Critical points for functionals with quasilinear singular Euler–Lagrange equations

For a bounded, open subset Ω of ${\mathbb{R}^{N}}$ with N > 2, and a measurable function a(x) satisfying 0 < α ≤ a(x) ≤ β, a.e. ${x \in \Omega}$ , we study the existence of positive solutions of the Euler–Lagrange equation associated to the non-differentiable functional $$\begin{array}{ll}J(v) = \frac{1}{2} \int \limits_{\Omega} [a(x)+|v|^{\gamma}]| \nabla v|^{2}- \frac{1}{p} \int \limits_{\Omega}(v_{+})^p,\end{array}$$ if γ > 0 and p > 1. Special emphasis is placed on the case ${2^{*} < p < \frac{2^{*}}{2} ( \gamma +2 )}$ .     

Multi-symplectic variational integrators for the Gross–Pitaevskii equations in BEC

In this paper, a variational integrator is constructed for Gross–Pitaevskii equations in Bose–Einstein condensate. The discrete multi-symplectic geometric structure is derived. The discrete mass and energy conservation laws are proved. The numerical tests show the effectiveness of the variational integrator, and the performance of the proved discrete conservation law.     

Euler–Lagrange Equations for Lagrangians Containing Complex-order Fractional Derivatives

Two variational problems of finding the Euler–Lagrange equations corresponding to Lagrangians containing fractional derivatives of real- and complex-order are considered. The first one is the unconstrained variational problem, while the second one is the fractional optimal control problem. The expansion formula for fractional derivatives of complex-order is derived in order to approximate the fractional derivative appearing in the Lagrangian. As a consequence, a sequence of approximated Euler–Lagrange equations is obtained. It is shown that the sequence of approximated Euler–Lagrange equations converges to the original one in the weak sense as well as that the sequence of the minimal values of approximated action integrals tends to the minimal value of the original one.     

On derivation of Euler–Lagrange equations for incompressible energy-minimizers

We prove that any distribution q satisfying the grad-div system \({\nabla q={\rm div}\,{\bf f}}\) for some tensor \({{\bf f}=(f^i_j), \,f^i_j\in h^r(U)\,(1\leq r < \infty}\)) -the local Hardy space; q is in h ^r and q is locally represented by the sum of singular integrals of \({f^i_j}\) with Calderón-Zygmund kernel. As a consequence, we prove the existence and the local representation of the hydrostatic pressure p (modulo constant) associated with incompressible elastic energy-minimizing deformation u satisfying \({|\nabla{\bf u}|^2,\,|{\rm cof}\,\nabla{\bf u}|^2\in h^1}\). We also derive the system of Euler–Lagrange equations for volume preserving local minimizers u that are in the space \({K^{1,3}_{\rm loc}}\) [defined in (1.2)]—partially resolving a long standing problem. In two dimensions we prove partial C ^1,α regularity of weak solutions provided their gradient is in L ³ and p is Hölder continuous.     

A new method of finding the fractional Euler–Lagrange and Hamilton equations within Caputo fractional derivatives

In this paper, we have investigated the fractional Caputo derivative of a composition function. The obtained results were applied to investigate the fractional Euler–Lagrange and Hamilton equations for constrained systems. The approach was applied within an illustrative.     

Cauchy problems for fractional differential equations with Riemann–Liouville fractional derivatives

In this paper, we are concerned with Cauchy problems of fractional differential equations with Riemann–Liouville fractional derivatives in infinite-dimensional Banach spaces. We introduce the notion of fractional resolvent, obtain some its properties, and present a generation theorem for exponentially bounded fractional resolvents. Moreover, we prove that a homogeneous α-order Cauchy problem is well posed if and only if its coefficient operator is the generator of an α-order fractional resolvent, and we give sufficient conditions to guarantee the existence and uniqueness of weak solutions and strong solutions of an inhomogeneous α-order Cauchy problem.     

Invariance of functionals and related Euler–Lagrange equations

We establish a connection between symmetries of functionals and symmetries of the corresponding Euler–Lagrange equations. A similar problem is investigated for equations with quasi-B _u -potential operators.     

Regularity results for fractional diffusion equations involving fractional derivative with Mittag–Leffler kernel

This paper studies partial differential equation model with the new general fractional derivatives involving the kernels of the extended Mittag–Leffler type functions. An initial boundary value problem for the anomalous diffusion of fractional order is analyzed and considered. The fractional derivative with Mittag–Leffler kernel or also called Atangana and Baleanu fractional derivative in time is taken in the Caputo sense. We obtain results on the existence, uniqueness, and regularity of the solution.     

