Numerical verification of solutions for nonlinear hyperbolic equations

Numerical Verifications of Solutions for Nonlinear Hyperbolic Equations

In this paper, we consider a numerical technique to enclose the solutions with guaranteed error bounds for nonlinear hyperbolic initial boundary value problems as well as to verify the existence of solutions. Using a finite element approximation and explicit error estimates for a certain simple linear hyperbolic problem, we construct, by computer, a set of functions which satisfies the conditio...

Numerical Existence and Uniqueness Proof for Solutions of Nonlinear Hyperbolic Equations Numerical Existence and Uniqueness Proof for Solutions of Nonlinear Hyperbolic Equations

We consider a numerical method to verify the existence and uniqueness of the solutions of nonlinear hyperbolic problems with guaranteed error bounds. Using a C 1 nite element solution and an inequality constituting a bound on the norm of the inverse operator of the linearized operator, we numerically construct a set of functions which satisfy the hypothesis of Banach's xed point theorem for a c...

Solutions of Nonlinear Hyperbolic Equations at Resonance

Numerical Treatment of Coupled Nonlinear Hyperbolic Klein-gordon Equations

E.H. DOHA1,a, A.H. BHRAWY2,3,b, D. BALEANU4,5,6,c, M.A. ABDELKAWY3,d 1Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt, E-mail: [email protected] 2Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia, E-mail: [email protected] 3Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef 62511, E...

On Finite Closures of Homogenized Solutions of Nonlinear Hyperbolic Equations

We study nonlinear equations subject to oscillatory initial data. The oscillatory solution of such problems tends to a homogenized weak limit that is characterized by the corresponding homogenized equations. Those equations usually involve an additional independent variable, so that the weak limit is an average of infinitely many functions. In certain cases, however, there is an alternative des...