PDEs, which will be explored in future work. In addition, it shows promise for obtaining numerical outcomes for fractional-order differential equations containing other types of fractional operators.

Acknowledgments

The authors thank the referees for their valuable comments that helped improve this paper.

Funding information: The authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript. Najeeb Alam Khan: formulated the problem, designed the algorithm, supervision, methodology. Mumtaz Ali: simulated the problem examples, analysis, writing the draft, writing – review and editing. Muhammad Ayaz: writing the draft, writing – review and editing, Nadeem Alam Khan: simulation, statistical analysis, validation.
Conflict of interest: The authors state no conflict of interest.
Ethics approval: This study does not involve ethical issues. No research involves Human Participants and/or Animals.
Data availability statement: There is no data reported for this work.

References

[1] Arora S, Mathur T, Agarwal S, Tiwari K, Gupta P. Applications of fractional calculus in computer vision: A survey. Neurocomputing. 2022;489:407–28.10.1016/j.neucom.2021.10.122Suche in Google Scholar

[2] Ghanbari B, Kumar S, Kumar R. A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative. Chaos Solitons Fractals. 2020;133:109619.10.1016/j.chaos.2020.109619Suche in Google Scholar

[3] Tenreiro Machado JA, Silva MF, Barbosa RS, Jesus IS, Reis CM, Marcos MG, et al. Some applications of fractional calculus in engineering. Math Probl Eng. 2010;2010:1–34.10.1155/2010/639801Suche in Google Scholar

[4] Abdeljawad T. On Riemann and Caputo fractional differences. Comput Math Appl. 2011;62(3):1602–11.10.1016/j.camwa.2011.03.036Suche in Google Scholar

[5] Caputo M, Fabrizio M. A new definition of fractional derivative without singular kernel. Prog Fract Differ Appl. 2015;1(2):73–85.10.18576/pfda/020101Suche in Google Scholar

[6] Losada J, Nieto JJ, Arabia S. Properties of a new fractional derivative without singular kernel. Progr Fract Differ Appl. 2015;1(2):87–92.Suche in Google Scholar

[7] Atangana A, Baleanu D. New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model. Therm Sci. 2016;20(2):763–9.10.2298/TSCI160111018ASuche in Google Scholar

[8] Atangana A, Alqahtani RT. New numerical method and application to Keller-Segel model with fractional order derivative. Chaos Solitons Fractals. 2018;116:14–21.10.1016/j.chaos.2018.09.013Suche in Google Scholar

[9] Heydari MH, Atangana A. A cardinal approach for nonlinear variable-order time fractional Schrödinger equation defined by Atangana–Baleanu–Caputo derivative. Chaos Solitons Fractals. 2019;128:339–48.10.1016/j.chaos.2019.08.009Suche in Google Scholar

[10] Tajadodi H. A Numerical approach of fractional advection-diffusion equation with Atangana–Baleanu derivative. Chaos Solitons Fractals. 2020;130:109527.10.1016/j.chaos.2019.109527Suche in Google Scholar

[11] Yadav S, Pandey RK, Shukla AK. Numerical approximations of Atangana–Baleanu Caputo derivative and its application. Chaos Solitons Fractals. 2018;118:58–64.10.1016/j.chaos.2018.11.009Suche in Google Scholar

[12] Cattani C. Haar wavelet splines. J Interdiscip Mathematics. 2001;4(1):35–47.10.1080/09720502.2001.10700287Suche in Google Scholar

[13] Chen CF, Hsiao CH. Haar wavelet method for solving lumped and distributed-parameter systems. IEE Proc-Control Theory Appl. 1997;144(1):87–94.10.1049/ip-cta:19970702Suche in Google Scholar

[14] Ahmed S, Jahan S, Nisar KS. Haar wavelet based numerical technique for the solutions of fractional advection diffusion equations. J Math Comput Sci. 2024;34(3):217–33.10.22436/jmcs.034.03.02Suche in Google Scholar

[15] Aruldoss R, Balaji K. A wavelet-based collocation method for fractional Cahn-Allen equations. J Fract Calc Appl. 2023;14(1):26–35.Suche in Google Scholar

[16] Khan NA, Hameed T. An implementation of Haar wavelet based method for numerical treatment of time-fractional Schrödinger and coupled Schrödinger systems. IEEE/CAA J Autom Sin. 2016;6(1):177–87.10.1109/JAS.2016.7510193Suche in Google Scholar

[17] ur Rehman M, Baleanu D, Alzabut J, Ismail M, Saeed U. Green–Haar wavelets method for generalized fractional differential equations. Adv Differ Equ. 2020;2020(1):515.10.1186/s13662-020-02974-6Suche in Google Scholar

[18] Amin R, Senu N, Hafeez MB, Arshad NI, Ahmadian AL, Salahshour S, et al. A computational algorithm for the numerical solution of nonlinear fractional integral equations. Fractals. 2021;30(1):2240030.10.1142/S0218348X22400308Suche in Google Scholar

[19] Ali M, Baleanu D. Haar wavelets scheme for solving the unsteady gas-flow in 4-D. Therm Sci. 2019;24(2 Part B):1357–67.10.2298/TSCI190101292ASuche in Google Scholar

[20] Amin R, Sitthiwirattham T, Hafeez MB, Sumelka W. Haar collocations method for nonlinear variable order fractional integro-differential equations. Prog Fract Differ Appl. 2023;9(2):223–9.10.18576/pfda/090203Suche in Google Scholar

[21] Majak J, Shvartsman BS, Kirs M, Pohlak M, Herranen H. Convergence theorem for the Haar wavelet based discretization method. Compos Struct. 2015;126:227–32.10.1016/j.compstruct.2015.02.050Suche in Google Scholar

[22] Abdeljawad T. A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel. J Inequal Appl. 2017;2017(1):130.10.1186/s13660-017-1400-5Suche in Google Scholar PubMed PubMed Central

[23] Yi M, Huang J. Wavelet operational matrix method for solving fractional differential equations with variable coefficients. Appl Math Comput. 2014;230:383–94.10.1016/j.amc.2013.06.102Suche in Google Scholar

[24] Price K, Storn RM, Lampinen JA. Differential evolution. Springer Science & Business Media; 2006.Suche in Google Scholar

[25] Wang L, Ma Y, Zhang M. Haar wavelet method for solving fractional partial differential equations numerically. Appl Math Comput. 2014;227:66–76.10.1016/j.amc.2013.11.004Suche in Google Scholar

Received: 2024-10-10

Revised: 2024-12-06

Accepted: 2025-10-03

Published Online: 2025-11-10

This work is licensed under the Creative Commons Attribution 4.0 International License.

Artikel in diesem Heft

https://doi.org/10.1515/eng-2025-0142

Schlagwörter für diesen Artikel

advection–diffusion equation; Atangana–Baleanu–Caputo derivative; operational matrix method; Haar wavelets, fractional calculus, evolutionary algorithm

Creative Commons

BY 4.0