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Solving fractional-order ordinary differential equations via Adomian decomposition method
Šustková, Apolena ; Řehák, Pavel (referee) ; Nechvátal, Luděk (advisor)
This master's thesis deals with solving fractional-order ordinary differential equations by the Adomian decomposition method. A part of the work is therefore devoted to the theory of equations containing differential operators of non-integer order, especially the Caputo operator. The next part is devoted to the Adomian decomposition method itself, its properties and implementation in the case of Chen system. The work also deals with bifurcation analysis of this system, both for integer and non-integer case. One of the objectives is to clarify the discrepancy in the literature concerning the fractional-order Chen system, where experiments based on the use of the Adomian decomposition method give different results for certain input parameters compared with numerical methods. The clarification of this discrepancy is based on recent theoretical knowledge in the field of fractional-order differential equations and their systems. The conclusions are supported by numerical experiments, own code implementing the Adomian decomposition method on the Chen system was used.
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Singular Initial Value Problem for Ordinary Differential and Integrodifferential Equations
Archalousová, Olga ; Beránek, Jaroslav (referee) ; Růžičková,, Miroslava (referee) ; Šmarda, Zdeněk (advisor)
The thesis deals with qualitative properties of solutions of singular initial value problems for ordinary differential and integrodifferential equations which occur in the theory of linear and nonlinear electrical circuits and the theory of therminionic currents. The research is concentrated especially on questions of existence and uniqueness of solutions, asymptotic estimates of solutions and modications of Adomian decomposition method for singular initial problems. Solution algoritms are derived for scalar differential equations of Lane-Emden type using Taylor series and modication of the Adomian decomposition method. For certain classes of nonlinear of integrodifferential equations asymptotic expansions of solutions are constructed in a neighbourhood of a singular point. By means of the combination of Wazewski's topological method and Schauder xed-point theorem there are proved asymptotic estimates of solutions in a region which is homeomorphic to a cone having vertex coinciding with the initial point. Using Banach xed-point theorem the uniqueness of a solution of the singular initial value problem is proved for systems of integrodifferential equations of Volterra and Fredholm type including implicit systems. Moreover, conditions of continuous dependence of a solution on a parameter are determined. Obtained results are presented in illustrative examples.
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Solving fractional-order ordinary differential equations via Adomian decomposition method
Šustková, Apolena ; Řehák, Pavel (referee) ; Nechvátal, Luděk (advisor)
This master's thesis deals with solving fractional-order ordinary differential equations by the Adomian decomposition method. A part of the work is therefore devoted to the theory of equations containing differential operators of non-integer order, especially the Caputo operator. The next part is devoted to the Adomian decomposition method itself, its properties and implementation in the case of Chen system. The work also deals with bifurcation analysis of this system, both for integer and non-integer case. One of the objectives is to clarify the discrepancy in the literature concerning the fractional-order Chen system, where experiments based on the use of the Adomian decomposition method give different results for certain input parameters compared with numerical methods. The clarification of this discrepancy is based on recent theoretical knowledge in the field of fractional-order differential equations and their systems. The conclusions are supported by numerical experiments, own code implementing the Adomian decomposition method on the Chen system was used.
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Singular Initial Value Problem for Ordinary Differential and Integrodifferential Equations
Archalousová, Olga ; Beránek, Jaroslav (referee) ; Růžičková,, Miroslava (referee) ; Šmarda, Zdeněk (advisor)
The thesis deals with qualitative properties of solutions of singular initial value problems for ordinary differential and integrodifferential equations which occur in the theory of linear and nonlinear electrical circuits and the theory of therminionic currents. The research is concentrated especially on questions of existence and uniqueness of solutions, asymptotic estimates of solutions and modications of Adomian decomposition method for singular initial problems. Solution algoritms are derived for scalar differential equations of Lane-Emden type using Taylor series and modication of the Adomian decomposition method. For certain classes of nonlinear of integrodifferential equations asymptotic expansions of solutions are constructed in a neighbourhood of a singular point. By means of the combination of Wazewski's topological method and Schauder xed-point theorem there are proved asymptotic estimates of solutions in a region which is homeomorphic to a cone having vertex coinciding with the initial point. Using Banach xed-point theorem the uniqueness of a solution of the singular initial value problem is proved for systems of integrodifferential equations of Volterra and Fredholm type including implicit systems. Moreover, conditions of continuous dependence of a solution on a parameter are determined. Obtained results are presented in illustrative examples.
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