Academic Positions

  • Present 2016

    Post Doctoral Research Associate

    Florida State University,Department of Scientific Computing

  • 2016 2015

    Post Doctoral Research Associate

    Florida State University, Computational Science & Engineering DEP.

Education & Training

  • Ph.D. 2015

    Ph.D. in Mathematics

    Mississippi State University,Department of Mathematics and Statistics

  • Ph.D. 2013

    Ph.D. in Applied Mathematics

    Alzahra University,Department of Mathematics,Tehran, Iran

  • M.S 2006

    M.S in Applied Mathematics

    Alzahra University,Department of Mathematics,Tehran, Iran

Research Projects

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    Population Genetics and Coalescent Theory

    Very short description of the project.

    Theoretical population genetics bridges mathematics and evolutionary biology. A key innovation in the field was the development of the n-coalescent by the probabilist JFC Kingman in 1982. The n-coalescent introduced a retrospective view of a population of individuals allowing probabilistic statements of the past and allowing us to infer parameters of complex population models using genomic data of many individuals. Coalescente theory is based on a Markov process using exponential distribution.

    I have expanded the theory by generalizing the underlying Poisson process of the coalescent process using a recent advancement in the description of the fractional Poisson process that is a semi-Markov process. I call my extension the fractional coalescent, or f-coalescent. The simulated data has been used to validate the model, and the real data has been used to show the advantages of fractional coalescent.

    When data is simulated using models with alpha<1 (a key parameter of the fractional Poisson process), or for real datasets (H1N1 influenza, Malaria parasites, Humpback whales), Bayes factor comparisons show an improved model fit of the f-coalescent over the n-coalescent. In this model, the distribution of the number of offspring depends on a parameter alpha which is a potential measure of the environmental heterogeneity that is commonly ignored in current inferences.

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    Fractional Viscoelasticity in Fractal Media

    Very short description of the project.

    I predicted complex phenomena in viscoelastic materials. Such materials exhibit unusual interactions between elastic and viscous forces. The unique properties of such materials make them ideal for the study of fractional derivatives but they also have practical importance for many smart damping applications (i.e., robotics, automotive, and aerospace structures). While the Scott-Blair fractional model of viscoelasticity has been introduced by using the idealized model of spring and dashpot, I introduced theoretical justification for this model where the results have been experimentally validated using Bayesian uncertainty analysis.

    This analysis shows superior results compared to the integer Maxwell model. Also, for the first time I reported using a nonlinear fractional model of viscoelasticity. Importantly, the use of fractional order calculus operations to describe the rate-dependent viscoelastic response yielded a model with self-consistent model parameters. Therefore, the model performs better at predictions across a broad range of experimental rates. This led me to discover a physical explanation for the fractional time derivative applied to viscoelasticity derived by using thermal diffusion and fractal dimensions in fractal media.

    The challenge of deriving this explanation has been long-standing (dating back to the work of Bagley and Torvik and Mandelbrot) and this shows the fractional model of viscoelasticity provides more information about the structure of a material. This area opens up an application of fractional calculus which may describe the multi-scale thermomechanical material behavior of many polymers.

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    Numerical Methods for Solving the Fractional System

    Very short description of the project.

    I introduced a new method for solving fractional differential systems. The method is based on hybrid-function approximation in which the hybrid functions consist of block-pulse functions and Bernoulli polynomials. The method uses the Riemann-Liouville fractional integral operator to reduce the solution of the system to a set of algebraic equations.

    This method can be applied to linear and nonlinear distributed fractional order differential equations, nonlinear fractional integro-differential equations, fractional Bagley-Torvik equations as well as fractional optimal control problems.

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    Hybrid Functions of Block-pulse and Bernoulli Polynomials

    Very short description of the project.

    I introduced hybrid functions of block-pulse and Bernoulli polynomials as a new base for solving differential systems. My work demonstrated that using Bernoulli polynomials has several advantages over the most commonly used method, Legendre polynomials.

