Page 54 - IJOCTA-15-4

MN. Khan et.al. / IJOCTA, Vol.15, No.4, pp.594-609 (2025)
in this method. The HWCM approximates the us
layout of the differential equations by select- 
κ − ς 1 ς 1 ≤ κ < ς 2 ,
ing the midpoints of Haar wavelets as colloca- 
ℜ n,1 (κ) = ς 3 − κ ς 2 ≤ κ < ς 3 ,
tion points, where the differential equations are 
 0 otherwise,
evaluated. The piecewise constant nature of

Haar wavelets, forming an orthonormal basis for  0 0 ≤ κ < ς 1 ,

square-integrable functions, allows them to be di-  (κ − ς 1 ) 2 ς 1 ≤ κ < ς 2 ,
 1

rectly combined to approximate the set of differ- ℜ n,2 (κ) = 2
1 2 − (ς 1 − κ) 2 ς 2 ≤ κ < ς 3 ,
1
ential equations. The collocation points are cho-  4q 2



sen at the midpoints of the Haar wavelets where  1 ς 3 ≤ κ < 1,
4q 2
the differential equation is evaluated. 
 0 0 ≤ κ < ς 1 ,
The content of the rest of the paper is or- 

 1 (κ − ς 1 ) 3 ς 1 ≤ κ < ς 2 ,

ganized as follows: in Section 2, the proposed 6
ℜ n,3 (κ) = 1 1 3
method is discussed briefly. In Section 3, the im- 2 (κ − ς 2 ) + (ς 3 − κ) ς 2 ≤ κ < ς 3 ,
 4q 6


plementation procedure of the proposed method  1 2 (κ − ς 2 ) ς 3 ≤ κ < 1,

4q
is given for 1D and 2D problems. In Section 4,
(7)
the numerical results and discussion are included.
(
Finally, in Section 5, some conclusions of the pro- 1 if n = 1,
ℜ n,2 (1) = 2
posed work are given. 1 if n > 1,
4q 2
(
1 if n = 1,
ℜ n,3 (1) = 6 (8)
2. Haar wavelets collocation method 1 2 (1 − ς 2 ) if n > 1,
4q
1
The paper 39 presents the Haar wavelet family for ( 24 if n = 1,
κ ∈ [0,1): ℜ n,4 (1) = 1 2 1 if n > 1,
4q 2 (1 − ς 2 ) + 192q 4

1 ς 1 ≤ κ < ς 2 ,
 where Equation (3) defines ς 1 , ς 2 , ς 3 , and m.
χ n (κ) = −1 ς 2 ≤ κ < ς 3 , Furthermore, we need to find:

0 otherwise, n = 2, 3, ...,
 Z 1
(2) ψ(κ)ℜ n,2 (t) dt, (9)
0
where
Consider a function ψ(κ) is provided. Evalu-
j (j + 0.5) (j + 1)
ς 1 = , ς 2 = , ς 3 = , ating the integral in Equation (9) is straightfor-
q q q
ward.
i
j = 0, 1, . . . , q − 1, q = 2 , i = 0, 1, . . . , I. The method’s complexity is linear with re-
(3) spect to the number of collocation points due
In this context, i represents the upper limit to the sparsity of Haar wavelet basis functions.
Compared to global spectral methods, the Haar
and j represents the lower limit. The index of χ n
is n = q + j + 1 in Equation (2). n = 2q = 2 I+1 method has lower memory requirements and
is the maximum value of n, where I is the high- avoids large, dense matrix systems.
est resolution level. For q = 1 and j = 0, the
smallest value of n is n = 2. The scaling function
3. Proposed methodology for 1D and
presumably corresponds to n = 1 as follows:
2D problems
(
1 0 ≤ κ < 1,
χ 1 (κ) = (4) 3.1. 1D problem with one integral BC
0 otherwise.
Consider the 1D equation 40 as follows :
In order to solve the PDE problems Equations
2
(32)-Equations (37), we must evaluate the follow- ∂s = γ ∂ s + f(κ, t), 0 < κ < 1, 0 < t ≤ T,
ing integrals using Haar wavelets. ∂t ∂κ 2
(10)
Z
κ
ℜ n,1 (κ) = χ n (t) dt, (5) with the initial, Dirichlet boundary, nonlocal in-
0 tegral boundary conditions
κ
Z
ℜ n,v (κ) = ℜ n,v−1 (t) dt, v = 2, 3, · · · . s(κ, 0) = ℜ(κ), 0 ≤ κ ≤ 1, (11)
0
(6) s(0, t) = g(t), 0 ≤ t ≤ T, (12)
Z 1
An analytical approach for computing these
ϑ(κ)s(κ, t) dκ = m(t). (13)
integrals is provided by Equation (2), which gives
0
596