A survey of local existence theories for abstract nonlinear initial value problems

1989 ◽  
pp. 136-184 ◽  
Author(s):  
Eric Schechter
Keyword(s):  
Initial Value Problems ◽  
Local Existence ◽  
Initial Value

Nonlinear analytic semiflows

1990 ◽  
Vol 115 (1-2) ◽  
pp. 91-107 ◽  
Author(s):  
Sigurd B. Angenent
Keyword(s):  
Existence Theorem ◽  
Initial Value Problems ◽  
Local Existence ◽  
Regularity Theory ◽  
Initial Value ◽  
Nonlinear Parabolic ◽  
Direct Corollary

SynopsisIn this paper a local existence and regularity theory is given for nonlinear parabolic initial value problems (x′(t) = f(x(t))), and quasilinear initial value problems (x′(t)=A(x(t))x(t) + f(x(t))). This theory extends the theory of DaPrato and Grisvard of 1979, and shows how various properties, like analyticity of solutions, can be derived as a direct corollary of the existence theorem.

Well-posedness of general Caputo-type fractional differential equations

2018 ◽  
Vol 21 (3) ◽  
pp. 819-832 ◽  
Author(s):  
Chung-Sik Sin
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Initial Value Problems ◽  
Fractional Derivatives ◽  
Local Existence ◽  
Well Posedness ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Fractional Differential

Abstract In the present paper, initial value problems of fractional differential equations are considered. The fractional derivatives are defined as the general Caputo-type fractional derivatives proposed by Anatoly Kochubei. By using the Schauder fixed point theorem, the local existence, the global existence and the uniqueness of solutions are obtained under appropriate conditions. In addition, it is proved that the solution depends on the parameters of the equation in a continuous way.

Mathematical Models of Papermaking

2001 ◽  
Vol 6 (1) ◽  
pp. 9-19 ◽  
Author(s):  
A. Buikis ◽  
J. Cepitis ◽  
H. Kalis ◽  
A. Reinfelds ◽  
A. Ancitis ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Initial Value Problems ◽  
Moisture Transfer ◽  
Bound Water ◽  
Drying Process ◽  
Initial Value ◽  
First Order ◽  
Heat And Moisture ◽  
The Mathematical Model ◽  
The Web

The mathematical model of wood drying based on detailed transport phenomena considering both heat and moisture transfer have been offered in article. The adjustment of this model to the drying process of papermaking is carried out for the range of moisture content corresponding to the period of drying in which vapour movement and bound water diffusion in the web are possible. By averaging as the desired models are obtained sequence of the initial value problems for systems of two nonlinear first order ordinary differential equations. 

scholarly journals Intrinsic Discontinuities in Solutions of Evolution Equations Involving Fractional Caputo–Fabrizio and Atangana–Baleanu Operators

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2023
Author(s):  
Christopher Nicholas Angstmann ◽  
Byron Alexander Jacobs ◽  
Bruce Ian Henry ◽  
Zhuang Xu
Keyword(s):  
Initial Value Problems ◽  
Fractional Derivatives ◽  
Evolution Equations ◽  
Integral Transform ◽  
Initial Value ◽  
Intrinsic Nature ◽  
Recent Interest ◽  
Special Cases ◽  
Singular Kernels

There has been considerable recent interest in certain integral transform operators with non-singular kernels and their ability to be considered as fractional derivatives. Two such operators are the Caputo–Fabrizio operator and the Atangana–Baleanu operator. Here we present solutions to simple initial value problems involving these two operators and show that, apart from some special cases, the solutions have an intrinsic discontinuity at the origin. The intrinsic nature of the discontinuity in the solution raises concerns about using such operators in modelling. Solutions to initial value problems involving the traditional Caputo operator, which has a singularity inits kernel, do not have these intrinsic discontinuities.

Sign in / Sign up

Export Citation Format

Share Document