Shiralasetti, S., S, K. (2018). Some results on Haar wavelets matrix through linear algebra. Wavelet and Linear Algebra, 4(2), 49-59. doi: 10.22072/wala.2018.53432.1093

Siddu Shiralasetti; Kumbinarasaiah S. "Some results on Haar wavelets matrix through linear algebra". Wavelet and Linear Algebra, 4, 2, 2018, 49-59. doi: 10.22072/wala.2018.53432.1093

Shiralasetti, S., S, K. (2018). 'Some results on Haar wavelets matrix through linear algebra', Wavelet and Linear Algebra, 4(2), pp. 49-59. doi: 10.22072/wala.2018.53432.1093

Shiralasetti, S., S, K. Some results on Haar wavelets matrix through linear algebra. Wavelet and Linear Algebra, 2018; 4(2): 49-59. doi: 10.22072/wala.2018.53432.1093

Some results on Haar wavelets matrix through linear algebra

Can we characterize the wavelets through linear transformation? the answer for this question is certainly YES. In this paper we have characterized the Haar wavelet matrix by their linear transformation and proved some theorems on properties of Haar wavelet matrix such as Trace, eigenvalue and eigenvector and diagonalization of a matrix.

[1] J. Astola and L. Yaroslavsky, Advances in Signal Transforms:Theory and Applications, Hindawi Publishing Corporation, 2007. [2] P. Chang and P. Piau, Haar wavelet matrices designation in numerical solution of ordinary differential equations, Int. J. Appl. Math., 38 (2008), 3-11. [3] C.F. Chen and C.H. Hsiao, Haar wavelet method for solving lumped and distributed-parameter systems, IEE Proc., Control Theory Appl., 144 (1997), 87-94. [4] C. Capilla, Application of the haar wavelet transform to detect microseismic signal arrivals, Journal of Applied Geophysics, 59 (2006), 36-46. [5] M.D. Choi, Completely positive linear maps on complex matrices, Linear Algebra Appl., 10 (1975), 285-290. [6] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992. [7] J. Eisenfeld, Block diagonalization and eigenvalues, Linear Algebra Appl., 15 (1976), 205-215. [8] G. Hariharan and K. Kannan, An overview of haar wavelet method for solving differential and integral equations, World Applied Sciences Journal, 23(12) (2013), 01-14. [9] M.H. Heydaria and F.M. Maalek Ghainia, Legendre wavelets method for numerical solution of time-fractional heat equation, Wavel. Linear Algebra, 1 (2014), 19-31. [10] R.D. HILL, Linear transformations which preserve hermitian matrices, Linear Algebra Appl., 6 (1973), 257-262. [11] C.H. Hsiao, Haar wavelet approach to linear stiff systems, Math. Comput. Simul., 64 (2004), 561-567. [12] I. Aziz and S.U. Islam, New algorithms for the numerical solution of nonlinear fredholm and volterra integral equations using Haar wavelets, J. Comput. Appl. Math., 239 (2013), 333-345. [13] R. Jiwari, A Haar wavelet quasilinearization approach for numerical simulation of burger's equation, Comput. Phys. Commun., 183 (2012), 2413-2423. [14] R. Jiwari, A hybrid numerical scheme for the numerical solution of the Burger's Equation, Comput. Phys. Commun., 188 (2015), 59-67. [15] K. Nouri, Application of shannon wavelet for solving boundary value problems of fractional differential equations, Wavel. Linear Algebra, 1 (2014), 33-42. [16] U. Lepik, Numerical solution of evolution equations by the Haar wavelet method, Appl. Math. Comput., 185 (2007), 695-704. [17] U. Lepik, Numerical solution of differential equations using Haar wavelets, Math. Comput. Simul., 68 (2005), 127-143. [18] U. Lepik, H. Hein, Haar Wavelets With Applications, Springer International publishing, 2014. [19] M. Marcus, Linear transformations on matrices, Journal of research of the notional bureau of standards-B. Mathematical sciences, 75B (1971), 107-113. [20] J. K. Merikoski, P.H. George and S.H. Wolkowicz, Bounds for ratios of eigenvalues using traces, Linear Algebra Appl., 55 (1983), 105-124. [21] A. Mohammed, M. Balarabe and A.T. Imam, Rhotrix linear transformation, Advances in Linear Algebra and Matrix Theory, 2 (2012), 43-47. [22] R.K. Mallik, The inverse of a tridiagonal matrix, Linear Algebra Appl., 325 (2001), 109-139. [23] U. Saeed and M. Rehman, Haar wavelet quasilinearization technique for fractional nonlinear differential equations, Appl. Math. Comput., 220 (2013), 630-648. [24] S.C. Shiralashetti and A.B. Deshi, An effective Haar wavelet collocation method for the numerical solution of multiterm fractional differential equation, Nonlinear Dyn., 83 (2016), 293-303. [25] S.C. Shiralashetti, A.B. Deshi and P.B. Mutalik Desai, Haar wavelet collocation method for the numerical solution of singular initial value problems, Ain Shams Engineering Journal, 7 (2016), 663-670. [26] S.C. Shiralashetti, L.M. Angadi, A.B. Deshi and M.H. Kantli, Haar wavelet method for the numerical solution of Klein-Gordan equations, Asian-Eur. J. Math., 9(1), (2016) 1-14. [27] S.C. Shiralashetti, M.H. Kantli and A.B. Deshi, Haar wavelet based numerical solution of nonlinear differential equations arising in fluid dynamics, International Journal of Computational Materials Science and Engineering, 5(2) (2016), 1-13. [28] S.C. Shiralashetti, L.M. Angadi, A.B. Deshi and M.H. Kantli, Haar wavelet method for the numerical solution of Benjamin-Bona-Mahony equations, Journal of Information and Computing Sciences, 11(2) (2016), 136-145. [29] S.C. Shiralashetti, L.M. Angadi, M.H. Kantli and A.B. Deshi, Numerical solution of parabolic partial differential equations using adaptive gird Haar wavelet collocation method, Asian-Eur. J. Math., 10(1) (2017), 1-11. [30] S.C. Shiralashetti, A.B. Deshi, S.S. Naregal and B. Veeresh, Wavelet series solutions of the nonlinear emden-flower type equations, International Journal of Scientific and Innovative Mathematical Research, 3(2) (2015), 558-567. [31] S.C. Shiralashetti, P.B. Mutalik Desai and A. B. Deshi, A comparative study of finite element method and Haar wavelet collocation method for the numerical solution of nonlinear ordinary differential equations, International Journal of Modern Mathematical Sciences, 13(3) (2015), 228-250. [32] S.U. Islam, I. Aziz, A. Fhaid and A. Shah, A numerical assessment of parabolic partial differential equations using Haar and legendre wavelets, Appl. Math. Modelling, 37 (2013), 9455-9481. [33] S.U. Islam, I. Aziz, and A.S. Al-Fhaid, An improved method based on Haar wavelets for numerical solution of nonlinear integral and Integro-differential equations of first and higher orders, J. Comput. Appl. Math., 260 (2014), 449-469. [34] G. Strang, Linear Algebra and Its Applications, Cengage Learning, (2005).