Search results
1 – 10 of 15
The purpose of this study is to investigate viscoelastic properties for the constitutive equation in terms of distributed-order derivatives.
Abstract
Purpose
The purpose of this study is to investigate viscoelastic properties for the constitutive equation in terms of distributed-order derivatives.
Design/methodology/approach
The authors considered the steady oscillatory shear flow between two parallel plates (one is fixed and another oscillates in its own plane), and then examined the effects of different forms of the order-weight functions.
Findings
The constitutive equation in terms of distributed-order derivatives can characterize viscoelastic properties. The order-weight functions can effectively describe viscoelasticity.
Originality/value
Model the viscoelastic constitutive equation in terms of distributed-order derivatives, where order-weight functions can select to respond viscoelastic properties.
Details
Keywords
Mostafa Abbaszadeh, AliReza Bagheri Salec and Shurooq Kamel Abd Al-Khafaji
The space fractional PDEs (SFPDEs) play an important role in the fractional calculus field. Proposing a high-order, stable and flexible numerical procedure for solving SFPDEs is…
Abstract
Purpose
The space fractional PDEs (SFPDEs) play an important role in the fractional calculus field. Proposing a high-order, stable and flexible numerical procedure for solving SFPDEs is the main aim of most researchers. This paper devotes to developing a novel spectral algorithm to solve the FitzHugh–Nagumo models with space fractional derivatives.
Design/methodology/approach
The fractional derivative is defined based upon the Riesz derivative. First, a second-order finite difference formulation is used to approximate the time derivative. Then, the Jacobi spectral collocation method is employed to discrete the spatial variables. On the other hand, authors assume that the approximate solution is a linear combination of special polynomials which are obtained from the Jacobi polynomials, and also there exists Riesz fractional derivative based on the Jacobi polynomials. Also, a reduced order plan, such as proper orthogonal decomposition (POD) method, has been utilized.
Findings
A fast high-order numerical method to decrease the elapsed CPU time has been constructed for solving systems of space fractional PDEs.
Originality/value
The spectral collocation method is combined with the POD idea to solve the system of space-fractional PDEs. The numerical results are acceptable and efficient for the main mathematical model.
Details
Keywords
Zain ul Abdeen and Mujeeb ur Rehman
The purpose of this paper is to obtain a numerical scheme for finding numerical solutions of linear and nonlinear Hadamard-type fractional differential equations.
Abstract
Purpose
The purpose of this paper is to obtain a numerical scheme for finding numerical solutions of linear and nonlinear Hadamard-type fractional differential equations.
Design/methodology/approach
The aim of this paper is to develop a numerical scheme for numerical solutions of Hadamard-type fractional differential equations. The classical Haar wavelets are modified to align them with Hadamard-type operators. Operational matrices are derived and used to convert differential equations to systems of algebraic equations.
Findings
The upper bound for error is estimated. With the help of quasilinearization, nonlinear problems are converted to sequences of linear problems and operational matrices for modified Haar wavelets are used to get their numerical solution. Several numerical examples are presented to demonstrate the applicability and validity of the proposed method.
Originality/value
The numerical method is purposed for solving Hadamard-type fractional differential equations.
Details
Keywords
Mehdi Dehghan, Baharak Hooshyarfarzin and Mostafa Abbaszadeh
This study aims to use the polynomial approximation method based on the Pascal polynomial basis for obtaining the numerical solutions of partial differential equations. Moreover…
Abstract
Purpose
This study aims to use the polynomial approximation method based on the Pascal polynomial basis for obtaining the numerical solutions of partial differential equations. Moreover, this method does not require establishing grids in the computational domain.
Design/methodology/approach
In this study, the authors present a meshfree method based on Pascal polynomial expansion for the numerical solution of the Sobolev equation. In general, Sobolev-type equations have several applications in physics and mechanical engineering.
Findings
The authors use the Crank-Nicolson scheme to discrete the time variable and the Pascal polynomial-based (PPB) method for discretizing the spatial variables. But it is clear that increasing the value of the final time or number of time steps, will bear a lot of costs during numerical simulations. An important purpose of this paper is to reduce the execution time for applying the PPB method. To reach this aim, the proper orthogonal decomposition technique has been combined with the PPB method.
Originality/value
The developed procedure is tested on various examples of one-dimensional, two-dimensional and three-dimensional versions of the governed equation on the rectangular and irregular domains to check its accuracy and validity.
Details
Keywords
Farshid Mehrdoust, Amir Hosein Refahi Sheikhani, Mohammad Mashoof and Sabahat Hasanzadeh
The purpose of this paper is to evaluate a European option using the fractional version of the Black-Scholes model.
Abstract
Purpose
The purpose of this paper is to evaluate a European option using the fractional version of the Black-Scholes model.
Design/methodology/approach
In this paper, the authors employ the block-pulse operational matrix algorithm to approximate the solution of the fractional Black-Scholes equation with the initial condition for a European option pricing problem.
Findings
The fractional derivative will be described in the Caputo sense in this paper. The authors show the accuracy and computational efficiency of the proposed algorithm through some numerical examples.
Originality/value
This is the first paper that considers an alternative algorithm for pricing a European option using the fractional Black-Scholes model.
Details
Keywords
The purpose of the present work is to propose a wavelet method for the numerical solutions of Caputo–Hadamard fractional differential equations on any arbitrary interval.
Abstract
Purpose
The purpose of the present work is to propose a wavelet method for the numerical solutions of Caputo–Hadamard fractional differential equations on any arbitrary interval.
