Non-standard Lagrange–Burman methods for the numerical integration of differential equations

We present new non-standard methods based on Mickens' ideas for constructing numerical schemes for differential equations. By analysing his schemes, we were able to give generalized versions by means of Lagrange–Burman expansions. Our generalization provides a theoretical frame to understand non-standard finite difference methods and also to analyse the errors in the approximations. We apply the new non-standard methods to several examples.     

Multi-soliton solution,rational solution of the Boussinesq–Burgers equations

In this paper we consider the Boussinesq–Burgers equations and establish the transformation which turns the Boussinesq–Burgers equations into the single nonlinear partial differential equation, then we obtain an auto-Bäcklund transformation and abundant new exact solutions, including the multi-solitary wave solution and the rational series solutions. Besides the new trigonometric function periodic solutions are obtained by using the generalized tan h method.     

On the pathwise exponential stability of non–linear stochastic partial differential equations

Sufficient conditions to get exponential stability for the sample paths (with probability one) of a non–linear monotone stochastic Partial Differential Equation are proved. In fact, we improve a stability criterion established in Chow [3] since, under the same hypotheses, we get pathwise exponential stability instead of stability of sample paths     

Nonhypoellipticity and comparison principle for partial differential equations of Black–Scholes type

This paper studies some less known properties of the Black–Scholes equation and of its nonlinear modifications arising in Finance. In particular, the nonhypoellipticity of the linear Black–Scholes equation is shown; a comparison principle is formulated for a class of nonlinear degenerate parabolic equations which incorporates the most relevant financial applications; finally, some comments on the properties of the viscosity solutions are given.     

Pure Lagrangian and semi-Lagrangian finite element methods for the numerical solution of Navier–Stokes equations

In this paper we propose a unified formulation to introduce Lagrangian and semi-Lagrangian velocity and displacement methods for solving the Navier–Stokes equations. This formulation allows us to state classical and new numerical methods. Several examples are given. We combine them with finite element methods for spatial discretization. In particular, we propose two new second-order characteristics methods in terms of the displacement, one semi-Lagrangian and the other one pure Lagrangian. The pure Lagrangian displacement methods are useful for solving free surface problems and fluid-structure interaction problems because the computational domain is independent of the time and fluid–solid coupling at the interphase is straightforward. However, for moderate to high-Reynolds number flows, they can lead to high distortion in the mesh elements. When this happens it is necessary to remesh and reinitialize the transformation to the identity. In order to assess the performance of the obtained numerical methods, we solve different problems in two space dimensions. In particular, numerical results for a sloshing problem in a rectangular tank and the flow in a driven cavity are presented.     

Hyers–Ulam stability of linear partial differential equations of first order

In this work, we will prove the Hyers–Ulam stability of linear partial differential equations of first order.     

Sturm–Picone comparison theorem of a kind of conformable fractional differential equations on time scales

In this paper, we consider the Sturm–Picone comparison theorem of conformable fractional differential equations on arbitrary time scales. Since the Picone identity plays an important role in discussing the Sturm comparison theorem. Firstly, we establish the Picone identity of conformable fractional differential equations on arbitrary time scales. By using this identity, we obtain our main result—the Sturm–Picone comparison theorem of conformable fractional differential equations on time scales. This result not only extends and improves the corresponding continuous and discrete time statement, but also contains the usual time scale case when the order of differentiation is one.     

A class of A(α)-stable numerical methods for stiff problems in ordinary differential equations

The A(α)-stable numerical methods (ANMs) for the number of steps k ≤ 7 for stiff initial value problems (IVPs) in ordinary differential equations (ODEs) are proposed. The discrete schemes proposed from their equivalent continuous schemes are obtained. The scaled time variable t in a continuous method, which determines the discrete coefficients of the discrete method, is chosen in such a way as to ensure that the discrete scheme attains a high order and A(α)-stability. We select the value of α for which the schemes proposed are absolutely stable. The new algorithms are found to have a comparable accuracy with that of the backward differentiation formula (BDF) discussed in [12] which implements the Ode15s in the Matlab suite.     