    First, I showed that Bernoulli polynomials have fewer errors in their operational matrix of integrals than in those of shifted Legendre polynomials. Second, Bernoulli polynomials have fewer terms than Legendre polynomials, and those individual terms have lower coefficients. Finally, I have shown that the operational matrix for the hybrid functions of block-pulse and Bernoulli polynomials is more sparse than those that can be achieved through a block-pulse hybrid with Legendre, Chebychev, or Taylor polynomials.

    This makes my method computationally more efficient than other methods, even the widely used Legendre polynomials. This method can be applied to nonlinear, constrained optimal control problems, optimal control of systems described by integro-differential equations, Duffing equation, multi-delay, and piecewise constant delay systems.

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    Least Squares Fit of Highly Oscillatory Functions

    Very short description of the project.

    Many physical phenomena exhibit a pronounced oscillatory character. Behavior of pendulum-like systems, vibrations, resonances or wave propagation are all phenomena of this type in classical mechanics, while the same is true for the typical behavior of quantum particles. Finding a suitable tool for a good approximation of these kinds of functions is therefore of acute interest from both a mathematical and an application perspective.

    I developed a new base for finding the least squares fit of oscillatory functions. The results obtained with the new base are compared to the ones obtained by means of the Legendre polynomials and with theoretical predictions. The new base is attractive to use in many other mathematical contexts where highly oscillatory functions are involved.

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Differential and Integral (In persian),

M. Shafiebeyk Mohammadi, S. Mashayekhi, H. Pourbashash
Book Noavaran Sharif Publication | 2011 | ISBN: 978-600-5953-00-8
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Differential and Integral

The present book is prepared in 11 chapters for the course of General Mathematics (1) in the fields of Mathematics, Statistics and engineering fields. In the first chapter, complex numbers and their properties are expressed. In the second chapter, relation and function, their properties and different types of functions are reviewed. In chapters three, four and five the concepts of limit, derivative and continuity are presented by using functions and different examples. In chapter six, derivative applications, maximum and minimum questions and the related theorems are presented.

In chapter seven, the concept of indefinite integral and methods of integration has been introduced. Definite integrations and their applications are presented in chapters eight and nine. Chapter ten includes polar coordinates and their properties, calculation of arc length, volume and area. In chapter 11 the concepts of sequences and series have been expressed by presenting some questions. For better comprehension, some examples have been presented in each chapter and at the end of each chapter there some questions that solving them shall help in better understanding of the concepts. It is hoped that this book would meet the students’ needs.

Fractional Viscoelasticity in Fractal Media: Theory, Experimental Validation, and Uncertainty Analysis

S.Mashayekhi,P.Miles, M.Yousuff Hussaini and W.Oates
Journal Paper Journal of the Mechanics and Physics of Solids,Volume 111, February 2018, Pages 134-156

Abstract

In this paper, fractional and non-fractional viscoelastic models for elastomeric materials are derived and analyzed in comparison to experimental results. The viscoelastic models are derived by expanding thermodynamic balance equations for both fractal and non-fractal media. The order of the fractional time derivative is shown to strongly affect the accuracy of the viscoelastic constitutive predictions. Model validation uses experimental data describing viscoelasticity of the dielectric elastomer Very High Bond (VHB) 4910. Since these materials are known for their broad applications in smart structures, it is important to characterize and accurately predict their behavior across a large range of time scales. Whereas integer order viscoelastic models can yield reasonable agreement with data, the model parameters often lack robustness in prediction at different deformation rates. Alternatively, fractional order models of viscoelasticity provide an alternative framework to more accurately quantify complex rate-dependent behavior. Prior research that has considered fractional order viscoelasticity lacks experimental validation and contains limited links between viscoelastic theory and fractional order derivatives. To address these issues, we use fractional order operators to experimentally validate fractional and non-fractional viscoelastic models in elastomeric solids using Bayesian uncertainty quantification. The fractional order model is found to be advantageous as predictions are significantly more accurate than integer order viscoelastic models for deformation rates spanning four orders of magnitude.

