Hany N. Hassan

An analytic solution of micropolar flow in a porous channel with mass injection using homotopy analysis method

Purpose – The paper aims to find an accurate analytic solution (series solution) for the micropolar flow in a porous channel with mass injection for different values of Reynolds number. Design/methodology/approach – In this paper, the homotopy analysis method (HAM) with different numbers of unknown convergence-control parameters has been used to derive accurate analytic solution for micropolar flow in a porous channel with mass injection. The possible optimal value of convergence-control parameter determined by minimizing the averaged residual error. Findings – The results obtained from HAM solution with two parameters are compared with numerical results and that obtained from HAM solution with only one parameter. The results show that this method gives an analytical solution with high order of accuracy with a few iterations. As shown in this paper, by minimizing the averaged residual error, the authors can get the possible optimal value of the convergence-control parameters which may give the fastest con...

A new technique of using homotopy analysis method for second order nonlinear differential equations

Abstract In this paper, a new technique of homotopy analysis method (nHAM) is proposed for solving second order nonlinear differential equations. This method improves the convergence of the series solution, eliminates the unneeded terms and reduces time consuming in the standard homotopy analysis method (HAM). The proposed provides an approximate solution by rewriting the second order nonlinear differential equation in the form of two first order differential equations. The solution of these two differential equations is obtained as a power series solution. This scheme is tested on four non-linear exactly solvable differential equations. Three of the examples are initial value problems and the fourth is boundary value problem. The results demonstrate reliability and efficiency of the algorithm developed.

A new approach for a class of nonlinear boundary value problems with multiple solutions

Abstract In this paper, an approach based on the variational iteration method (VIM) is proposed with an auxiliary parameter to predict the multiplicity of the solutions of homogeneous nonlinear ordinary differential equations with boundary conditions. The proposed approach is capable to predict and calculate all branches of the solutions simultaneously. Four practical problems are chosen to show the efficiency and importance of the proposed method. The proposed approach successfully detects multiple solutions to Bratu’s problem, the model of mixed convection flows in a vertical channel, the nonlinear model of diffusion and reaction in porous catalysts and the nonlinear reactive transport model.

Fundamentals of fractional-order LTI circuits and systems: number of poles, stability, time and frequency responses

This paper investigates some basic concepts of fractional-order linear time invariant systems related to their physical and non-physical transfer functions, poles, stability, time domain, frequency domain, and their relationships for different fractional-order differential equations. The analytical formula that calculates the number of poles in physical and non-physical s-plane for different orders is achieved and verified using many practical examples. The stability contour versus the number of poles in the physical s-plane for different fractional-order systems is discussed in addition to the effect of the non-physical poles on the steady state responses. Moreover, time domain responses based on Mittag-Leffler functions for both physical and non-physical transfer functions are discussed for different cases, which confirm the stability analysis. Many fractional-order linear time invariant systems based on fractional-order differential equations have been discussed numerically in both time and frequency domains to validate the previous fundamentals. Copyright

Controlled Picard Method for Solving Nonlinear Fractional Reaction–Diffusion Models in Porous Catalysts

This paper discusses the diffusion and reaction behaviors of catalyst pellets in the fractional-order domain as well as the case of nth-order reactions. Two generic models are studied to calculate the concentration of reactant in a porous catalyst in the case of a spherical geometric pellet and a flat-plate particle with different examples. A controlled Picard analytical method is introduced to obtain an approximated solution for these systems in both linear and nonlinear cases. This method can cover a wider range of problems due to the extra auxiliary parameter, which enhances the convergence and is suitable for higher-order differential equations. Moreover, the exact solution in the linear fractional-order system is obtained using the Mittag–Leffler function where the conventional solution is a special case. For nonlinear models, the proposed method gives matched responses with the homotopy analysis method (HAM) solutions for different fractional orders. The effect of fractional-order parameter on the dimensionless concentration of the reactant in a porous catalyst is analyzed graphically for different cases of order reactions and Thiele moduli. Moreover, the proposed method has been applied numerically for different cases to predict and calculate the dual solutions of a nonlinear fractional model when the reaction order n = −1.

