scholarly journals A Parabolic Transform and Averaging Methods for General Partial Differential Equations

2019 ◽  
Vol 17 ◽  
pp. 352-361
Author(s):  
Mahmoud Mohammed Mostafa El-Borai ◽  
Hamed Kamal Awad Awad ◽  
Randa Hamdy. M. Ali Ali
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Differential Operators ◽  
Averaging Method ◽  
Partial Differential ◽  
Averaging Methods ◽  
Partial Differential Operators ◽  
Existence And Stability ◽  
Special Case ◽  
Stability Results

Averaging method of the fractional general partial differential equations and a special case of these equations are studied, without any restrictions on the characteristic forms of the partial differential operators. We use the parabolic transform, existence and stability results can be obtained.

scholarly journals Infinite-dimensional linear algebra and solvability of partial differential equations

2021 ◽  
Vol 13 ◽  
Author(s):  
Todor D. Todorov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Linear Algebra ◽  
Differential Operators ◽  
Test Functions ◽  
Partial Differential ◽  
Infinite Dimensional ◽  
Partial Differential Operators ◽  
Linear Partial Differential Operators ◽  
Regular Operators

  We discuss linear algebra of infinite-dimensional vector spaces in terms of algebraic (Hamel) bases. As an application we prove the surjectivity of a large class of linear partial differential operators with smooth ($\mathcal C^\infty$-coefficients) coefficients, called in the article \emph{regular}, acting on the algebraic dual $\mathcal D^*(\Omega)$ of the space of test-functions $\mathcal D(\Omega)$. The surjectivity of the partial differential operators guarantees solvability of the corresponding partial differential equations within $\mathcal D^*(\Omega)$. We discuss our result in contrast to and comparison with similar results about the restrictions of the regular operators on the space of Schwartz distribution $\mathcal D^\prime(\Omega)$, where these operators are often non-surjective. 

scholarly journals The Extension of the Physical and Stochastic Problems to Space-Time-Fractional Differential Equations

2021 ◽  
Vol 2090 (1) ◽  
pp. 012031
Author(s):  
E.A. Abdel-Rehim
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Differential Operators ◽  
Space Time ◽  
Partial Differential ◽  
Stochastic Problems ◽  
Fractional Differential ◽  
Fractional Differential Operators ◽  
Time Fractional Differential Equations

Abstract The fractional calculus gains wide applications nowadays in all fields. The implementation of the fractional differential operators on the partial differential equations make it more reality. The space-time-fractional differential equations mathematically model physical, biological, medical, etc., and their solutions explain the real life problems more than the classical partial differential equations. Some new published papers on this field made many treatments and approximations to the fractional differential operators making them loose their physical and mathematical meanings. In this paper, I answer the question: why do we need the fractional operators?. I give brief notes on some important fractional differential operators and their Grünwald-Letnikov schemes. I implement the Caputo time fractional operator and the Riesz-Feller operator on some physical and stochastic problems. I give some numerical results to some physical models to show the efficiency of the Grünwald-Letnikov scheme and its shifted formulae. MSC 2010: Primary 26A33, Secondary 45K05, 60J60, 44A10, 42A38, 60G50, 65N06, 47G30,80-99

scholarly journals Wiener-Hopf equation and Fredholm property of the Goursat problem in Gevrey space

1994 ◽  
Vol 135 ◽  
pp. 165-196 ◽  
Author(s):  
Masatake Miyake ◽  
Masafumi Yoshino
Keyword(s):  
Differential Equations ◽  
Differential Operators ◽  
Newton Polygon ◽  
Index Theorem ◽  
Fredholm Property ◽  
Goursat Problem ◽  
Point Of View ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Gevrey Spaces

In the study of ordinary differential equations, Malgrange ([Ma]) and Ramis ([R1], [R2]) established index theorem in (formal) Gevrey spaces, and the notion of irregularity was nicely defined for the study of irregular points. In their studies, a Newton polygon has a great advantage to describe and understand the results in visual form. From this point of view, Miyake ([M2], [M3], [MH]) studied linear partial differential operators on (formal) Gevrey spaces and proved analogous results, and showed the validity of Newton polygon in the study of partial differential equations (see also [Yn]).

V.— On the Spectrum of Second Order Partial Differential Operators.

1970 ◽  
Vol 69 (1) ◽  
pp. 77-84
Author(s):  
K. J. Brown ◽  
I. M. Michael
Keyword(s):  
Differential Equations ◽  
Spectral Properties ◽  
Differential Operators ◽  
Second Order ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Ordinary Differential Operators

SynopsisIn a recent paper, Jyoti Chaudhuri and W. N. Everitt linked the spectral properties of certain second order ordinary differential operators with the analytic properties of the solutions of the corresponding differential equations. This paper considers similar properties of the spectrum of the corresponding partial differential operators.

ON TAYLOR'S SCRAPING PROBLEM AND FLOW OF A SISKO FLUID

2009 ◽  
Vol 14 (4) ◽  
pp. 515-529 ◽  
Author(s):  
Abdul M. Siddiqui ◽  
Ali R. Ansari ◽  
Ahmed Ahmad ◽  
N. Ahmad
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Viscous Fluid ◽  
Normal Stress ◽  
Partial Differential ◽  
Sisko Fluid ◽  
Linear Partial Differential Equations ◽  
Non Linear ◽  
Special Case

The aim of the present investigation is to study the properties of a Sisko fluid flowing between two intersecting planes. The problem is similar to Taylor's scraping problem for a viscous fluid. We determine the solution of the complicated set of non‐linear partial differential equations describing the flow analytically. The analysis is carried out in detail reflecting the effects of varying the angle of the scraper on the flow. In addition, the tangential and normal stress are also computed. We also show the well known Taylor scraper problem as a special case.

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