Explicit pseudo two-step exponential Runge–Kutta methods for the numerical integration of first-order differential equations

Numerical Algorithms - This paper is devoted to the explicit pseudo two-step exponential Runge–Kutta (EPTSERK) methods for the numerical integration of first-order ordinary differential...     

High-order compact solution of the one-dimensional heat and advection–diffusion equations

In this work, we propose a high-order accurate method for solving the one-dimensional heat and advection–diffusion equations. We apply a compact finite difference approximation of fourth-order for discretizing spatial derivatives of these equations and the cubic C¹$C^1$-spline collocation method for the resulting linear system of ordinary differential equations. The cubic C¹$C^1$-spline collocation method is an A-stable method for time integration of parabolic equations. The proposed method has fourth-order accuracy in both space and time variables, i.e. this method is of order O(h⁴,k⁴)$O(h^4\text{,}{k}^4)$. Additional to high-order of accuracy, the proposed method is unconditionally stable which will be proved in this paper. Numerical results show that the compact finite difference approximation of fourth-order and the cubic C¹$C^1$-spline collocation method give an efficient method for solving the one-dimensional heat and advection–diffusion equations.     

Green–Samoilenko operator in the theory of invariant sets of nonlinear differential equations

We establish conditions for the existence of an invariant set of the system of differential equations

Fourier–Taylor parameterization of unstable manifolds for parabolic partial differential equations: Formalism,implementation and rigorous validation

We study polynomial expansions of local unstable manifolds attached to equilibrium solutions of parabolic partial differential equations. Due to the smoothing properties of parabolic equations, these manifolds are finite dimensional. Our approach is based on an infinitesimal invariance equation and recovers the dynamics on the manifold in addition to its embedding. The invariance equation is solved, to any desired order in space and time, using a Newton scheme on the space of formal Fourier–Taylor series. Under mild non-resonance conditions we show that the formal series converge in some small enough neighborhood of the equilibrium. An a-posteriori computer assisted argument is given which, when successful, provides mathematically rigorous convergence proofs in explicit and much larger neighborhoods. We give example computations and applications.     

On the almost-periodic solution of Hasegawa–Wakatani equations

In order to describe the resistive drift wave turbulence appearing in nuclear fusion plasma, the Hasegawa–Wakatani equations were proposed in 1983. In this paper, we consider the zero-resistivity limit for the Hasegawa–Wakatani equations in a cylindrical domain when the initial data are Stepanov-almost-periodic in the axial direction. We prove two results: one is the existence and uniqueness of a strong global in time Stepanov-almost-periodic solution to the initial boundary value problem for the Hasegawa–Mima-like equation; another is the convergence of the solution of the Hasegawa–Wakatani equations to that of the Hasegawa–Mima-like equation established at the first stage as the resistivity tends to zero. In order to obtain a priori estimates of the Stepanov-almost-periodic solutions to our problems, we have to overcome some difficulties. In the proof, we prepare some lemmas for the Stepanov-almost-periodic functions and then obtain a priori estimates.     

On the numerical approximations of stiff convection–diffusion equations in a circle

Our aim in this article is to study the numerical solutions of singularly perturbed convection–diffusion problems in a circular domain and provide as well approximation schemes, error estimates and numerical simulations. To resolve the oscillations of classical numerical solutions due to the stiffness of our problem, we construct, via boundary layer analysis, the so-called boundary layer elements which absorb the boundary layer singularities. Using a $P_1$ classical finite element space enriched with the boundary layer elements, we obtain an accurate numerical scheme in a quasi-uniform mesh.     

Comment on: Multi soliton solution,rational solution of the Boussinesq–Burgers equations

We demonstrate that all “new” exact solutions of the Boussinesq-Burgers equations by Rady et al. [Rady ASA, Osman ES, Khalfallah M. Communications in Nonlinear Science and Numerical Simulation; 2009. doi:10.10016/j.cnsns.2009.05.053] are well known and were obtained many years ago.