Fractional derivatives, differential and integral equations with singularities and ill-posed problems

Fractional (non-integer order) differential equations often describe the behavior of various materials and process better than the classical differential equations with integer order derivatives. However, it is usually not possible to find an exact solution to these equations, and so we need to find their solutions approximately, which requires the development of special methods. The question, which functions are fractionally differentiable, also continues to be relevant.
Fuzzy integral and differential equations contain functions with fuzzy values, which describe the situation where the information is incomplete or approximate, e.g. due to measurement errors or noise. The existence and uniqueness and smoothness of solutions to such problems together with numerical methods are of interest.
The solution of the ill-posed problem does not depend continuously on the data, and to reduce the effect of data errors (e.g. measurement errors), special regularization methods are used to solve such problems. The main problem of using regularization methods is the choice of a suitable regularization parameter depending on the information on the noise level of the data. In practice, the noise level information is often unknown, and parameter selection rules that do not use this information are of particular interest.

Research Group staff

• Professor Arvet Pedas (Head of Research Group)
• Professor Emeritus and Academician of Estonian Academy of Science Gennadi Vainikko (Scientific Supervisor of the Research Group)
• Associate Professor Uno Hämarik
• Associate Professor Urve Kangro
• Associate Professor Toomas Raus
• Lecturer Kaido Lätt
• Research Fellow Mikk Vikerpuur
• Junior Research Fellow Margus Lillemäe
• Junior Research Fellow Hanna Britt Soots
• PhD student Britt Kalam

Topics:

• conditions for fractional differentiability of the function;
• effective methods for solution of differential equations with fractional derivatives, fuzzy differential and integral equations, weakly singular integral equations and cordial Volterra integral equations;
• methods for solving ill-posed problems and rules for choice of the regularization parameter.

Research Group projects

Current projects

• PRG 864 Theoretical and numerical analysis of differential equations with fractional order derivatives, integral equations and ill-posed problems

Main past projects

• ETF9104 Integral and differential equations with singularities (01.2012-06.2016)
• ETF9120 Ill-posed problems (01.2012-06.2016)
• PUTJD840 Singular fractional integro-differential equations (12.2019-02.2021)

Relevant publications

• Diogo, Teresa; Pedas, Arvet; Vainikko, Gennadi (2020). Integral equations of the third kind in Lp spaces.  Journal of integral equations and applications, 32 (4), 417-427.
• Raus, Toomas; Hämarik, Uno (2020). Q-curve and area rules for choosing heuristic parameter in Tikhonov regularization. Mathematics, 8 (7, 1166), 1−21.
• Kangro, Urve (2017). Cordial Volterra integral equations and singular fractional integro-differential equations in spaces of analytic functions. Mathematical Modelling and Analysis, 22 (4), 548−567.
• Pedas, Arvet; Tamme, Enn; Vikerpuur, Mikk (2017). Smoothing transformation and spline collocation for nonlinear fractional initial and boundary value problems. Journal of Computational and Applied Mathematics, 317, 1−16.
• Vainikko, Gennadi (2016). Which functions are fractionally differentiable? Zeitschrift für Analysis und Ihre Anwendungen, 35 (4), 465−487.
• Hämarik, Uno; Kaltenbacher, Barbara; Kangro, Urve; Resmerita, Elena (2016). Regularization by discretization in Banach spaces. Inverse Problems, 32 (3, 035004), 1−28.
• Lätt, Kaido; Pedas, Arvet; Vainikko, Gennadi (2015). A Smooth Solution of a Singular Fractional Differential Equation. Zeitschrift für Analysis und Ihre Anwendungen, 34 (2), 127−146.
• Hämarik, Uno; Kangro, Urve; Palm, Reimo; Raus, Toomas; Tautenhahn, Ulrich (2014). Monotonicity of error of regularized solution and its use for parameter choice. Inverse Problems in Science and Engineering, 22 (1), 10−30.
• Pedas, Arvet; Tamme, Enn (2012). Piecewise polynomial collocation for linear boundary value problems of fractional differential equations. Journal of Computational and Applied Mathematics, 236 (13), 3349−3359.

More information about group's activites in studying ill-posed problems can be found on the website.