Abstract: In this work, we present an efficient approach for
solving variable-order time-fractional partial differential equations,
which are based on Legendre and Laguerre polynomials. First, we
introduced the pseudo-operational matrices of integer and variable
fractional order of integration by use of some properties of
Riemann-Liouville fractional integral. Then, applied together with
collocation method and Legendre-Laguerre functions for solving
variable-order time-fractional partial differential equations. Also, an
estimation of the error is presented. At last, we investigate numerical
examples which arise in physics to demonstrate the accuracy of the
present method. In comparison results obtained by the present method
with the exact solution and the other methods reveals that the method
is very effective.
Abstract: In this study, one dimensional phase change problem
(a Stefan problem) is considered and a numerical solution of this
problem is discussed. First, we use similarity transformation to
convert the governing equations into ordinary differential equations
with its boundary conditions. The solutions of ordinary differential
equation with the associated boundary conditions and interface
condition (Stefan condition) are obtained by using a numerical
approach based on operational matrix of differentiation of shifted
second kind Chebyshev wavelets. The obtained results are compared
with existing exact solution which is sufficiently accurate.
Abstract: A numerical method for solving nonlinear Fredholm integral equations of second kind is proposed. The Fredholm type equations which have many applications in mathematical physics are then considered. The method is based on hybrid function approximations. The properties of hybrid of block-pulse functions and Chebyshev polynomials are presented and are utilized to reduce the computation of nonlinear Fredholm integral equations to a system of nonlinear. Some numerical examples are selected to illustrate the effectiveness and simplicity of the method.
Abstract: To increase precision and reliability of automatic control systems, we have to take into account of random factors affecting the control system. Thus, operational matrix technique is used for statistical analysis of first order plus time delay system with uniform random parameter. Examples with deterministic and stochastic disturbance are considered to demonstrate the validity of the method. Comparison with Monte Carlo method is made to show the computational effectiveness of the method.
Abstract: A numerical method for Riccati equation is presented in this work. The method is based on the replacement of unknown functions through a truncated series of hybrid of block-pulse functions and Chebyshev polynomials. The operational matrices of derivative and product of hybrid functions are presented. These matrices together with the tau method are then utilized to transform the differential equation into a system of algebraic equations. Corresponding numerical examples are presented to demonstrate the accuracy of the proposed method.
Abstract: In this paper, we have proposed a Haar wavelet quasilinearization
method to solve the well known Blasius equation. The
method is based on the uniform Haar wavelet operational matrix
defined over the interval [0, 1]. In this method, we have proposed the
transformation for converting the problem on a fixed computational
domain. The Blasius equation arises in the various boundary layer
problems of hydrodynamics and in fluid mechanics of laminar
viscous flows. Quasi-linearization is iterative process but our
proposed technique gives excellent numerical results with quasilinearization
for solving nonlinear differential equations without any
iteration on selecting collocation points by Haar wavelets. We have
solved Blasius equation for 1≤α ≤ 2 and the numerical results are
compared with the available results in literature. Finally, we
conclude that proposed method is a promising tool for solving the
well known nonlinear Blasius equation.