Using matrix-based rationalized Haar wavelet method for solving consolidation equation

Settlement and pore water pressure in soils during the consolidation process are of great concern for geotechnical and structural engineers. Therefore, different analytical and numerical methods have been proposed for its solution considering the complexity of soil behavior and geometry of soil layers. In this research study, consolidation equation has been solved using the matrix-based rationalized Haar wavelet transform method. Two typical and most commonly used cases are solved and compared with the classical solution method which is based on Taylor series expansion. The proposed method is fast, convenient and requires less computational effort due to its matrix calculation implementation and it yields accurate results when compared with the classical solution. Furthermore, this method may be used for implementing explicit functions rather than a constant value for the consolidation coefficient (c[Formula: see text]).

Solving of nonlinear Fredholm integro-differential equation in a complex plane with rationalized Haar wavelet bases

Everyone knows about the complicated solution of the nonlinear Fredholm integro-differential equation in general. Hence, often, authors attempt to obtain the approximate solution. In this paper, a numerical method for the solutions of the nonlinear Fredholm integro-differential equation (NFIDE) of the second kind in the complex plane is presented. In fact, by using the properties of Rationalized Haar (RH) wavelet, we try to give the solution of the problem. So far, as we know, no study has yet been attempted for solving the NFIDE in the complex plane. For this purpose, we introduce the continuous integral operator and real valued function. The Banach fixed point theorem guarantees that, under certain assumptions, the integral operator has a unique solution. Furthermore, we give an upper bound for the error analysis. An algorithm is presented to compute and illustrate the solutions for some numerical examples.