On the Euler–Lagrange equation for a variational problem: the general case II

In this paper we study the existence of a solution in ${L^\infty_{\rm loc}(\Omega)}In this paper we study the existence of a solution in L^￥_loc(W){L^\infty_{\rm loc}(\Omega)} to the Euler–Lagrange equation for the variational problem

inf[`(u)] + W1,￥0(W) òW (ID(?u) + g(u)) dx,                   (0.1)\inf_{\bar u + W^{1,\infty}_0(\Omega)} \int\limits_{\Omega} ({\bf I}_D(\nabla u) + g(u)) dx,\quad \quad \quad \quad \quad(0.1)      11. Improved blowup results for the Euler and Euler–Poisson equations with repulsive forces      Rui Li  Xing LinZongwei Ma  Jingjun Zhang 《Journal of Mathematical Analysis and Applications》2014 Some results are presented on the formation of singularities in the solutions of the radially-symmetric N-dimensional Euler or Euler–Poisson equations with repulsive forces. Based on the integration method of M.W. Yuen, we generalize the blowup results with constant compact radius R   of solutions to the case with general compact radius R(t)$R(t)$ and to the case with no compact support restriction.      12. Vanishing viscosity limit for Navier–Stokes equations with general viscosity to Euler equations      Tingting Zhang 《Applicable analysis》2018,97(11):1967-1982 In this paper, we study vanishing viscosity limit of 1-D isentropic compressible Navier–Stokes equations with general viscosity to isentropic Euler equations. Firstly, we improve estimates of the entropy flux, then we obtain that the weak solution of the isentropic Euler equations is the inviscid limit of the isentropic compressible Navier–Stokes equations with general viscosity using the compensated compactness frame recently established by G.-Q. Chen and M. Perepelitsa.      13. An approximate solution method for ordinary fractional differential equations with the Riemann–Liouville fractional derivatives      《Communications in Nonlinear Science & Numerical Simulation》2014,19(2):390-400       14. From quantum Euler–Maxwell equations to incompressible Euler equations      Jianwei Yang  Zhiping Ju 《Applicable analysis》2013,92(11):2201-2210 The combined quasi-neutral and non-relativistic limit of compressible quantum Euler–Maxwell equations for plasmas is studied in this paper. For well-prepared initial data, it is shown that the smooth solution of compressible quantum Euler–Maxwell equations converges to the smooth solution of incompressible Euler equations by using the modulated energy method. Furthermore, the associated convergence rates are also obtained.      15. Minimizers of Kirchhoff's plate functional: Euler–Lagrange equations and regularity      Peter Hornung 《Comptes Rendus Mathematique》2009,347(11-12):647-650       16. Fractional stochastic integration and Black–Scholes equation for fractional Brownian model with stochastic volatility      Yuliya Mishura 《Stochastics An International Journal of Probability and Stochastic Processes》2013,85(4):363-381 We modify the Hu-Øksendal and Elliot-van der Hoek approach to arbitrage-free financial markets driven by a fractional Brownian motion that is defined on a white noise space. We deduce and solve a Black–Scholes fractional equation for constant volatility and outline the corresponding equation with stochastic volatility. As an auxiliary result, we produce some simple conditions implying the existence of the Wick integral w.r.t. fractional noise.      17. Fractional integral problems for Hadamard–Caputo fractional Langevin differential inclusions      Sotiris K. Ntouyas  Jessada Tariboon 《Journal of Applied Mathematics and Computing》2016,51(1-2):13-33       18. Strong predictor–corrector Euler–Maruyama methods for stochastic differential equations with Markovian switching      Haibo Li  Lili XiaoJun Ye 《Journal of Computational and Applied Mathematics》2013 In this paper numerical methods for solving stochastic differential equations with Markovian switching (SDEwMSs) are developed by pathwise approximation. The proposed family of strong predictor–corrector Euler–Maruyama methods is designed to overcome the propagation of errors during the simulation of an approximate path. This paper not only shows the strong convergence of the numerical solution to the exact solution but also reveals the order of the error under some conditions on the coefficient functions. A natural analogue of the p$p$-stability criterion is studied. Numerical examples are given to illustrate the computational efficiency of the new predictor–corrector Euler–Maruyama approximation.      19. Conservation of energy for the Euler–Korteweg equations      Tomasz?D?biec  Piotr?Gwiazda  Agnieszka??wierczewska-GwiazdaEmail author  Athanasios?Tzavaras 《Calculus of Variations and Partial Differential Equations》2018,57(6):160 In this article we study the principle of energy conservation for the Euler–Korteweg system. We formulate an Onsager-type sufficient regularity condition for weak solutions of the Euler–Korteweg system to conserve the total energy. The result applies to the system of Quantum Hydrodynamics.      20. Application of variational iteration method for Hamilton–Jacobi–Bellman equations      B. Kafash  A. Delavarkhalafi  S.M. Karbassi 《Applied Mathematical Modelling》2013 In this paper, we use the variational iteration method (VIM) for optimal control problems. First, optimal control problems are transferred to Hamilton–Jacobi–Bellman (HJB) equation as a nonlinear first order hyperbolic partial differential equation. Then, the basic VIM is applied to construct a nonlinear optimal feedback control law. By this method, the control and state variables can be approximated as a function of time. Also, the numerical value of the performance index is obtained readily. In view of the convergence of the method, some illustrative examples are presented to show the efficiency and reliability of the presented method.

