Differential Equations and Applications Research Group
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Ordinary, partial and delay differential equations (ODEs, PDEs and DDEs) are fundamental tools in mathematical modelling. However, these equations in general cannot be solved directly, only via numerical methods. Our group focus includes both of the above topics.
▶ Develop numerical methods for non-linear systems of ODEs to preserve certain qualitative properties, such as positivity, monotonicity, TVD or SSP properties, or range boundedness, in the context of one-step methods (e.g., Runge–Kutta methods) and multistep methods
Related publications:
☞ Dense output for strong stability preserving Runge–Kutta methods
☞ Positivity for convective semi-discretizations
☞ General relaxation methods for initial-value problems with application to multistep schemes
▶ Develop optimal, computationally efficient numerical methods for ODEs via maximizing the allowable step size while simultaneously guaranteeing the preservation of certain qualitative properties
Related publication:
☞ Strong stability preserving explicit linear multistep methods with variable step size
▶ Stability analysis of numerical methods for ODEs: optimal regions of absolute stability and optimal step-size coefficients
Related publications:
☞ On the absolute stability regions corresponding to partial sums of the exponential function
☞ Rational functions with maximal radius of absolute monotonicity
☞ Exact optimal values of step-size coefficients for boundedness of linear multistep methods
▶ Study the error propagation of Runge–Kutta methods with many stages
Related publication:
☞ Internal error propagation in explicit Runge–Kutta methods
▶ Convergence acceleration of multistep methods via extrapolation
Related publication:
☞ Linear multistep methods and global Richardson extrapolation
▶ Perturbation analysis of PDEs with periodic coefficients
Related publication:
☞ A multiscale model for weakly nonlinear shallow water waves over periodic bathymetry
▶ Pattern formation in PDE models
(ide nagyon jó lenne kép)Related publication:
☞ Turing bifurcation in a system with cross diffusion
▶ Mathematical biology and epidemiology: investigating equilibria, stability, certain qualitative properties (positivity or boundedness), oscillations, and bifurcations in the context of PDE and DDE models
Related publications:
☞ Bifurcations in a human migration model of Scheurle-Seydel type-I: Turing bifurcation
☞ Stability of delayed ratio-dependent predator-prey system
☞ Bifurcations in a human migration model of Scheurle-Seydel type-II: Rotating waves
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▶ Study the behavior of numerical methods for ODEs near bifurcation points: topological equivalence and preservation of the phase portrait
Related publications:
☞ On gradient enriched elasticity theories
☞ On the convergence of a class of finite difference schemes for the numerical solution of the time-fractional diffusion equation
☞ On the convergence of a class of finite difference schemes for the numerical solution of the time-fractional diffusion equation
▶ Fractional calculus: fractional-order derivatives, fractional-order operators, numerical solutions of fractional-order PDEs, and applications, e.g., in relativistic quantum mechanics
Related publications:
☞ Convergence of the matrix transformation method for the finite difference approximation of fractional order diffusion problems
☞ New approximate dynamic programming algorithms for undiscounted Markov decision processes
☞ Finite element approximation of fractional order elliptic boundary value problems
☞ Numerical analysis of the meshless element-free Galerkin method for fractional diffusion problems
☞ Models of space-fractional diffusion: A critical review
☞ Direct computation of the quantum partition function by path-integral nested sampling
☞ Models of space-fractional diffusion: A critical review
☞ Efficient numerical solution of space-fractional diffusion problems
▶ Various applications of the gradient (i.e., steepest descent) method: (a) gradient methods in Sobolev spaces via operator preconditioning to solve non-linear elliptic partial differential equations iteratively; (b) investigation of the gradient method in the training process of implicit neural networks; (c) generalizations, such as fractional gradient descent methods
Related publications:
☞ Numerical solution of a non-classical parabolic problem: an integro-differential approach
☞ Operator preconditioning with efficient applications for nonlinear elliptic problems
Under construction...
- Dr. Kovács Sándor (research group leader, bibliography: MTMT)
- Dr. György Szilvia
- Dr. Lóczi Lajos
- Dr. Szekeres Béla
- Tóth Gergő PhD hallgató
- Ottó Panna MSc hallgató
Under construction...
Under construction...
Dr. Lóczi Lajos: lloczi@inf.elte.hu