Papers by Mehrdad Lakestani
Qualitative Theory of Dynamical Systems, Jun 29, 2023
Zenodo (CERN European Organization for Nuclear Research), May 11, 2022
Here, we present a numerical scheme to solve optimal control problems with time-varying delay sys... more Here, we present a numerical scheme to solve optimal control problems with time-varying delay system. This method is based on Lucas wavelets and Galerkin method. Operational matrices of integration and delay for Lucas wavelets are proposed. Then, Galerkin method is used to solve the mentioned problems. Numerical results are included to demonstrate the efficiency of the present technique.
Acta Physica Polonica A, Jul 1, 2019
This paper retrieves new combo, dark and singular optical soliton solutions along with singular p... more This paper retrieves new combo, dark and singular optical soliton solutions along with singular periodic, combined hyperbolic and rational solutions to the Fokas-Lenells equation in birefringent fibres by integration tools such as the improved tan(φ(ξ)/2)-expansion method, improved Bernoulli sub-ODE method, and generalized (G /G)expansion method. The existence criterions of these solutions are also listed.
This paper is concerned the using of biorthogonal Flatlet oblique multiwavelet system to solve li... more This paper is concerned the using of biorthogonal Flatlet oblique multiwavelet system to solve linear Fredholm integral equations. The biorthogonality and high vanishing moments properties of this system result in efficient and accurate solutions. Finally, numerical results and the absolute errors for some test problems with known solutions are presented.
Physica Scripta, Aug 9, 2006
This paper presents a numerical method for solving the controlled Duffing oscillator. The method ... more This paper presents a numerical method for solving the controlled Duffing oscillator. The method can be extended to nonlinear calculus of variations and optimal control problems. The method is based upon compactly supported linear semiorthogonal B-spline wavelets. The differential and integral expressions which arise in the system dynamics, the performance index and the boundary conditions are converted into some algebraic equations which can be solved for the unknown coefficients. Illustrative examples are included to demonstrate the validity and applicability of the technique.
Optical and Quantum Electronics, Jan 20, 2016
In this study, the new extended direct algebraic method is exerted for constructing more general ... more In this study, the new extended direct algebraic method is exerted for constructing more general exact solutions of the three nonlinear evolution equations with physical interest namely, the Tzitzéica equation, the Dodd-Bullough-Mikhailor equation and the Liouville equation. By using of an appropriate traveling wave transformation reduces these equations to ODE. We state that this method is excellently a generalized form to obtain solitary wave solutions of the nonlinear evolution equations that are widely used in theoretical physics. The method appears to be easier and faster by means of symbolic computation system.
Pramana, Nov 4, 2016
In this paper, we find exact solutions of some nonlinear evolution equations by using generalized... more In this paper, we find exact solutions of some nonlinear evolution equations by using generalized tanh-coth method. Three nonlinear models of physical significance, i.e. the Cahn-Hilliard equation, the Allen-Cahn equation and the steady-state equation with a cubic nonlinearity are considered and their exact solutions are obtained. From the general solutions, other well-known results are also derived. Also in this paper, we shall compare the generalized tanh-coth method and generalized (G /G)-expansion method to solve partial differential equations (PDEs) and ordinary differential equations (ODEs). Abundant exact travelling wave solutions including solitons, kink, periodic and rational solutions have been found. These solutions might play important roles in engineering fields. The generalized tanh-coth method was used to construct periodic wave and solitary wave solutions of nonlinear evolution equations. This method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. It is shown that the generalized tanh-coth method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear problems.
International Journal of Systems Science, Mar 30, 2020
In this article, a pair of wavelets for Hermite cubic spline bases are presented. These wavelets ... more In this article, a pair of wavelets for Hermite cubic spline bases are presented. These wavelets are in C 1 and supported on [−1, 1]. These spline wavelets are then adapted to the interval [0, 1] and we prove that they form a Riesz wavelet in L 2 ([0, 1]). The wavelet bases are used to solve the linear optimal control problems. The operational matrices of integration and product are then utilised to reduce the given optimisation problems to the system of algebraic equations. Because of the sparsity nature of these matrices, this method is computationally very attractive and reduces CPU time and computer memory. In order to save the memory requirement and computation time, a threshold procedure is applied to obtain algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.
Computers & mathematics with applications, Dec 1, 2022
Asian Journal of Control, Jun 28, 2017
In this paper, a numerical method for solving nonlinear quadratic optimal control problems with i... more In this paper, a numerical method for solving nonlinear quadratic optimal control problems with inequality constraints is presented. The method is based upon cardinal Hermite interpolant multiscaling function approximation. The properties of these multiscaling functions are presented first. These properties are then utilized to reduce the solution of the nonlinear constrained optimal control to a nonlinear programming one, to which existing algorithms may be applied. Illustrative examples are included to demonstrate the efficiency and applicability of the technique.