Improved blowup results for the Euler and Euler–Poisson equations with repulsive forces

Some results are presented on the formation of singularities in the solutions of the radially-symmetric N-dimensional Euler or Euler–Poisson equations with repulsive forces. Based on the integration method of M.W. Yuen, we generalize the blowup results with constant compact radius R   of solutions to the case with general compact radius R(t)$R(t)$ and to the case with no compact support restriction.     

Vanishing viscosity limit for Navier–Stokes equations with general viscosity to Euler equations

In this paper, we study vanishing viscosity limit of 1-D isentropic compressible Navier–Stokes equations with general viscosity to isentropic Euler equations. Firstly, we improve estimates of the entropy flux, then we obtain that the weak solution of the isentropic Euler equations is the inviscid limit of the isentropic compressible Navier–Stokes equations with general viscosity using the compensated compactness frame recently established by G.-Q. Chen and M. Perepelitsa.     

From quantum Euler–Maxwell equations to incompressible Euler equations

The combined quasi-neutral and non-relativistic limit of compressible quantum Euler–Maxwell equations for plasmas is studied in this paper. For well-prepared initial data, it is shown that the smooth solution of compressible quantum Euler–Maxwell equations converges to the smooth solution of incompressible Euler equations by using the modulated energy method. Furthermore, the associated convergence rates are also obtained.     

Fractional stochastic integration and Black–Scholes equation for fractional Brownian model with stochastic volatility

We modify the Hu-Øksendal and Elliot-van der Hoek approach to arbitrage-free financial markets driven by a fractional Brownian motion that is defined on a white noise space. We deduce and solve a Black–Scholes fractional equation for constant volatility and outline the corresponding equation with stochastic volatility. As an auxiliary result, we produce some simple conditions implying the existence of the Wick integral w.r.t. fractional noise.     

Strong predictor–corrector Euler–Maruyama methods for stochastic differential equations with Markovian switching

In this paper numerical methods for solving stochastic differential equations with Markovian switching (SDEwMSs) are developed by pathwise approximation. The proposed family of strong predictor–corrector Euler–Maruyama methods is designed to overcome the propagation of errors during the simulation of an approximate path. This paper not only shows the strong convergence of the numerical solution to the exact solution but also reveals the order of the error under some conditions on the coefficient functions. A natural analogue of the p$p$-stability criterion is studied. Numerical examples are given to illustrate the computational efficiency of the new predictor–corrector Euler–Maruyama approximation.     

Conservation of energy for the Euler–Korteweg equations

In this article we study the principle of energy conservation for the Euler–Korteweg system. We formulate an Onsager-type sufficient regularity condition for weak solutions of the Euler–Korteweg system to conserve the total energy. The result applies to the system of Quantum Hydrodynamics.     

Application of variational iteration method for Hamilton–Jacobi–Bellman equations

In this paper, we use the variational iteration method (VIM) for optimal control problems. First, optimal control problems are transferred to Hamilton–Jacobi–Bellman (HJB) equation as a nonlinear first order hyperbolic partial differential equation. Then, the basic VIM is applied to construct a nonlinear optimal feedback control law. By this method, the control and state variables can be approximated as a function of time. Also, the numerical value of the performance index is obtained readily. In view of the convergence of the method, some illustrative examples are presented to show the efficiency and reliability of the presented method.