lipschitz conditions
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2022 ◽  
Vol 40 ◽  
pp. 1-18
Author(s):  
J. R. Sharma ◽  
Ioannis K. Argyros ◽  
Deepak Kumar

We introduce a new faster  King-Werner-type derivative-free method for solving nonlinear equations. The local as well as semi-local  convergence analysis is presented under weak center Lipschitz and Lipschitz conditions. The convergence order as well as the convergence radii are also provided. The radii are compared to the corresponding ones from similar methods. Numerical examples further validate the theoretical results.


2021 ◽  
Vol 5 (4) ◽  
pp. 239
Author(s):  
Mahmoud Abouagwa ◽  
Rashad A. R. Bantan ◽  
Waleed Almutiry ◽  
Anas D. Khalaf ◽  
Mohammed Elgarhy

In this manuscript, a new class of impulsive fractional Caputo neutral stochastic differential equations with variable delay (IFNSDEs, in short) perturbed by fractional Brownain motion (fBm) and Poisson jumps was studied. We utilized the Carathéodory approximation approach and stochastic calculus to present the existence and uniqueness theorem of the stochastic system under Carathéodory-type conditions with Lipschitz and non-Lipschitz conditions as special cases. Some existing results are generalized and enhanced. Finally, an application is offered to illustrate the obtained theoretical results.


2021 ◽  
Author(s):  
◽  
Graeme C Wake

<p>A study is made of the equations of heat conduction with slow combustion. A mathematical model is established from an interpretation of the physical model, with a few simplifying assumptions. This gives rise to a coupled pair of partial differential equations which are the direct concern of this thesis, the dependent variables being the temperature and reactant concentration as functions of position and time. The model is shown to possess a unique solution for which some properties, such as Lipschitz conditions etc., are established. An investigation into the use of a comparison theorem is given, in which it is shown that no direct comparison theorem is possible for this and related systems. However, it is also shown that it is possible to obtain upper and lower estimates by appealing to the physical model. A discussion of the boundary layer is given and this is followed by a detailed discussion of stability. The latter has been one of the main concerns of earlier authors on this system. Their use of a space-averaging process to establish a criterion for stability is also discussed. Probably one of the most interesting features of this system is the subclass of problems for which the reactant is exhausted in a finite time. These hare been named the "cut-off" problems and they can be likened to the free boundary problems in fluid dynamics. A discussion of the cut-off problem is given with particular examples chosen to illustrate the main features. This thesis, contains no material which has been accepted for the award of any other degree or diploma in any University and to the best of my knowledge and belief, the thesis contains no material previously published or written by another person, except where due reference is made in the text of the thesis.</p>


2021 ◽  
Author(s):  
◽  
Graeme C Wake

<p>A study is made of the equations of heat conduction with slow combustion. A mathematical model is established from an interpretation of the physical model, with a few simplifying assumptions. This gives rise to a coupled pair of partial differential equations which are the direct concern of this thesis, the dependent variables being the temperature and reactant concentration as functions of position and time. The model is shown to possess a unique solution for which some properties, such as Lipschitz conditions etc., are established. An investigation into the use of a comparison theorem is given, in which it is shown that no direct comparison theorem is possible for this and related systems. However, it is also shown that it is possible to obtain upper and lower estimates by appealing to the physical model. A discussion of the boundary layer is given and this is followed by a detailed discussion of stability. The latter has been one of the main concerns of earlier authors on this system. Their use of a space-averaging process to establish a criterion for stability is also discussed. Probably one of the most interesting features of this system is the subclass of problems for which the reactant is exhausted in a finite time. These hare been named the "cut-off" problems and they can be likened to the free boundary problems in fluid dynamics. A discussion of the cut-off problem is given with particular examples chosen to illustrate the main features. This thesis, contains no material which has been accepted for the award of any other degree or diploma in any University and to the best of my knowledge and belief, the thesis contains no material previously published or written by another person, except where due reference is made in the text of the thesis.</p>


2021 ◽  
pp. 2250003
Author(s):  
Chengfeng Sun ◽  
Qianqian Huang ◽  
Hui Liu

The stochastic two-dimensional Cahn–Hilliard–Navier–Stokes equations under non-Lipschitz conditions are considered. This model consists of the Navier–Stokes equations controlling the velocity and the Cahn–Hilliard model controlling the phase parameters. By iterative techniques, a priori estimates and weak convergence method, the existence and uniqueness of an energy weak solution to the equations under non-Lipschitz conditions have been obtained.


2021 ◽  
Vol 359 (6) ◽  
pp. 675-685
Author(s):  
Salah El Ouadih ◽  
Radouan Daher
Keyword(s):  

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1917
Author(s):  
Junmei Wang ◽  
James Hoult ◽  
Yubin Yan

Spatial discretization of the stochastic semi-linear subdiffusion equations driven by fractionally integrated multiplicative space-time white noise is considered. The nonlinear terms f and σ satisfy the global Lipschitz conditions and the linear growth conditions. The space derivative and the fractionally integrated multiplicative space-time white noise are discretized by using the finite difference methods. Based on the approximations of the Green functions expressed by the Mittag–Leffler functions, the optimal spatial convergence rates of the proposed numerical method are proved uniformly in space under some suitable smoothness assumptions of the initial value.


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