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In this paper the symmetry operators of the fractional Klein-Gordon (KG) equation are found. The generalized fractional Noether’s theorem is used in order to find the conservation laws of the considered equation. Finally, the Chebyshev spectral collocation method is extended to space–time-fractional case for giving some numerical results and conservation laws for the equation.

The scope of this work is based on symmetry analysis of perturbed fractional Boussinesq equation. For beginning the group formalism, invariance properties and conservation laws of the nonlinear perturbed fractional Boussinesq equation have been explored generally. This method was first described by Lukashchuk [Commun. Nonlinear Sci. Numer. Simul. 68 (2019) 147–159]. The key subject is that when the order of fractional derivative in a fractional differential equation (FDE) is nearly integers, we can approximate it to a perturbed integer-order differential equation with a small perturbation parameter. For obtaining the results, perturbed and unperturbed symmetries are computed. Then, the methodology of reduction is applied for finding some new solutions by the symmetry operators of the equation. These solutions are obtained by the similarity transformations of the symmetries. Another exact solution will be found with constructing one-dimensional optimal system of the symmetries. Finally, the meaning of nonlinear self-adjointness concept is attended in order to find conservation laws with informal Lagrangians.

In this paper, we deal with the diffusivity equation, by using the Lie group analysis method. The infinitesimal generators of this equation are investigated. The concept of non-linear self-adjointn...

In this paper the Lie group analysis of the time-fractional cylindrical Burgers equation (time-FCB), which is a fundamental PDE occurring in various areas of applied mathematics, such as fluid mechanics, non-linear acoustics, gas dynamics, traffic flow and etc. is given. For this purpose the Riemann–Liouville derivative is used to implement the Lie algorithm for finding the symmetry operators. A reduced form of the equation is given by using the similarity variables obtained from a symmetry and Erdelyi–Kober operator. In the next step conservation laws are derived via a generalization of Noether’s theorem. Finally the Chebyshev wavelets for time-fractional differential equations (FDEs) is applied for solving the considered equation.

The present work considers the Lie group analysis of a system of linear wave type perturbed systems. The methodology is based on finding approximate symmetry operators of a given system. Approximate conservation laws are found via an approximate version of Noether’s theorem. This is based on the modified Noether’s method provided by Ibragimov. Finally a numerical method is applied to solve the considered system.Key words: Approximate symmetry, approximate conservation laws, Noether’s theorem, non-linear self-adjointness, Legendre polynomials.

In this paper the transition joint probability density function of the solution of the Ornstein–Uhlenbeck process is presented by a deterministic parabolic time-fractional PDE (FPDE), named time-fractional Fokker–Planck equation. The article generally is divided to two parts: theoretical and approximated analysis. In the theoretical sections Lie group method and invariant subspace method are applied for finding exact solutions and conservation laws of the considered equation. In the next part, by Chebyshev wavelets’s method the numerical solutions are driven. Then the usefulness of this approximated method is comparing with the exact solutions by some plotted graphs.

The present paper considers the group analysis of the Newell–Whitehead–Segel equation of fractional order. The important advantage of the article is that for the first time we analyzed the equation for time, space and space–time derivatives of Riemann fractional together. This is based on similarity variables obtained from Lie point symmetry operators of the equation in triple cases. These variables help us to find the reduced equations. Finally, we use the generalized Noether’s theorem of fractional order for finding conservation laws of the equations.

The topic of the present paper concentrates on a systematic investigation of Lie group analysis of non-linear time-fractional Black–Scholes equation including numerical approximations. Lie point symmetries of the equation are found via Riemann–Liouville derivative with four different volatility models. Also the invariant solutions as a kind of exact solutions are constructed by a useful explained method. Conservation laws for the intended equation are derived by using a modified version of Noether’s theorem. A numerical simulation by Chebyshev pseudo-spectral (CPS) method is computed for the equation. Approximate solutions together with their absolute errors are found and the graphs of solutions are plotted by the action of order of derivative α .

