In the present paper,the numerical solution of It?type stochastic parabolic equation with a timewhite noise process is imparted based on a stochastic finite difference scheme.At the beginning,an implicit stochastic fi...In the present paper,the numerical solution of It?type stochastic parabolic equation with a timewhite noise process is imparted based on a stochastic finite difference scheme.At the beginning,an implicit stochastic finite difference scheme is presented for this equation.Some mathematical analyses of the scheme are then discussed.Lastly,to ascertain the efficacy and accuracy of the suggested technique,the numerical results are discussed and compared with the exact solution.展开更多
In this work,a system of three masses on the vertices of equilateral triangle is investigated.This system is known in the literature as a planar system.We first give a description to the system by constructing its cla...In this work,a system of three masses on the vertices of equilateral triangle is investigated.This system is known in the literature as a planar system.We first give a description to the system by constructing its classical Lagrangian.Secondly,the classical Euler-Lagrange equations(i.e.,the classical equations of motion)are derived.Thirdly,we fractionalize the classical Lagrangian of the system,and as a result,we obtain the fractional Euler-Lagrange equations.As the final step,we give the numerical simulations of the fractional model,a new model which is based on Caputo fractional derivative.展开更多
This paper presents a new approach to find an approximate solution for the nonlinear path planning problem. In this approach, first by defining a new formulation in the calculus of variations, an optimal control probl...This paper presents a new approach to find an approximate solution for the nonlinear path planning problem. In this approach, first by defining a new formulation in the calculus of variations, an optimal control problem, equivalent to the original problem, is obtained. Then, a metamorphosis is performed in the space of problem by defining an injection from the set of admissible trajectory-control pairs in this space into the space of positive Radon measures. Using properties of Radon measures, the problem is changed to a measure-theo- retical optimization problem. This problem is an infinite dimensional linear programming (LP), which is approximated by a finite dimensional LP. The solution of this LP is used to construct an approximate solution for the original path planning problem. Finally, a numerical example is included to verify the effectiveness of the proposed approach.展开更多
In this study,a harmonic oscillator with position-dependent mass is investigated.Firstly,as an introduction,we give a full description of the system by constructing its classical Lagrangian;thereupon,we derive the rel...In this study,a harmonic oscillator with position-dependent mass is investigated.Firstly,as an introduction,we give a full description of the system by constructing its classical Lagrangian;thereupon,we derive the related classical equations of motion such as the classical Euler–Lagrange equations.Secondly,we fractionalize the classical Lagrangian of the system,and then we obtain the corresponding fractional Euler–Lagrange equations(FELEs).As a final step,we give the numerical simulations corresponding to the FELEs within different fractional operators.Numerical results based on the Caputo and the Atangana-Baleanu-Caputo(ABC)fractional derivatives are given to verify the theoretical analysis.展开更多
This paper presents a new approach for solving a class of infinite horizon nonlinear optimal control problems (OCPs).In this approach,a nonlinear two-point boundary value problem (TPBVP),derived from Pontryagin's ...This paper presents a new approach for solving a class of infinite horizon nonlinear optimal control problems (OCPs).In this approach,a nonlinear two-point boundary value problem (TPBVP),derived from Pontryagin's maximum principle,is transformed into a sequence of linear time-invariant TPBVPs.Solving the latter problems in a recursive manner provides the optimal control law and the optimal trajectory in the form of uniformly convergent series.Hence,to obtain the optimal solution,only the techniques for solving linear ordinary differential equations are employed.An efficient algorithm is also presented,which has low computational complexity and a fast convergence rate.Just a few iterations are required to find an accurate enough suboptimal trajectory-control pair for the nonlinear OCP.The results not only demonstrate the efficiency,simplicity,and high accuracy of the suggested approach,but also indicate its effectiveness in practical use.展开更多
文摘In the present paper,the numerical solution of It?type stochastic parabolic equation with a timewhite noise process is imparted based on a stochastic finite difference scheme.At the beginning,an implicit stochastic finite difference scheme is presented for this equation.Some mathematical analyses of the scheme are then discussed.Lastly,to ascertain the efficacy and accuracy of the suggested technique,the numerical results are discussed and compared with the exact solution.
文摘In this work,a system of three masses on the vertices of equilateral triangle is investigated.This system is known in the literature as a planar system.We first give a description to the system by constructing its classical Lagrangian.Secondly,the classical Euler-Lagrange equations(i.e.,the classical equations of motion)are derived.Thirdly,we fractionalize the classical Lagrangian of the system,and as a result,we obtain the fractional Euler-Lagrange equations.As the final step,we give the numerical simulations of the fractional model,a new model which is based on Caputo fractional derivative.
文摘This paper presents a new approach to find an approximate solution for the nonlinear path planning problem. In this approach, first by defining a new formulation in the calculus of variations, an optimal control problem, equivalent to the original problem, is obtained. Then, a metamorphosis is performed in the space of problem by defining an injection from the set of admissible trajectory-control pairs in this space into the space of positive Radon measures. Using properties of Radon measures, the problem is changed to a measure-theo- retical optimization problem. This problem is an infinite dimensional linear programming (LP), which is approximated by a finite dimensional LP. The solution of this LP is used to construct an approximate solution for the original path planning problem. Finally, a numerical example is included to verify the effectiveness of the proposed approach.
文摘In this study,a harmonic oscillator with position-dependent mass is investigated.Firstly,as an introduction,we give a full description of the system by constructing its classical Lagrangian;thereupon,we derive the related classical equations of motion such as the classical Euler–Lagrange equations.Secondly,we fractionalize the classical Lagrangian of the system,and then we obtain the corresponding fractional Euler–Lagrange equations(FELEs).As a final step,we give the numerical simulations corresponding to the FELEs within different fractional operators.Numerical results based on the Caputo and the Atangana-Baleanu-Caputo(ABC)fractional derivatives are given to verify the theoretical analysis.
文摘This paper presents a new approach for solving a class of infinite horizon nonlinear optimal control problems (OCPs).In this approach,a nonlinear two-point boundary value problem (TPBVP),derived from Pontryagin's maximum principle,is transformed into a sequence of linear time-invariant TPBVPs.Solving the latter problems in a recursive manner provides the optimal control law and the optimal trajectory in the form of uniformly convergent series.Hence,to obtain the optimal solution,only the techniques for solving linear ordinary differential equations are employed.An efficient algorithm is also presented,which has low computational complexity and a fast convergence rate.Just a few iterations are required to find an accurate enough suboptimal trajectory-control pair for the nonlinear OCP.The results not only demonstrate the efficiency,simplicity,and high accuracy of the suggested approach,but also indicate its effectiveness in practical use.