Communications in Nonlinear Science and Numerical Simulation, Mar 1, 2012
Fractional calculus has been used to model physical and engineering processes that are found to b... more Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. For that reason we need a reliable and efficient technique for the solution of fractional differential equations. Here we construct the operational matrix of fractional derivative of order a in the Caputo sense using the linear B-spline functions. The main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of algebraic equations thus we can solve directly the problem. The method is applied to solve two types of fractional differential equations, linear and nonlinear. Illustrative examples are included to demonstrate the validity and applicability of the new technique presented in the current paper.
International Journal of Control
Numerical Methods for Partial Differential Equations, Apr 27, 2021
In this work, we design, analyze, and test the multiwavelets Galerkin method to solve the two‐dim... more In this work, we design, analyze, and test the multiwavelets Galerkin method to solve the two‐dimensional Burgers equation. Using Crank–Nicolson scheme, time is discretized and a PDE is obtained for each time step. We use the multiwavelets Galerkin method for solving these PDEs. Multiwavelets Galerkin method reduces these PDEs to sparse systems of algebraic equations. The cost of this method is proportional to the number of nonzero coefficients at each time step. The results illustrate, by selecting the appropriate threshold while the number of nonzero coefficients reduces, the error will not be less than a certain amount. The L2 stability and convergence of the scheme have been investigated by the energy method. Illustrative examples are provided to verify the efficiency and applicability of the proposed method.
Mathematics
An efficient algorithm based on the wavelet collocation method is introduced in order to solve no... more An efficient algorithm based on the wavelet collocation method is introduced in order to solve nonlinear fractional optimal control problems (FOCPs) with inequality constraints. By using the interpolation properties of Hermite cubic spline functions, we construct an operational matrix of the Caputo fractional derivative for the first time. Using this matrix, we reduce the nonlinear fractional optimal control problem to a nonlinear programming problem that can be solved with some suitable optimization algorithms. Illustrative examples are examined to demonstrate the important features of the new method.
Iranian Journal of Science
Computational and Applied Mathematics
Optical and Quantum Electronics, 2017
Under investigation in this paper is a nonlinear conformable time-fractional Boussinesq equations... more Under investigation in this paper is a nonlinear conformable time-fractional Boussinesq equations as an important class of fractional differential equations in mathematical physics. The extended trial equation method, the expðÀXðgÞÞ-expansion method and the tanð/ðgÞ=2Þ-expansion method are used in examining the analytical solution of the nonlinear fractional equations. The proposed methods are based on the integration method and a wave transformation. The fractional derivative in the sense of conformable timefractional derivative is defined. Fractional complex transform is implemented to change fractional differential equations into ordinary differential equations in this paper. In addition, explicit new exact solutions are derived in different form such as dark solitons, bright solitons, solitary wave, periodic solitary wave, rational function, and elliptic function solutions of nonlinear conformable time-fractional Boussinesq equations. Keywords Conformable time-fractional Boussinesq equations Á Conformable time-fractional derivative Á The extended trial equation method Á The expðÀXðgÞÞexpansion method Á The tanð/ðgÞ=2Þ-expansion method & Jalil Manafian
Modern Physics Letters B
This paper aims to compute solitary wave solutions and soliton wave solutions based on the ansatz... more This paper aims to compute solitary wave solutions and soliton wave solutions based on the ansatz methods to the perturbed nonlinear Schrödinger equation (NLSE) arising in nano-fibers. The improved [Formula: see text]-expansion method and the rational extended sinh–Gordon equation expansion method are used for the first time to obtain the new optical solitons of this equation. The obtained results give an accuracy interpretation of the propagation of solitons. We held a comparison between our results and those are in the previous work. The outcome indicates that perturbed NLSE arising nano-fibers is used in optical problems. Finally, via symbolic computation, their dynamic structure and physical properties were vividly shown by three-dimensional, density, and [Formula: see text]-curves plots. These solutions have greatly enriched the exact solutions of (2+1)-dimensional perturbed nonlinear Schrödinger equation in the existing literatures.
Applied Numerical Mathematics, 2021
Abstract In this article, we develop a new set of functions called fractional-order Alpert multiw... more Abstract In this article, we develop a new set of functions called fractional-order Alpert multiwavelet functions to obtain the numerical solution of fractional pantograph differential equations (FPDEs). The fractional derivative of Caputo type is considered. Here we construct the Riemann–Liouville fractional operational matrix of integration (Riemann–Liouville FOMI) using the fractional-order Alpert multiwavelet functions. The most important feature behind the scheme using this technique is that the pantograph equation reduces to a system of linear or nonlinear algebraic equations. We perform the error analysis for the proposed technique. Illustrative examples are examined to demonstrate the important features of the new method.
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Papers by Mehrdad Lakestani