Fractional Hyperbolic Equations Research Articles

$$H^1$$-analysis of H3N3-2$$_\sigma $$-based difference method for fractional hyperbolic equations

$$H^1$$-analysis of H3N3-2$$_\sigma $$-based difference method for fractional hyperbolic equations

Boundary state and output feedback stabilisation of a coupled time fractional hyperbolic equation

In this paper, the boundary stabilisation of a coupled time fractional hyperbolic equation system is investigated by both state and output feedback. A target system is first introduced, which, by the semigroup method and the Lyapunov one, is proved to admit a unique solution and is Mittag–Leffler stable. A boundary state feedback controller and state observer are designed via the backstepping technique, with explicit formulae for the gain functions. An observer-based output feedback stabilising controller is then proposed for the controlled system. Some rigorous theoretical proofs are given to demonstrate that the resulting closed-loop system under the state and output feedback controllers has a unique solution and is Mittag–Leffler stable. Some numerical simulations are performed to validate the effectiveness of the proposed control approach.

A fully explicit variational integrator for multidimensional systems of coupled nonlinear fractional hyperbolic equations

In this work, we investigate a multidimensional system consisting of a finite (though arbitrary) number of coupled hyperbolic partial differential equations with fractional diffusion, constant damping and inertial times, and nonlinear reaction terms. Under suitable analytical conditions, the model has conserved quantities which are preserved in the absence of damping. We establish rigorously the conservation of the proposed quantities and, assuming that solutions of the model exist, we prove their boundedness. Motivated by these facts, we propose a finite-difference methodology to approximate the solutions of the continuous system. As its continuous counterpart, the discrete model has associated discrete quantities that estimate the Hamiltonian functional. Moreover, these quantities are preserved in the absence of damping, and they are dissipated when damping is present. To prove this feature of our finite-difference scheme, a new approximation form of the nonlinear reaction terms is proposed. This approach allows for the scheme to mimic the properties of the continuous system. The numerical properties of consistency, stability, boundedness and convergence of the scheme are proved rigorously. Some illustrative simulations confirm that the scheme is capable of preserving or dissipating the quantities, in agreement with the analytical results.

A parallelized computational model for multidimensional systems of coupled nonlinear fractional hyperbolic equations

In this work, we consider a general multidimensional system of hyperbolic partial differential equations with fractional diffusion of the Riesz type, constant damping and coupled nonlinear reaction terms. The system generalizes many particular models from the physical sciences (including inhibitor-activator models in chemistry, diffusive nonlinear systems in population dynamics and relativistic wave equations), and considers the presence of an arbitrary number of both spatial dimensions and dependent variables. Motivated by the wide range of applications, we propose an explicit four-step finite-difference methodology to approximate the solutions of the continuous system. The properties of stability, boundedness and convergence of the scheme are proved rigorously using a discrete form of the fractional energy method. An efficient computational implementation of the scheme is also proposed in this work. It is important to recall that algorithms for space-fractional systems are computationally highly demanding. To alleviate this problem, a parallel implementation of our scheme is proposed using a vector reformulation of the numerical method. We provide some illustrative simulations on the formation of complex patterns in the two-dimensional scenario, and even in the computationally intense three-dimensional case. For the sake of convenience, an algorithmic presentation of our computational model is provided in this manuscript.

Global Nonexistence for the Cauchy Problem of Nonlinear Fractional Hyperbolic Equations and Systems

Global Nonexistence for the Cauchy Problem of Nonlinear Fractional Hyperbolic Equations and Systems

Finite Difference and Iteration Methods for Fractional Hyperbolic Partial Differential Equations with the Neumann Condition

The numerical and analytic solutions of the mixed problem for multidimensional fractional hyperbolic partial differential equations with the Neumann condition are presented. The stable difference scheme for the numerical solution of the mixed problem for the multidimensional fractional hyperbolic equation with the Neumann condition is presented. Stability estimates for the solution of this difference scheme and for the first‐ and second‐order difference derivatives are obtained. A procedure of modified Gauss elimination method is used for solving this difference scheme in the case of one‐dimensional fractional hyperbolic partial differential equations. He′s variational iteration method is applied. The comparison of these methods is presented.

A note on the fractional hyperbolic differential and difference equations

The initial boundary value problem for the fractional differential equation. d 2 u ( t ) dt 2 + D t 1 2 u ( t ) + Au ( t ) = f ( t ) , 0 < t < 1 , u ( 0 ) = 0 , u ′ ( 0 ) = ψ , in a Hilbert space H with the self-adjoint positive definite operator A is considered. The stability estimates for the solution of this problem and its first and second order derivatives are established. The first order of accuracy difference scheme for the approximate solution of this problem is presented. The stability estimates for the solution of this difference scheme and its first and second order difference derivatives are established. In practice, the stability estimates for the solution of difference schemes for one dimensional fractional hyperbolic equation with nonlocal boundary conditions in space variable and multidimensional fractional hyperbolic equation with Dirichlet condition in space variables are obtained.

On the Numerical Solution of Fractional Hyperbolic Partial Differential Equations

The stable difference scheme for the numerical solution of the mixed problem for the multidimensional fractional hyperbolic equation is presented. Stability estimates for the solution of this difference scheme and for the first and second orders difference derivatives are obtained. A procedure of modified Gauss elimination method is used for solving this difference scheme in the case of one‐dimensional fractional hyperbolic partial differential equations.