Haar wavelet kolokační metoda vyššího řádu pro systémy integro-diferenciálních rovnic

Abstract

In this thesis, the Haar Wavelet Collocation Method (HWCM) and its higher-order extension (HHWCM) are used to solve a coupled system of second-order integro-differential equations (IDEs) numerically in an efficient manner. The piecewise constant nature and orthogonality of the Haar wavelet basis functions provide a sparse system and computational simplicity. To improve accuracy and smoothness, two techniques are developed: one based on the standard HWCM, which approximates the second-order derivatives, and the other on the HHWCM, which approximates the fourth-order derivatives. Collocation points are used to convert the system of IDEs into a collection of algebraic equations, and integration constants are then calculated by applying the initial conditions. To confirm the precision and dependability of the suggested techniques, a benchmark example with known exact solutions is examined. The numerical findings reveal that HHWCM produces better precision than the traditional HWCM, particularly for smooth solutions, and they also show great agreement with the exact solutions. In applied mathematics and engineering, the suggested methodology shows promise for solving intricate systems of integro-differential equations.In this thesis, the Haar Wavelet Collocation Method (HWCM) and its higher-order extension (HHWCM) are used to solve a coupled system of second-order integro-differential equations (IDEs) numerically in an efficient manner. The piecewise constant nature and orthogonality of the Haar wavelet basis functions provide a sparse system and computational simplicity. To improve accuracy and smoothness, two techniques are developed: one based on the standard HWCM, which approximates the second-order derivatives, and the other on the HHWCM, which approximates the fourth-order derivatives. Collocation points are used to convert the system of IDEs into a collection of algebraic equations, and integration constants are then calculated by applying the initial conditions. To confirm the precision and dependability of the suggested techniques, a benchmark example with known exact solutions is examined. The numerical findings reveal that HHWCM produces better precision than the traditional HWCM, particularly for smooth solutions, and they also show great agreement with the exact solutions. In applied mathematics and engineering, the suggested methodology shows promise for solving intricate systems of integro-differential equations.