Solutions of nonlinear fractional coupled Hirota-Satsuma-KdV Equation

Our interest in the present work is in implementing the FPSM to stress it power in handing the nonlinear fractional coupled Hirota-Satsuma-KdV Equation. The approximate analytical solution of this type equations are obtained

Application of fractional sub-equation method to nonlinear evolution equations

In this paper, we constructed a traveling wave solutions expressed by three types of functions, which are hyperbolic, trigonometric, and rational functions. By using a fractional sub-equation method for some space-time fractional nonlinear partial differential equations (FNPDE), which are considered models for different phenomena in natural and social sciences fields like engineering, physics, geology, etc. This method is a very effective and easy to investigate exact traveling wave solutions to FNPDE with the aid of the modified Riemann–Liouville derivative

A Domain Decomposition Method for Time Fractional Reaction-Diffusion Equation

The computational complexity of one-dimensional time fractional reaction-diffusion equation is O(N2M) compared with O(NM) for classical integer reaction-diffusion equation. Parallel computing is used to overcome this challenge. Domain decomposition method (DDM) embodies large potential for parallelization of the numerical solution for fractional equations and serves as a basis for distributed, parallel computations. A domain decomposition algorithm for time fractional reaction-diffusion equation with implicit finite difference method is proposed. The domain decomposition algorithm keeps the same parallelism but needs much fewer iterations, compared with Jacobi iteration in each time step. Numerical experiments are used to verify the efficiency of the obtained algorithm

On fractional differential equations: the generalised Cattaneo equations

The aim of this dissertation is to determine numerical solutions to fractional di usion and fractional Cattaneo equations using nite di erence formula and other de ned schemes. The spatial derivatives and time derivatives of integer order are approximated by a nite di erence approximation. Spatial derivatives of fractional order are approximated using the Gr unwald formula. Fractional time derivatives are approximated using the Gr unwald-Letnikov de nition of the Riemann-Liouville fractional derivative. The resulting di erence schemes are evaluated using Mathematica. The results obtained show that the fractional Cattaneo equaions have propagation and di usive properties. When the fractional exponent is 0:1 with the di usivity coe cient being greater than 0:1 one obtains numerical results that are unstable and display oscillatory behaviour. For other combinations of values, numerical results are stable and consistent with di usive behaviour

Fractional Heat Conduction Models and Thermal Diffusivity Determination

The contribution deals with the fractional heat conduction models and their use for determining thermal diffusivity. A brief historical overview of the authors who have dealt with the heat conduction equation is described in the introduction of the paper. The one-dimensional heat conduction models with using integer- and fractional-order derivatives are listed. Analytical and numerical methods of solution of the heat conduction models with using integer- and fractional-order derivatives are described. Individual methods have been implemented in MATLAB and the examples of simulations are listed. The proposal and experimental verification of the methods for determining thermal diffusivity using half-order derivative of temperature by time are listed at the conclusion of the paper

Numerical Simulation

Nowadays mathematical modeling and numerical simulations play an important role in life and natural science. Numerous researchers are working in developing different methods and techniques to help understand the behavior of very complex systems, from the brain activity with real importance in medicine to the turbulent flows with important applications in physics and engineering. This book presents an overview of some models, methods, and numerical computations that are useful for the applied research scientists and mathematicians, fluid tech engineers, and postgraduate students

Recent Trends in Coatings and Thin Film–Modeling and Application

Over the past four decades, there has been increased attention given to the research of fluid mechanics due to its wide application in industry and phycology. Major advances in the modeling of key topics such Newtonian and non-Newtonian fluids and thin film flows have been made and finally published in the Special Issue of coatings. This is an attempt to edit the Special Issue into a book. Although this book is not a formal textbook, it will definitely be useful for university teachers, research students, industrial researchers and in overcoming the difficulties occurring in the said topic, while dealing with the nonlinear governing equations. For such types of equations, it is often more difficult to find an analytical solution or even a numerical one. This book has successfully handled this challenging job with the latest techniques. In addition, the findings of the simulation are logically realistic and meet the standard of sufficient scientific value

Improved modeling for fluid flow through porous media

Petroleum production is one of the most important technological challenges in the current world. Modeling and simulation of porous media flow is crucial to overcome this challenge. Recent years have seen interest in investigation of the effects of history of rock, fluid, and flow properties on flow through porous media. This study concentrates on the development of numerical models using a ‘memory’ based diffusivity equation to investigate the effects of history on porous media flow. In addition, this study focusses on developing a generalized model for fluid flow in packed beds and porous media. The first part of the thesis solves a memory-based fractional diffusion equation numerically using the Caputo, Riemann-Liouville (RL), and Grünwald-Letnikov (GL) definitions for fractional-order derivatives on uniform meshes in both space and time. To validate the numerical models, the equation is solved analytically using the Caputo, and Riemann-Liouville definitions, for Dirichlet boundary conditions and a given initial condition. Numerical and analytical solutions are compared, and it is found that the discretization method used in the numerical model is consistent, but less than first order accurate in time. The effect of the fractional order on the resulting error is significant. Numerical solutions found using the Caputo, Riemann-Liouville, and Grünwald-Letnikov definitions are compared in the second part. It is found that the largest pressure values are found from Caputo definition and the lowest from Riemann-Liouville definition. It is also found that differences among the solutions increase with increasing fractional order

Numerical solution methods for fractional partial differential equations

Fractional partial differential equations have been developed in many different fields such as physics, finance, fluid mechanics, viscoelasticity, engineering and biology. These models are used to describe anomalous diffusion. The main feature of these equations is their nonlocal property, due to the fractional derivative, which makes their solution challenging. However, analytic solutions of the fractional partial differential equations either do not exist or involve special functions, such as the Fox (H-function) function (Mathai & Saxena 1978) and the Mittag-Leffler function (Podlubny 1998) which are diffcult to evaluate. Consequently, numerical techniques are required to find the solution of fractional partial differential equations. This thesis can be considered as two parts, the first part considers the approximation of the Riemann-Liouville fractional derivative and the second part develops numerical techniques for the solution of linear and nonlinear fractional partial differential equations where the fractional derivative is defied as a Riemann-Liouville derivative. In the first part we modify the L1 scheme, developed initially by Oldham & Spanier (1974), to develop the three schemes which will be defined as the C1, C2 and C3 schemes. The accuracy of each method is considered. Then the memory effect of the fractional derivative due to nonlocal property is discussed. Methods of reduction of the computation L1 scheme are proposed using regression approximations. In the second part of this study, we consider numerical solution schemes for linear fractional partial differential equations. Here the numerical approximation schemes are developed using an approximation of the fractional derivative and a spatial discretization scheme. In this thesis the L1, C1, C2, C3 fractional derivative approximation schemes, developed in the first part of the thesis, are used in conjunction with either the Centred-finite difference scheme, the Dufort-Frankel scheme or the Keller Box scheme. The stability of these numerical schemes are investigated via the technique of the Fourier analysis (Von Neumann stability analysis). The convergence of each the numerical schemes is also discussed. Numerical tests were used to conform the accuracy and stability of each proposed method. In the last part of the thesis numerical schemes are developed to handle nonlinear partial differential equations and systems of nonlinear fractional partial differential equations. We considered two models of a reversible reaction in the presence of anomalous subdiffusion. The Centred-finite difference scheme and the Keller Box methods are used to spatially discretise the spatial domain in these schemes. Here the L1 scheme and a modification of the L1 scheme are used to approximate the fractional derivative. The accuracy of the methods are discussed and the convergence of the scheme are demonstrated by numerical experiments. We also give numerical examples to illustrate the e�ciency of the proposed scheme