Abstract
Bu makalede, diferansiyel denklemlerin Haar dalgacık yöntemi ile sayısal çözümü,
çözüm aralığının üniform (eşit alt aralıklı) veya üniform olmayan (eşit olmayan alt
aralıklı) olmasına göre iki kategoride incelenmiştir.Sayısal çözümün yapıldığı aralıkta
artımların üniform olup olmaması Haar dalgacıkları ve integrallerinin
hesaplanmasında etkili olmaktadır. Haar dalgacıkları [0, 1] aralığında tanımlanır.
[0, 1] aralığından farklı bir aralıktaki bir diferansiyel problem için, çözüm aralığı alt
sınır ve üst sınır farkı ve bu farkın kuvvetlerinin kullanılması ile [0, 1] aralığındaki
Haar matrislerinin çözüm aralığına taşınabildiği görülmüştür. Böylece farklı bir
dönüşüme gerek kalmaz. Üniform olmayan Haar matrisleriyle, çözüm aralığının kritik
bölgelerinde hassasiyet artırılabilir. Kesinliği iyileştirmek için, çözüm bölgesinde bir
kollokasyon noktası sıklaştırma tekniği geliştirilmiştir. Hem geliştirilen hem de
literatürde mevcut olan sıklaştırma teknikleri ile yapılan sayısal çözümler, kesin
çözümle karşılaştırılmıştır. Bu çalışmada incelenen diferansiyel denklemler için
geliştirilen sıklaştırma yöntemi kullanılarak elde edilen sayısal sonuçlar ile kesin
çözümler arasında yüksek bir uyum gözlemlenmiştir.
Keywords
Haar dalgacıkları Diferansiyel Denklemler sayısal Çözüm Haar Matrisleri Üniform ve Üniform Olmayan Çözüm Aralığı
References
- Berwal N., Panchal D., Parihar CL. Haar wavelet method for numerical solution of telegraph equations. Italian Journal of Pure and Applied Mathematics 2013; (30): 317–328.
- Cattani C., Pecoraro M. Nonlinear differential equations in wavelet bases 2000; 3(4): 4–10.
- Cattani C. Haar wavelets based technique in evolution problems. Proceedings of the Estonian Academy of Sciences, Physics, Mathematics 2004; 53(1): 45.
- Chen CF., Hsiao CH. Haar wavelet method for solving lumped and distributed-parameter systems. IEE Proceedings - Control Theory and Applications 1997; 144(1).
- Graps A. An introduction to wavelets. IEEE Computational Science and Engineering 1995; 2(2): 50–61.
- Haar A. Zur theorie der orthogonalen funktionensysteme. Mathematische Annalen 1910; 69(3): 331–371.
- Heydari M., Avazzadeh Z., Hosseinzadeh N. Haar wavelet method for solving high-order differential equations with multi-point boundary conditions. Journal of Applied and Computational Mechanics 2022; 8(2): 528–544.
- Lepik Ü. Numerical solution of differential equations using Haar wavelets. Mathematics and Computers in Simulation 2005; 68(2): 127–143.
- Lepik Ü. Numerical solution of evolution equations by the Haar wavelet method. Applied Mathematics and Computation 2007; 185(1): 695–704.
- Lepik Ü. Solving integral and differential equations by the aid of non-uniform Haar wavelets. Applied Mathematics and Computation 2008; 198(1): 326–332.
- Lepik, Ü. Haar wavelet method for solving stiff differential equations. Mathematical Modelling and Analysis 2009a; 14(4): 467–481.
- Lepik Ü. Solving fractional integral equations by the Haar wavelet method. Applied Mathematics and Computation 2009b; 214(2): 468–478.
- Lepik Ü. Solving PDEs with the aid of two-dimensional Haar wavelets. Computers & Mathematics with Applications 2011; 61(7): 1873–1879.
Abstract
In this paper, the numerical solution of differential equations with the Haar wavelet
method is examined in two categories according to whether the solution interval is
uniform (equal sub-spaced) or non-uniform (unequal sub-spaced). Whether the
increments are uniform or not in the range of numerical solution is effective in the
calculation of Haar wavelets and their integrals. Haar wavelets are defined in the range
[0, 1]. A differential problem is defined in an interval different from the interval [0, 1],
it is seen that the matrices defined in the solution interval can be defined by
multiplying a coefficient obtained depending on the interval and the powers of this
coefficient with the Haar matrices. Thus, there is no need for a different
transformation. With non-uniform Haar matrices, the precision can be increased in the
critical regions of the solution range. To improve the precision a collocation point
increment method was developed in the solution region. Numerical solutions both
developed and available in the literature are compared to exact solution. For the
examined differential equations in this work, good agreements have been observed
between the obtained numerical results that employ developed increment method and
the exact solutions.
Keywords
Haar Wavelets Differential Equations Numerical Solutions Haar Matricies Uniform and non-uniform interval solution
References
- Berwal N., Panchal D., Parihar CL. Haar wavelet method for numerical solution of telegraph equations. Italian Journal of Pure and Applied Mathematics 2013; (30): 317–328.
- Cattani C., Pecoraro M. Nonlinear differential equations in wavelet bases 2000; 3(4): 4–10.
- Cattani C. Haar wavelets based technique in evolution problems. Proceedings of the Estonian Academy of Sciences, Physics, Mathematics 2004; 53(1): 45.
- Chen CF., Hsiao CH. Haar wavelet method for solving lumped and distributed-parameter systems. IEE Proceedings - Control Theory and Applications 1997; 144(1).
- Graps A. An introduction to wavelets. IEEE Computational Science and Engineering 1995; 2(2): 50–61.
- Haar A. Zur theorie der orthogonalen funktionensysteme. Mathematische Annalen 1910; 69(3): 331–371.
- Heydari M., Avazzadeh Z., Hosseinzadeh N. Haar wavelet method for solving high-order differential equations with multi-point boundary conditions. Journal of Applied and Computational Mechanics 2022; 8(2): 528–544.
- Lepik Ü. Numerical solution of differential equations using Haar wavelets. Mathematics and Computers in Simulation 2005; 68(2): 127–143.
- Lepik Ü. Numerical solution of evolution equations by the Haar wavelet method. Applied Mathematics and Computation 2007; 185(1): 695–704.
- Lepik Ü. Solving integral and differential equations by the aid of non-uniform Haar wavelets. Applied Mathematics and Computation 2008; 198(1): 326–332.
- Lepik, Ü. Haar wavelet method for solving stiff differential equations. Mathematical Modelling and Analysis 2009a; 14(4): 467–481.
- Lepik Ü. Solving fractional integral equations by the Haar wavelet method. Applied Mathematics and Computation 2009b; 214(2): 468–478.
- Lepik Ü. Solving PDEs with the aid of two-dimensional Haar wavelets. Computers & Mathematics with Applications 2011; 61(7): 1873–1879.
Details
| Primary Language | Turkish |
|---|---|
| Journal Section | RESEARCH ARTICLES |
| Authors | |
| Publication Date | January 22, 2024 |
| Submission Date | April 6, 2023 |
| Acceptance Date | July 21, 2023 |
| Published in Issue | Year 2024 Volume: 7 Issue: 1 |
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