Population divergence estimation using individual lineage label switching

P. Beerli,S. Mashayekhi,H. Ashki, , and M. Palczewski
Journal Paper Journal of Systematic Biology, 2018, Accepted

Abstract

Population divergence estimation using individual lineage label switching

Numerical solutions of fractional differential equations by using fractional Taylor basis

V.S. Krishnasamy, S. Mashayekhi, M.Razzaghi
Journal PaperIEEE/CAA Journal of Automatica Sinica,Volume: 4 , Issue: 1 ,Jan.2017

Abstract

In this paper, a new numerical method for solving fractional differential equations (FDEs) is presented. The method is based upon the fractional Taylor basis approximations. The operational matrix of the fractional integration for the fractional Taylor basis is introduced. This matrix is then utilized to reduce the solution of the fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of this technique.

An approximate method for solving fractional optimal control problems by hybrid functions

S. Mashayekhi, M.Razzaghi
Journal PaperJournal of Vibration and Control,Volume 24,Issue 9,August 2016 , Pages 1621-1631

Abstract

In this paper, a new numerical method for solving fractional optimal control problems by using hybrid functions is presented. The Riemann–Liouville fractional integral operator for hybrid functions is utilized to reduce the solution of optimal control problems to a nonlinear programming one, to which existing, well-developed algorithms may be applied. The method is computationally very attractive and gives very accurate results.

Numerical solution of distributed order fractional differential equations by hybrid functions

S. Mashayekhi, M.Razzaghi
Journal Paper Journal of Computational Physics, Volume 315, June 2016, Pages 169-181

Abstract

In this paper, a new numerical method for solving the distributed fractional differential equations is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The Riemann–Liouville fractional integral operator for hybrid functions is introduced. This operator is then utilized to reduce the solution of the distributed fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Numerical solution of the fractional Bagley‐Torvik equation by using hybrid functions approximation

S. Mashayekhi, M.Razzaghi
Journal Paper Journal of Mathematical Methods in the Applied Sciences, Volume39, Issue3,February 2016,Pages 353-365

Abstract

In this paper, a new numerical method for solving the fractional Bagley‐Torvik equation is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block‐pulse functions and Bernoulli polynomials are presented. The Riemann‐Liouville fractional integral operator for hybrid functions is introduced. This operator is then utilized to reduce the solution of the initial and boundary value problems for the fractional Bagley‐Torvik differential equation to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Solution of Lane–Emden type equations using rational Bernoulli functions

Velinda Calvert,S. Mashayekhi, M.Razzaghi
Journal Paper Journal of Mathematical Methods in the Applied Sciences, Volume39, Issue5,April 2016,Pages 1268-1284

Abstract

In this paper, a numerical method for solving Lane‐Emden type equations, which are nonlinear ordinary differential equations on the semi‐infinite domain, is presented. The method is based upon the modified rational Bernoulli functions; these functions are first introduced. Operational matrices of derivative and product of modified rational Bernoulli functions are then given and are utilized to reduce the solution of the Lane‐Emden type equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Analysis of multi-delay and piecewise constant delay systems by hybrid functions approximation

S. Mashayekhi, M.Razzaghi,M. Wattanataweekul
Journal Paper Journal of Differential Equations and Dynamical Systems, Volume24, Issue1,January 2016,Pages 1-20

Abstract

In this paper, a new numerical method for solving multi-delay and piecewise constant delay systems is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The operational matrices of integration, product and delay are given. These matrices are then utilized to reduce the solution of multi-delay systems and the piecewise constant delay systems to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Numerical solution of nonlinear fractional integro-differential equations by hybrid functions

S. Mashayekhi, M.Razzaghi
Journal Paper Journal of Engineering Analysis with Boundary Elements, Volume 56, July 2015, Pages 81-89

Abstract

In this paper, a new numerical method for solving nonlinear fractional integro-differential equations is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The Riemann–Liouville fractional integral operator for hybrid functions is given. This operator is then utilized to reduce the solution of the nonlinear fractional integro-differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Hybrid functions approach for optimal control of systems described by integro-differential equations

S. Mashayekhi, Y Ordokhani, M Razzaghi
Journal Paper Journal of Applied Mathematical Modelling, Volume 37,Issue 5, March 2013, Pages 3355-3368

Abstract

In this paper, a new numerical method for solving the optimal control of a class of systems described by integro-differential equations with quadratic performance index is presented. This optimization problem plays an important role in describing the dynamics of an elastic aircraft with allowance for non-steady flow past its profile. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The operational matrices of integration and product are given. These matrices are then utilized to reduce the solution of optimization problem to a nonlinear programming one to which existing well-developed algorithms may be applied. Illustrative examples are included to demonstrate the validity and applicability of the technique.

