RECENT ADVANCES ON NUMERICAL METHODS FOR EVOLUTIONARY PROBLEMS

This contribution regards the numerical solution of evolutionary problems related to natural phenomena end physical processes modelled by functional equationsofvarioustype, withspecificcharacteristicssuch as: oscillations, songularities,presenceofstochasticterms,byfocusingexpeciallyonstabilitypropertiesof the methods. We will describe the construction of ’adapted’ numerical methods, that is, strongly oriented to the problem and the stability will be also interpreted as preservation of the intrinsic qualitative characteristics of the problem itself. We will consider systems of ordinary differential equations with oscillating solution, deriving for example from the spatial semi-discretization of partial differential equations of advection-diffusion in hydrodynamic problems (2). In this case, the use of non-polynomial bases reveals a powerful tool for adapting numerical methods to the known behavior of the solution. We will also discuss stochastic Volterra integral equations, deriving, for example, from the modelling of economic problems (3). The focus will be on the construction of numerical methods that can inherit the stability properties of the methods for stochastic differential equations. Finally, we will consider multi-value methods for fractional differential equations, which model, for example, the behaviour of viscoelastic materials (1), and will analyse the relative stability properties. Acknowledgements The presented results have been obtained in collaboration with Beatrice Paternoster and Angelamaria Cardone (University of Salerno), Raffaele D’Ambrosio (University of L’Aquila), Zdzislaw Jackiewicz (Arizona State University), Liviu Ixaru(InstituteofPhysicsandNuclearEngineering,Bucharest),LeilaMoradiand Fakhrodin Mohamadi (University of Hormozgan, Iran). References [1] M. Di Paola, A. Pirrotta and A. Valenza. (2011). Visco-elastic behavior through fractional calculus: an easier method for best fitting experimental results, Mech. Mater. 43, 799-806. [2] N. Su, F. Liu, V. Anh. (2003). Tides as phase-modulated waves inducing periodic groundwaterowincoastalaquifersoverlayingaslopingimperviousbase,Environ. Model. Softw. 18, 937–942. [3] Zhao, Q., Wang, R., Wei, J. (2016). Exponential utility maximization for an insurer with time-inconsistent preferences, Insur. Math. and Econ. 70, 89–104.