The present paper is intended for the investigation of the fractional integro-differential system called Vlasov–Maxwell system. The method is based on using the geometric vector fields of the symmetries. This system arises for interaction of charged particles in plasma. The fractional derivative is considered in both the Caputo and Riemann–Liouville sense. Under some suitable conditions, we construct the infinitesimal criterion of invariance for detecting Lie symmetries of these equations. In this study, Lie symmetry method for constructing the similarity solutions of the considered system is implemented. The theory is constructed step by step and carefully to apply in the Vlasov–Maxwell system.

In the present paper, Lie point symmetries of the time-fractional generalized Hirota–Satsuma coupled KdV (HS-cKdV) system based on the Riemann–Liouville derivative are obtained. Using the derived Lie point symmetries, we obtain similarity reductions and conservation laws of the considered system. Finally, some analytic solutions are furnished by means of the invariant subspace method in the Caputo sense.

In this paper, we deal with the complete algebra of Lie point symmetries for the generalized model of an irrigation system of fractional order. By means of Lie symmetry method, the vector fields ha...

This paper is concerned with the time fractional derivatives (Riemann–Liouville) of non-linear anomalous diffusion equation. Using Lie symmetry method, we show this equation can be reduced to Erdelyi–Kober fractional derivatives type. Then all of the symmetry vector fields and some exact solutions of our time fractional non-linear equation are obtained.

The non-linear anomalous diffusion in anisotropic media is using in various fields, e.g. in molecular dynamics, hydrology, financial systems, porous media analysis, etc. The Lie group method is developed to study the time-fractional diffusion equation with a source term in anisotropic media. As an application, the complete Lie group classification is performed up to the equivalence transformations for all special cases of the coefficients. Some similarity reductions are obtained for implicit cases. The analytical invariant subspace method is used in order to find some exact solutions. The work is concluded by the fractional Chebyshev pseudo-spectral (FCPS) method for constructing a numerical simulation for some of the reduced equations in the symmetry analysis section.

The KdV equation is one of the most famous PDE which has a vast field of applications in fluid dynamics. The scope of our research is the Lie group analysis of time-fractional KdV-type equation including a perturbation term. Thus, the Lie group theory is extended to both cases simultaneously. Geometric vector fields of symmetries and one-dimensional optimal system are found in order to reduce the equation. Finally, approximated conservation laws are given via modified Ibragimov’s theorem.

Instability and excessive volatilities in the various financial and non-financial markets have increased the importance of risk hedging in investments. So in this paper, the hypothetical options pricing have been done based on fractional Heston model in Iran’s gold market. In fact power option pricing has been driven using fast Fourier transformation (FFT), and maximum likelihood method. The study period has been classified to periods of three months and the profit and loss resulting from this option in each period has been calculated using discount rate based on the consumer price index. Findings indicated in most periods of three months, the model’s determination has been done correctly and the estimated price of power option has prevented the creation of arbitrage opportunity and caused a secure position of investment. But in the two periods related to 2018 due to high volatilities, the power option has created an arbitrage opportunity.

In this paper, group analysis of the fourth-order time-fractional Burgers–Korteweg–de Vries (KdV) equation is considered. Geometric vector fields of Lie point symmetries of the equation are investigated and the corresponding optimal system is found. Similarity solutions of the equation are presented by using the obtained optimal system. Finally, a useful method called invariant subspaces is applied in order to find another solutions.

A supply chain (SC) is a coordinated system involved in moving products or services from supplier to customer. With growing focus on supply chain management, it is realized that inventories across the supply chain should be efficiently managed through greater cooperation and better coordination. Multi-agent systems (MAS) offer a new dimension for cooperation and coordination. MAS consist of a collection of automated agents who can perform a specific task intelligently without any human supervision and communicate with each other cooperatively. In this research, a multi-agent system has been developed to simulate a multi echelon SC. These agents are coordinated to control inventories and minimize the total cost of SC by sharing information, forecasting knowledge and using negotiation mechanism in special situation. The result showed that in comparison to a traditional supply chain network, this system decreases the total cost and bullwhip effect.

