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Solving parabolic differential equations via Haar wavelets: A focus on integral boundary conditions
with integral boundary conditions related to heat
∂s(1, t) transfer, emphasizing accurate results with mini-
= g 2 (t), t ∈ [0, T],
∂κ mal computational cost. The study highlights the
b method’s capability to effectively handle sharp
Z (1)
s(κ, t)dκ = m(t), b ∈ (0, 1), t ∈ [0, T], transitions over time, including boundary layers,
0 21
shock layers, and wave fronts. Additionally,
where a, b, and c are constants, and f, g 1 , g 2 , and
discusses error estimates for discretization, fur-
m are prescribed functions.
ther reflecting the method’s established reliability.
Various physical and chemical processes The article 22 introduces an operational matrix
can be modeled through the problem (1), as method utilizing hybrid Legendre Block-Pulse
demonstrated in. 5–11 For the purpose of self- functions to solve PDEs with nonlocal boundary
containment, we briefly describe some examples integral conditions. By applying operational ma-
here. For instance, if s denotes the concentration trices and analyzing convergence through theo-
of a chemical in a diffusion process, then m(t) rep- rems and lemmas, the method reduces integro-
resents the total mass of the chemical in the region PDEs to algebraic systems, demonstrating accu-
0 < κ < b at time t. Similarly, if s represents the racy and effectiveness through numerical exam-
temperature in a heat conduction problem, then ples compared to established methods. In, 23 the
m(t) corresponds to the internal energy content of authors used two efficient numerical methods to
the region 0 < κ < b at time t. In another exam- examine the Poisson equation under distinct non-
ple, if s describes the distribution of impurities in local boundary conditions. Additionally, 24 em-
a plate over the interval 0 < κ < 1, the problem ploys meshless method utilizing radial basis func-
(1) models a technological process for the exter- tion approximations to interpolate field variables
nal elimination of gas. This process is applied, across subdomain boundaries. In, 25 the author
for example, in refining silicon plates to remove addressed a specific type of Schr”odinger equa-
impurities, where m(t) represents the total mass tion featuring logarithmic nonlinearity. In, 26 the
of impurities in the plate 0 < κ < 1. A sim- complex nonlinear Lorenz system is analyzed for
ilar problem arises in biochemistry when b = 1 accurate parameter estimation and effective re-
and m are constants. In this case, the condition construction of the system dynamics, 27 the fo-
b
R
0 s(κ, t)dκ = m(t) reflects the conservation of cus was on higher-order finite-volume methods
the protein. applied to solve elliptic boundary value prob-
Numerical methods play a critical role in lems. These methods ensure accurate solutions
solving complex mathematical models that arise to PDEs by demonstrating that their bilinear
in science and engineering. 12,13 They enable re- forms are uniformly elliptic, thereby achieving op-
searchers to simulate real-world phenomena with timal error estimates. The Haar wavelets col-
high accuracy. 14,15 In biomedical sciences, numer- location method (HWCM) has recently gained
ical techniques are essential for modeling biolog- traction and is extensively applied in various
ical systems, analyzing treatment strategies, and fields such as signal processing, numerical anal-
predicting disease progression. 16 Moreover, ad- ysis, and engineering. This method employs the
vancements in numerical algorithms continue to Haar function in several approaches, including the
enhance computational efficiency, 17 making large- wavelet Galerkin technique, 28 meshless wavelet
scale simulations feasible in scientific and indus- schemes, 29 wavelet collocation-based schemes, 30
trial applications. In view of the applications Daubechies wavelet procedure 31 and other weak
of partial differential equations (PDEs) with in- and strong formulations. 32–35
tegral boundary conditions, several methods are Many research efforts have employed diverse
used for the efficient solution of these problems. Haar function-based methods to solve different
For example, a finite difference method for solv- problems in science and engineering. 36,37 In, 38
ing second-order non-linear PDEs with two-point wavelet decomposition is used to filter the orig-
boundary conditions is discussed in, 18 while in, 19 inal data, effectively removing the negative im-
the author applied the finite difference approach pact of noise, and enhancing the anomaly detec-
to novel boundary value problems. This paper tion model’s performance. Recently, the Haar
introduces a finite difference method for address- wavelet approach has been employed as a math-
ing second-order boundary value problems in or- ematical technique for handling differential sce-
dinary differential equations (ODEs) with an in- narios, particularly effective in solving parabolic
terior singularity. PDEs with integral boundary conditions. Haar
In, 20 the authors used the finite element wavelets, which form an orthonormal basis for
method to solve parabolic differential equations square-integrable functions, are utilized directly
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