We deal with the Wick-type stochastic fractional Korteweg de–Vries(KdV) equation with conformable derivatives.With the aid of the Exp-function method, white noise theory, and Hermite transform, we produce a novel set...We deal with the Wick-type stochastic fractional Korteweg de–Vries(KdV) equation with conformable derivatives.With the aid of the Exp-function method, white noise theory, and Hermite transform, we produce a novel set of exact soliton and periodic wave solutions to the fractional KdV equation with conformable derivatives. With the help of inverse Hermite transform, we get stochastic soliton and periodic wave solutions of the Wick-type stochastic fractional KdV equation with conformable derivatives. Eventually, by an application example, we show how the stochastic solutions can be given as Brownian motion functional solutions.展开更多
A modified fractional sub-equation method is applied to Wick-type stochastic fractional two-dimensional （2D） KdV equations. With the help of a Hermit transform, we obtain a new set of exact stochastic solutions to W...A modified fractional sub-equation method is applied to Wick-type stochastic fractional two-dimensional （2D） KdV equations. With the help of a Hermit transform, we obtain a new set of exact stochastic solutions to Wick-type stochastic fractional 2D KdV equations in the white noise space. These solutions include exponential decay wave solutions, soliton wave solutions, and periodic wave solutions. Two examples are explicitly given to illustrate our approach.展开更多
Variable coefficients and Wick-type stochastic fractional coupled KdV equations are investigated. By using the mod- ified fractional sub-equation method, Hermite transform, and white noise theory the exact travelling ...Variable coefficients and Wick-type stochastic fractional coupled KdV equations are investigated. By using the mod- ified fractional sub-equation method, Hermite transform, and white noise theory the exact travelling wave solutions and white noise functional solutions are obtained, including the generalized exponential, hyperbolic, and trigonometric types.展开更多
In this paper,the non-stationary incompressible fluid flows governed by the Navier-Stokes equations are studied in a bounded domain.This study focuses on the time-fractional Navier-Stokes equations in the optimal cont...In this paper,the non-stationary incompressible fluid flows governed by the Navier-Stokes equations are studied in a bounded domain.This study focuses on the time-fractional Navier-Stokes equations in the optimal control subject,where the control is distributed within the domain and the time-fractional derivative is proposed as Riemann-Liouville sort.In addition,the control object is to minimize the quadratic cost functional.By using the Lax-Milgram lemma with the assistance of the fixed-point theorem,we demonstrate the existence and uniqueness of the weak solution to this system.Moreover,for a quadratic cost functional subject to the time-fractional Navier-Stokes equations,we prove the existence and uniqueness of optimal control.Also,via the variational inequality upon introducing the adjoint linearized system,some inequalities and identities are given to guarantee the first-order necessary optimality conditions.A direct consequence of the results obtained here is that when a→1,the obtained results are valid for the classical optimal control of systems governed by the Navier-Stokes equations.展开更多
基金the Deanship of Scientific Research at King Khalid University for funding their work through Research Group Program under grant number(G.P.1/160/40)。
文摘We deal with the Wick-type stochastic fractional Korteweg de–Vries(KdV) equation with conformable derivatives.With the aid of the Exp-function method, white noise theory, and Hermite transform, we produce a novel set of exact soliton and periodic wave solutions to the fractional KdV equation with conformable derivatives. With the help of inverse Hermite transform, we get stochastic soliton and periodic wave solutions of the Wick-type stochastic fractional KdV equation with conformable derivatives. Eventually, by an application example, we show how the stochastic solutions can be given as Brownian motion functional solutions.
文摘A modified fractional sub-equation method is applied to Wick-type stochastic fractional two-dimensional （2D） KdV equations. With the help of a Hermit transform, we obtain a new set of exact stochastic solutions to Wick-type stochastic fractional 2D KdV equations in the white noise space. These solutions include exponential decay wave solutions, soliton wave solutions, and periodic wave solutions. Two examples are explicitly given to illustrate our approach.
文摘Variable coefficients and Wick-type stochastic fractional coupled KdV equations are investigated. By using the mod- ified fractional sub-equation method, Hermite transform, and white noise theory the exact travelling wave solutions and white noise functional solutions are obtained, including the generalized exponential, hyperbolic, and trigonometric types.
基金The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding their work through General Research Project under grant number(GRP-114-41).
文摘In this paper,the non-stationary incompressible fluid flows governed by the Navier-Stokes equations are studied in a bounded domain.This study focuses on the time-fractional Navier-Stokes equations in the optimal control subject,where the control is distributed within the domain and the time-fractional derivative is proposed as Riemann-Liouville sort.In addition,the control object is to minimize the quadratic cost functional.By using the Lax-Milgram lemma with the assistance of the fixed-point theorem,we demonstrate the existence and uniqueness of the weak solution to this system.Moreover,for a quadratic cost functional subject to the time-fractional Navier-Stokes equations,we prove the existence and uniqueness of optimal control.Also,via the variational inequality upon introducing the adjoint linearized system,some inequalities and identities are given to guarantee the first-order necessary optimality conditions.A direct consequence of the results obtained here is that when a→1,the obtained results are valid for the classical optimal control of systems governed by the Navier-Stokes equations.