Design/methodology/approach
The author has modified the CAS wavelets (mCAS) and utilized it for the solution of Caputo–Hadamard fractional linear/nonlinear initial and boundary value problems. The author has derived and constructed the new operational matrices for the mCAS wavelets. Furthermore, The author has also proposed a method which is the combination of mCAS wavelets and quasilinearization technique for the solution of nonlinear Caputo–Hadamard fractional differential equations.
Findings
The author has proved the orthonormality of the mCAS wavelets. The author has constructed the mCAS wavelets matrix, mCAS wavelets operational matrix of Hadamard fractional integration of arbitrary order and mCAS wavelets operational matrix of Hadamard fractional integration for Caputo–Hadamard fractional boundary value problems. These operational matrices are used to make the calculations fast. Furthermore, the author works out on the error analysis for the method. The author presented the procedure of implementation for both Caputo–Hadamard fractional initial and boundary value problems. Numerical simulation is provided to illustrate the reliability and accuracy of the method.
Originality/value
Many scientist, physician and engineers can take the benefit of the presented method for the simulation of their linear/nonlinear Caputo–Hadamard fractional differential models. To the best of the author’s knowledge, the present work has never been proposed and implemented for linear/nonlinear Caputo–Hadamard fractional differential equations.
Details
Keywords
Hamid Mesgarani, Mahya Kermani and Mostafa Abbaszadeh
The purpose of this study is to use the method of lines to solve the two-dimensional nonlinear advection–diffusion–reaction equation with variable coefficients.
Abstract
Purpose
The purpose of this study is to use the method of lines to solve the two-dimensional nonlinear advection–diffusion–reaction equation with variable coefficients.
Design/methodology/approach
The strictly positive definite radial basis functions collocation method together with the decomposition of the interpolation matrix is used to turn the problem into a system of nonlinear first-order differential equations. Then a numerical solution of this system is computed by changing in the classical fourth-order Runge–Kutta method as well.
Findings
Several test problems are provided to confirm the validity and efficiently of the proposed method.
Originality/value
For the first time, some famous examples are solved by using the proposed high-order technique.
Details
Keywords
Abstract
Purpose
The purpose of this study is to investigate the unsteady stagnation-point flow and heat transfer of fractional Maxwell fluid towards a time power-law-dependent stretching plate. Based on the characteristics of pressure in the boundary layer, the momentum equation with the fractional Maxwell model is firstly formulated to analyze unsteady stagnation-point flow. Furthermore, generalized Fourier’s law is considered in the energy equation and boundary condition of convective heat transfer.
Design/methodology/approach
The nonlinear fractional differential equations are solved by the newly developed finite difference scheme combined with L1-algorithm, whose convergence is verified by constructing a numerical example.
Findings
Some interesting results can be revealed. The larger fractional derivative parameter of velocity promotes the flow, while the smaller fractional derivative parameter of temperature accelerates the heat transfer. The temperature boundary layer is thicker than the velocity boundary layer, and the velocity enlarges as the stagnation parameter raises. This is because when Prandtl number < 1, the capacity of heat diffusion is greater than that of momentum diffusion. It is to be observed that all the temperature profiles first enhance a little and then reduce rapidly, which indicates the thermal retardation of Maxwell fluid.
Originality/value
The unsteady stagnation-point flow model of Maxwell fluid is extended from integral derivative to fractional derivative, which has more flexibility to describe viscoelastic fluid’s complex dynamic process and provide a theoretical basis for industrial processing.
Details
Keywords
This paper aims to present an interpolating element-free Galerkin (IEFG) method for the numerical study of the time-fractional diffusion equation, and then discuss the stability…
Abstract
Purpose
This paper aims to present an interpolating element-free Galerkin (IEFG) method for the numerical study of the time-fractional diffusion equation, and then discuss the stability and convergence of the numerical solutions.
Design/methodology/approach
In the time-fractional diffusion equation, the time fractional derivatives are approximated by L1 method, and the shape functions are constructed by the interpolating moving least-squares (IMLS) method. The final system equations are obtained by using the Galerkin weak form. Because the shape functions have the interpolating property, the unknowns can be solved by the iterative method after imposing the essential boundary condition directly.
Findings
Both theoretical and numerical results show that the IEFG method for the time-fractional diffusion equation has high accuracy. The stability of the fully discrete scheme of the method on the time step is stable unconditionally with a high convergence rate.
Originality/value
This work will provide an interpolating meshless method to study the numerical solutions of the time-fractional diffusion equation using the IEFG method.
Details
Keywords
Creep behavior of concrete at high temperature has become a major concern in building structures, such as factories, bridges, tunnels, airports and nuclear buildings. Therefore, a…
Abstract
Purpose
Creep behavior of concrete at high temperature has become a major concern in building structures, such as factories, bridges, tunnels, airports and nuclear buildings. Therefore, a simple and accurate prediction model for the high-temperature creep behavior of concrete is crucial in engineering applications.
Design/methodology/approach
In this paper, the variable-order fractional operator is introduced to capture the high-temperature creep behavior of concrete. By assuming that the variable-order function is a linear function with time, the proposed model benefits from the advantages of both formal simplicity and the physical significance for macroscopic intermediate materials. The effectiveness of the model is demonstrated by data fitting with existing experimental results of high-temperature creep of two representative concretes.
Findings
The results show that the proposed model fits well with the experimental data, and the value of order is increasing with the increase of the applied stress levels, which meets the fact that higher stress can accelerate the rate of creep. Furthermore, the relationship between the model parameters and loading conditions is deeply analyzed. It is found that the material coefficients are constant at a constant temperature, while the order function parameters are determined by the applied stress levels. Finally, the variable-order fractional model can be further written into a general equation of time and applied stress.
Originality/value
This paper provides a simple and practical variable-order fractional model for predicting the creep behavior of concrete at high temperature.
Details