29 research outputs found
Numerical investigation of Differential Biological-Models via GA-Kansa Method Inclusive Genetic Strategy
In this paper, we use Kansa method for solving the system of differential
equations in the area of biology. One of the challenges in Kansa method is
picking out an optimum value for Shape parameter in Radial Basis Function to
achieve the best result of the method because there are not any available
analytical approaches for obtaining optimum Shape parameter. For this reason,
we design a genetic algorithm to detect a close optimum Shape parameter. The
experimental results show that this strategy is efficient in the systems of
differential models in biology such as HIV and Influenza. Furthermore, we prove
that using Pseudo-Combination formula for crossover in genetic strategy leads
to convergence in the nearly best selection of Shape parameter.Comment: 42 figures, 23 page
A New Approach in Solving Regular and Singular Conformable Fractional Coupled Burger's Equations
The conformable double ARA decomposition approach is presented in this
current study to solve one-dimensional regular and singular conformable
functional Burger's equations. We investigate the conformable double ARA
transform's definition, existence requirements, and some basic properties. In
this study, we introduce a novel interesting method that combines the double
ARA transform with Adomian's decomposition method, in order to find the precise
solutions of some nonlinear fractional problems. Moreover, we use the new
approach to solve Burgers' equations for both regular and singular conformable
fractional coupled systems. We also provide several instances to demonstrate
the usefulness of the current study. Mathematica software has been used to get
numerical results
Numerical study of chemical reaction effects in magnetohydrodynamic Oldroyd B oblique stagnation flow with a non-Fourier heat flux model
Reactive magnetohydrodynamic (MHD) flows arise in many areas of nuclear reactor transport. Working fluids in such systems may be either Newtonian or non-Newtonian. Motivated by these applications, in the current study, a mathematical model is developed for electrically-conducting viscoelastic oblique flow impinging on stretching wall under transverse magnetic field. A non-Fourier Cattaneo-Christov model is employed to simulate thermal relaxation effects which cannot be simulated with the classical Fourier heat conduction approach. The Oldroyd-B non-Newtonian model is employed which allows relaxation and retardation effects to be included. A convective boundary condition is imposed at the wall invoking Biot number effects. The fluid is assumed to be chemically reactive and both homogeneous-heterogeneous reactions are studied. The conservation equations for mass, momentum, energy and species (concentration) are altered with applicable similarity variables and the emerging strongly coupled, nonlinear non-dimensional boundary value problem is solved with robust well-tested Runge-Kutta-Fehlberg numerical quadrature and a shooting technique with tolerance level of 10−4. Validation with the Adomian decomposition method (ADM) is included. The influence of selected thermal (Biot number, Prandtl number), viscoelastic hydrodynamic (Deborah relaxation number), Schmidt number, magnetic parameter and chemical reaction parameters, on velocity, temperature and concentration distributions are plotted for fixed values of geometric (stretching rate, obliqueness) and thermal relaxation parameter. Wall heat transfer rate (local heat flux) and wall species transfer rate (local mass flux) are also computed and it is observed that local mass flux increases with strength of heterogeneous reactions whereas it decreases with strength of homogeneous reactions. The results provide interesting insights into certain nuclear reactor transport phenomena and furthermore a benchmark for more general CFD simulations
A modern analytic method to solve singular and non-singular linear and non-linear differential equations
This article circumvents the Laplace transform to provide an analytical solution in a power series form for singular, non-singular, linear, and non-linear ordinary differential equations. It introduces a new analytical approach, the Laplace residual power series, which provides a powerful tool for obtaining accurate analytical and numerical solutions to these equations. It demonstrates the new approach’s effectiveness, accuracy, and applicability in several ordinary differential equations problem. The proposed technique shows the possibility of finding exact solutions when a pattern to the series solution obtained exists; otherwise, only rough estimates can be given. To ensure the accuracy of the generated results, we use three types of errors: actual, relative, and residual error. We compare our results with exact solutions to the problems discussed. We conclude that the current method is simple, easy, and effective in solving non-linear differential equations, considering that the obtained approximate series solutions are in closed form for the actual results. Finally, we would like to point out that both symbolic and numerical quantities are calculated using Mathematica software
An application of modern analytical solution techniques to nonlinear partial differential equations.
Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2013.Many physics and engineering problems are modeled by differential equations. In
many instances these equations are nonlinear and exact solutions are difficult to
obtain. Numerical schemes are often used to find approximate solutions. However,
numerical solutions do not describe the qualitative behaviour of mechanical systems
and are insufficient in determining the general properties of certain systems of
equations. The need for analytical methods is self-evident and major developments
were seen in the 1990’s. With the aid of faster processing equipment today, we are
able to compute analytical solutions to highly nonlinear equations that are more
accurate than numerical solutions.
In this study we discuss solutions to nonlinear partial differential equations with
focus on non-perturbation analytical methods. The non-perturbation methods of
choice are the homotopy analysis method (HAM) developed by Shijun Liao and the
variational iteration method (VIM) developed by Ji-Huan He. The aim is to compare the solutions obtained by these modern day analytical methods against each other
focusing on accuracy, convergence and computational efficiency.
The methods were applied to three test problems, namely, the heat equation, Burgers
equation and the Bratu equation. The solutions were compared against both the exact
results as well as solutions generated using the finite difference method, in some cases.
The results obtained show that the HAM successfully produces solutions which are
accurate, faster converging and requires less computational resources than the VIM.
However, the VIM still provides accurate solutions that are also in good agreement
with the closed form solutions of the test problems. The FDM also produced good
results which were used as a further comparison to the analytical solutions. The
findings of this study is in agreement with those published in the literature