Addition theorems for $${\mathcal {C}}^k$$ real functions and applications in ordinary differential equations

2021 ◽  
Author(s):  
Francisco Crespo ◽  
Salomón Rebollo-Perdomo ◽  
Jorge L. Zapata
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Addition Theorems ◽  
Real Functions

scholarly journals Solution of Linear Fuzzy Fractional Differential Equations Using Fuzzy Natural Transform

2021 ◽  
pp. 41-65
Author(s):  
Hameeda Oda Al-Humedi ◽  
Shaimaa Abdul-Hussein Kadhim
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fractional Differential Equations ◽  
Wide Range ◽  
Real Functions ◽  
Fractional Models ◽  
Fractional Differential ◽  
Fuzzy Fractional Differential Equations

The purpose of this paper is to apply the fuzzy natural transform (FNT) for solving linear fuzzy fractional ordinary differential equations (FFODEs) involving fuzzy Caputo’s H-difference with Mittag-Leffler laws. It is followed by proposing new results on the property of FNT for fuzzy Caputo’s H-difference. An algorithm was then applied to find the solutions of linear FFODEs as fuzzy real functions. More specifically, we first obtained four forms of solutions when the FFODEs is of order α∈(0,1], then eight systems of solutions when the FFODEs is of order α∈(1,2] and finally, all of these solutions are plotted using MATLAB. In fact, the proposed approach is an effective and practical to solve a wide range of fractional models.

scholarly journals SMT Encoding of Hybrid Systems in dReal

10.29007/s3b9 ◽  
2018 ◽  
Author(s):  
Kyungmin Bae ◽  
Soonho Kong ◽  
Sicun Gao
Keyword(s):  
Model Checking ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Hybrid Systems ◽  
Bounded Model Checking ◽  
Satisfiability Modulo Theories ◽  
Real Numbers ◽  
The Real ◽  
Real Functions ◽  
Nonlinear Hybrid

Analysis problems of hybrid systems, involving nonlinear real functions and ordinary differential equations, can be reduced to SMT (satisfiability modulo theories) problems over the real numbers. The dReal solver can automatically check the satisfiability of such SMT formulas up to a given precision δ > 0. This paper explains how bounded model checking problems of hybrid systems are encoded in dReal. In particular, a novel SMT syntax of dReal enables to effectively represent networks of hybrid systems in a modular way. We illustrate SMT encoding in dReal with simple nonlinear hybrid systems.

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