In this paper, power options pricing is driven via time-fractional PDE when the dynamic of underlying asset price follows a regime switching model in which the risky underlying asset depends on a continuous-time hidden Markov chain process. An exact solution for power options pricing is driven under our considered model.

In this paper, the transition joint probability density function of the solution of geometric Brownian motion (GBM) equation is obtained via Lie group theory of differential equations (DEs). Lie symmetry analysis is applied to find new solutions for time-fractional Fokker–Planck–Kolmogorov equation of GBM. This analysis classifies the forms of the solutions for the equation by the similarity variables arising from the symmetry operators. Finally, an analytic method called invariant subspace method is applied in order to find another exact solution.

This paper presents the initial results of a project to investigate the organizational antecedents of entrepreneurship at public research and technology institutions (RTIs) in Iran. In line with the policies of the government to increase the role of publicly funded R&D in solving the problems of domestic industry, the research centers within public RTIs are encouraged to direct their research efforts towards practical applications, thus creating value from their R&D efforts. We chose to utilize a widely-used construct from corporate entrepreneurship literature (Entrepreneurial Orientation, EO) to explain undeniable differences among these research centers in their success in academic entrepreneurship. Noting that EO is a context-sensitive construct, the items to measure EO in research centers were generated by in-depth interviews; and the relationship between the contextualized EO and financial performances of research centers was investigated. The results confirmed a notable association between the EO level of research centers and their financial performance.

In this paper, the Lie approximate symmetry analysis is applied to investigate the new exact solutions of the Rayleigh-wave equation. The power series method is employed to solve some of the obtained reduced ordinary differential equations with a small parameter. We yield the new analytical solutions with small parameter which is effectively obtained by the proposed method. The concept of nonlinear self-adjointness is used to construct the conservation laws for Rayleigh-wave equation. It is shown that this equation is approximately nonlinearly self-adjoint and therefore desired conservation laws can be found using appropriate formal Lagrangians.

This paper is a complete survey of flowshop-scheduling problems and contributions from early works of Johnson of 1954 to recent approaches of metaheuristics of 2004. It mainly considers a flowshop problem with a makespan criterion and it surveys some exact methods (for small size problems), constructive heuristics and developed improving metaheuristic and evolutionary approaches as well as some well-known properties and rules for this problem. Each part has a brief literature review of the contributions and a glimpse of that approach before discussing the implementation for a flowshop problem. Moreover, in the first section, a complete literature review of flowshop-related scheduling problems with different assumptions as well as contributions in solving these other aspects is considered. This paper can be seen as a reference to past contributions (particularly in n/m/p/c max or equivalently F/prmu/c max) for future research needs of improving and developing better approaches to flowshop-related schedulin...

This paper investigates the invariance properties of the time fractional Benny–Lin equation with Riemann–Liouville and Caputo derivatives. This equation can be reduced to the Kawahara equation, fifth-order Kdv equation, the Kuramoto–Sivashinsky equation and Navier–Stokes equation. By using the Lie group analysis method of fractional differential equations (FDEs), we derive Lie symmetries for the Benny–Lin equation. Conservation laws for this equation are obtained with the aid of the concept of nonlinear self-adjointness and the fractional generalization of the Noether’s operators. Furthermore, by means of the invariant subspace method, exact solutions of the equation are also constructed.

In this paper, some properties of the time fractional Boussinesq equation are presented. Group analysis of the time fractional Boussinesq equation with Riemann–Liouville derivative is performed and the corresponding optimal system of subgroups are determined. Next, we apply the obtained optimal systems for constructing reduced fractional ordinary differential equations (FODEs). Finally, we show how to derive exact solutions to time fractional Boussinesq equation via invariant subspace method.

This paper uses Lie symmetry group method to study a special kind of PDE. By using the Lie symmetry analysis, all of the geometric vector fields of the equation are obtained; the symmetry reductions are also presented. Some new nonlinear wave solutions, involving differentiable arbitrary functions are obtained.