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A robust framework for solving PDEs: Biorthogonal spline wavelet methods Cover

A robust framework for solving PDEs: Biorthogonal spline wavelet methods

International Journal of Mathematics and Computer in Engineering
AHEAD OF PRINT
By: Mutaz Mohammad and  Alexander Trounev  
Open Access
|Feb 2026

Figures & Tables

Fig. 1

Biorthogonal spline scaling and wavelet functions of order 2 and dual order 4, and of order 3 and dual order 5.
Biorthogonal spline scaling and wavelet functions of order 2 and dual order 4, and of order 3 and dual order 5.

Fig. 2

Graphs of the exact solution, numerical solution at different time steps, and comparison between the exact and numerical solutions for n = 2 and M = 8.
Graphs of the exact solution, numerical solution at different time steps, and comparison between the exact and numerical solutions for n = 2 and M = 8.

Fig. 3

Graphs of the numerical solution at different time steps and the error bound for n = 2 and M = 8.
Graphs of the numerical solution at different time steps and the error bound for n = 2 and M = 8.

Fig. 4

Graphs of the numerical solution at different time steps, and comparison between the exact and numerical solutions for n = 3 and M = 16.
Graphs of the numerical solution at different time steps, and comparison between the exact and numerical solutions for n = 3 and M = 16.

Fig. 5

Graphs of the numerical solution at different time steps, and the error bound for n = 3 and M = 16.
Graphs of the numerical solution at different time steps, and the error bound for n = 3 and M = 16.

Fig. 6

Convergence rate graphs of the numerical solutions for the Haar and BSW-2 wavelet methods, respectively.
Convergence rate graphs of the numerical solutions for the Haar and BSW-2 wavelet methods, respectively.

Fig. 7

Log-log plots of numerical solutions for Haar and BSW-2 wavelet methods, respectively.
Log-log plots of numerical solutions for Haar and BSW-2 wavelet methods, respectively.

Numerical values for xl,t and corresponding u[xl,t] for two methods_

t=0t=0t=0t=0.3t=0.3t=0.6t=0.6t=0.9t=0.9
xlBSW-2HaarBSW-2HaarBSW-2HaarBSW-2Haar
116{1 \over {16}}000.18675610.18500200.21581500.21407600.07522770.0745150
316{3 \over {16}}000.14514830.14401100.17057200.16952200.05487610.0542550
516{5 \over {16}}000.11305180.11202700.13302900.13206800.04340340.0430950
716{7 \over {16}}000.08834590.08752800.10379560.10302300.03363080.0332450
916{9 \over {16}}000.06888530.06826100.08101880.08035600.02637300.0261660
1116{{11} \over {16}}000.05377680.05329200.06320590.06273200.02050800.0203022
1316{{13} \over {16}}000.04195920.04150600.04932540.04882100.01604450.0159001
1516{{15} \over {16}}000.03274240.03240600.03849100.03810900.01249740.0124034
DOI: https://doi.org/10.2478/ijmce-2026-0005 | Journal eISSN: 2956-7068
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Language: English
Submitted on: Oct 7, 2024
|
Accepted on: Feb 14, 2025
|
Published on: Feb 2, 2026
Published by: Harran University
In partnership with: Paradigm Publishing Services
Publication frequency: 2 issues per year
Keywords:
PDEs,
biorthogonal spline wavelets,
Diffusion equation,
collocation methods,
boundary conditions
Related subjects:
Computer sciences,
Computer sciences, other,
Engineering,
Introductions and overviews,
Engineering, other,
Mathematics,
General mathematics,
Physics,
Physics, other

© 2026 Mutaz Mohammad, Alexander Trounev, published by Harran University
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

AHEAD OF PRINT