A hybrid functions approach for the Duffing equation

S. Mashayekhi, Y Ordokhani, M Razzaghi
Journal Paper Journal of Physica Scripta, Volume 88,Issue 2,July 2013, 025002

Abstract

In this paper, a new numerical method for solving the Duffing equation is presented. We consider this equation in two forms, with integral boundary conditions and involving both integral and non-integral forcing terms. The method is based on a hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The operational matrix of integration is given. This matrix is then utilized to reduce the solution of the Duffing equation to a nonlinear equation. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Hybrid functions approach for nonlinear constrained optimal control problems

S. Mashayekhi, Y Ordokhani, M Razzaghi
Journal Paper Journal of Communications in Nonlinear Science and Numerical Simulation, Volume 17,Issue 4 , April 2012, Pages 1831-1843

Abstract

In this paper, a new numerical method for solving the nonlinear constrained optimal control with quadratic performance index is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The operational matrix of integration is introduced. This matrix is then utilized to reduce the solution of the nonlinear constrained optimal control to a nonlinear programming one to which existing well-developed algorithms may be applied. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Rate dependent constitutive behavior of dielectric elastomers and applications in legged robotics

W. Oates, P. Miles, W. Gao, J. Clark,S. Mashayekhi, M.Y. Hussaini
ProceedingsProceedings Volume 10163, Electroactive Polymer Actuators and Devices (EAPAD) 2017, 1016316 (2017) .
Proc. SPIE Smart Structures and Materials+ Nondestructive Evaluation and Health Monitoring , Portland, Oregan, March 26-29, 2017

Abstract

Dielectric elastomers exhibit novel electromechanical coupling that has been exploited in many adaptive structure applications. Whereas the quasi-static, one-dimensional constitutive behavior can often be accurately quantified by hyperelastic functions and linear dielectric relations, accurate predictions of electromechanical, rate-dependent deformation during multiaxial loading is non-trivial. In this paper, an overview of multiaxial electromechanical membrane finite element modeling is formulated. Viscoelastic constitutive relations are extended to include fractional order. It is shown that fractional order viscoelastic constitutive relations are superior to conventional integer order models. This knowledge is critical for transition to control of legged robotic structures that exhibit advanced mobility.

The fractional coalescent

S. Mashayekhi, P. Beerli
Journal Submitted Journal of , Volume , 20--, Pages

Abstract

The fractional coalescent

On the least squares fit of highly oscillatory functions

S. Mashayekhi, L. Gr. Ixaru
Journal Submitted Journal of , Volume , 20--, Pages

Abstract

On the least squares fit of highly oscillatory functions

An efficient technique based on Bernoulli hybrid for solving neutral delay differential equations

S. Sedaghat, S. Mashayekhi
Journal Submitted Journal of , Volume , 20--, Pages

Abstract

An efficient technique based on Bernoulli hybrid for solving neutral delay differential equations

A physical interpretation of fractional viscoelasticity based on the fractal structure of media: Theory and experimental validation

S. Mashayekhi,W. Oates, M.Y. Hussaini
Journal Submitted Journal of , Volume , 20--, Pages

Abstract

A physical interpretation of fractional viscoelasticity based on the fractal structure of media: Theory and experimental validation.

Teaching History

  • 2015 2014

    Business calculus (Multiple sections)

    Instructor: Mississippi State University, USA

  • 2013 2003

    Complex variables, Numerical analysis, Ordinary differential equations, Engineering mathematics, Computer programming, Introductory probability, Probability and statistics for scientists and engineers, Advanced calculus, Precalculus algebra, College algebra

    Instructor: Alzahra university, Payame Noor university and Science and Culture university, Iran