## The numerical solution of second-order boundary-value problems by collocation method with the Haar wavelets.(English)Zbl 1205.74187

Summary: An efficient numerical method based on uniform Haar wavelets is proposed for the numerical solution of second-order boundary-value problems (BVPs) arising in the mathematical modeling of deformation of beams and plate deflection theory, deflection of a cantilever beam under a concentrated load, obstacle problems and many other engineering applications. The Haar wavelet basis permits to enlarge the class of functions used so far in the collocation framework. The performance of the Haar wavelets is compared with the Walsh wavelets, semi-orthogonal B-spline wavelets, spline functions, Adomian decomposition method (ADM), finite difference method, and Runge-Kutta method coupled with nonlinear shooting method. A more accurate solution can be obtained by wavelet decomposition in the form of a multi-resolution analysis of the function which represents the solution of a given problem. Through this analysis the solution is found on the coarse grid points, and then refined towards higher accuracy by increasing the level of the Haar wavelets. Neumann’s boundary conditions which are problematic for most of the numerical methods are automatically coped with. The main advantage of the Haar wavelet based method is its efficiency and simple applicability for a variety of boundary conditions. The convergence analysis of the proposed method alongside numerical procedure for multi-point boundary-value problems are given to test wider applicability and accuracy of the method.

### MSC:

 74S30 Other numerical methods in solid mechanics (MSC2010) 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations 65T60 Numerical methods for wavelets 74K10 Rods (beams, columns, shafts, arches, rings, etc.)
Full Text:

### References:

 [1] Na, T.Y., Computational methods in engineering boundary value problems, (1979), Academic Press New York · Zbl 0456.76002 [2] Bisshopp, K.; Drucker, D.C., Appl. math., 3, 272-275, (1945) [3] Glabisz, W., The use of Walsh-wavelets packets in linear boundary value problems, Comput. struct., 82, 131-141, (2004) [4] Siraj-ul-Islam; Noor, M.A.; Tirmizi, I.A.; Khan, M.A., Quadratic non-polynomial spline approach to the solution of a system of second-order boundary-value problems, Appl. math. comput., 179, 153-160, (2006) · Zbl 1100.65067 [5] Robert, S.; Shipman, J., Solution of troesch’s two-point boundary value problems by shooting techniques, J. comput. phys., 10, 232-241, (1972) · Zbl 0247.65052 [6] Wiebel, E., Confinement of a plasma column by radiation pressure in the plasma in a magnetic field, (1958), Stanford University Press California [7] Keller, H.H.; Holdrege, E.S., Radiation heat transfer for annular fins of trapezoidal profile, Int. J. high perform. comput. appl., 92, 113-116, (1970) [8] Tatari, M.; Dehgan, M., The use of the Adomian decomposition method for solving multipoint boundary value problems, Phys. scr., 73, 672-676, (2006) [9] Al-Said, E., The use of cubic splines in the numerical solution of system of second-order boundary-value problems, Int. J. comput. math. appl., 42, 861-869, (2001) · Zbl 0983.65089 [10] Lakestani, M.; Dehgan, M., The solution of a second-order nonlinear differential equation with Neumann boundary conditions using semi-orthogonal $$B$$-spline wavelets, Int. J. comput. math., 83, 685-694, (2006) · Zbl 1114.65090 [11] Tirmizi, I.A.; Twizell, E.H., Higher-order finite-difference methods for nonlinear second-order two-point boundary-value problems, Appl. math. lett., 15, 897-902, (2002) · Zbl 1013.65078 [12] Katti, C.P.; Baboo, S., Radiation heat transfer for annular fins of trapezoidal profile, Appl. math. comput., 75, 287-302, (1996) · Zbl 0849.65061 [13] Siraj-ul-Islam; Tirmizi, I.A.; Haq, F., Quartic non-polynomial splines approach to the solution of a system of second-order boundary-value problems, Int. J. high perform. comput. appl., 21, 42-49, (2007) [14] Hsiao, C.H.; Wang, W.-J., Haar wavelet approach to nonlinear stiff systems, Math. comput. simulation, 57, 347-353, (2001) · Zbl 0986.65062 [15] Hsiao, C.H., Haar wavelet approach to linear stiff systems, Math. comput. simulation, 64, 561-567, (2004) · Zbl 1039.65058 [16] Lepik, U., Numerical solution of differential equations using Haar wavelets, Math. comput. simulation, 68, 127-143, (2005) · Zbl 1072.65102 [17] Lepik, U., Numerical solution of evolution equations by the Haar wavelet method, Appl. math. comput., 185, 695-704, (2007) · Zbl 1110.65097 [18] Goswami, J.C.; Chan, Fundamentals of wavelets. theory, algorithms, and applications, (1999), John Wiley and Sons New York · Zbl 1209.65156 [19] Mallat, S., A wavelet tour of signal processing, (1999), Academic Press New York · Zbl 0998.94510 [20] Jameson, L.; Waseda, T., Error estimation using wavelet analysis for data assimilation: eewadai, J. atmos. Ocean. technol., 17, 1235-1246, (2000) [21] Strang, G.; Nguyen, T., Wavelets and filter banks, (1996), Wellesley-Cambridge Press · Zbl 1254.94002 [22] Hashih, H.; Behiry, S.; El-Shamy, N., Numerical integration using wavelets, Appl. math. comput., (2009) · Zbl 1162.65322
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.