An optimal linear system approximation of nonlinear fractional-order memristor–capacitor charging circuit

Abstract The analysis of nonlinear fractional-order circuits is a challenging problem. This is due to the lack of nonlinear circuit theorems and designs particularly in the presence of memristive elements. The response of a series connection of a simple resistor with fractional order capacitor and its analytical formulation in both charging and discharging phases is considered. The numerical simulation of fractional order HP memristor in series with a fractional order capacitor is also discussed. It is a demonstration of a simple nonlinear fractional-order memristive circuit in both charging and discharging cases. Furthermore, this paper introduces an approach to approximate nonlinear fractional-order memrisitve circuits by linear circuits using a minimax optimization technique. Hence, the new circuit can be analyzed using the conventional linear circuit theorems. The charging and discharging of a series fractional-order memristor with a fractional-order capacitor are discussed numerically. The effect of fractional-order parameters and memristor polarity are also investigated. Using a suitable optimization technique, an accurate approximation by a circuit that include a resistor and a fractional-capacitor is obtained for both charging and discharging cases. A great matching was observed between the frequency responses of the fractional-order nonlinear low pass filter based on fractional-order memristor and fractional-order capacitor and that of the optimized linear fractional order case. Similar matching is observed for the nonlinear and optimized cases when a periodic triangular waveform is applied using Fourier series expansion.

Modified methods for solving two classes of distributed order linear fractional differential equations

This paper introduces two methods for the numerical solution of distributed order linear fractional differential equations. The first method focuses on initial value problems (IVPs) and based on the αth Caputo fractional definition with the shifted Chebyshev operational matrix of fractional integration. By applying this method, the IVPs are converted into simple linear differential equations which can be easily handled. The other method focuses on boundary value problems (BVPs) based on Picards method frame. This method is based on iterative formula contains an auxiliary parameter which provides a simple way to control the convergence region of solution series. Several numerical examples are used to illustrate the accuracy of the proposed methods compared to the existing methods. Also, the response of mechanical system described by such equations is studied.

Single and dual solutions of fractional order differential equations based on controlled Picard’s method with Simpson rule

Abstract This paper presents a semi-analytical method for solving fractional differential equations with strong terms like (exp, sin, cos, …). An auxiliary parameter is introduced into the well-known Picard’s method and so called controlled Picard’s method. The proposed approach is based on a combination of controlled Picard’s method with Simpson rule. This approach can cover a wider range of integer and fractional orders differential equations due to the extra auxiliary parameter which enhances the convergence and is suitable for higher order differential equations. The proposed approach can be effectively applied to Bratu’s problem in fractional order domain to predict and calculate all branches of problem solutions simultaneously. Also, it is tested on other fractional differential equations like nonlinear fractional order Sine-Gordon equation. The results demonstrate reliability, simplicity and efficiency of the approach developed.

Nonlinear Fractional Order Boundary-Value Problems With Multiple Solutions

Abstract It is well-known that discovering and then calculating all branches of solutions of fractional order nonlinear differential equations with boundary conditions can be difficult even by numerical methods. To overcome this difficulty, in this chapter two semianalytic methods are presented to predict and obtain multiple solutions of nonlinear boundary value problems. These methods are based on the homotopy analysis method (HAM) and Picard method namely, predictor HAM and controlled Picard method. The used techniques are capable of predicting and calculating all branches of the solutions simultaneously. Four problems are solved, three of them are practical problems which are generalized in fractional order domain to show the efficiency and importance of these methods. And the solutions are calculated by simple procedures without any need for special transformations or perturbation techniques.

An efficient numerical method for the modified regularized long wave equation using Fourier spectral method

Abstract The modified regularized long wave (MRLW) equation is numerically solved using Fourier spectral collection method. The MRLW equation is discretized in space variable by the Fourier spectral method and Leap-Frog method for time dependence. To validate the efficiency, accuracy and simplicity of the used method, four cases study are solved. The single soliton wave motion, interaction of two solitary waves, interaction of three solitary waves and a Maxwellian initial condition pulse are studied. The and error norms are computed for the motion of single solitary waves. To determine the conservation properties of the MRLW equation three invariants of motion are evaluated for all test problems.