scholarly journals MULTIPLICITIES OF EIGENVALUES OF THE DIFFUSION OPERATOR WITH RANDOM JUMPS FROM THE BOUNDARY

2018 ◽  
Vol 99 (1) ◽  
pp. 101-113 ◽  
Author(s):  
JUN YAN ◽  
GUOLIANG SHI
Keyword(s):  
Differential Operator ◽  
Characteristic Function ◽  
Diffusion Process ◽  
Algebraic Multiplicity ◽  
Diffusion Operator ◽  
Random Jumps ◽  
New Criterion ◽  
Multiplicity Of An Eigenvalue

This paper deals with a non-self-adjoint differential operator which is associated with a diffusion process with random jumps from the boundary. Our main result is that the algebraic multiplicity of an eigenvalue is equal to its order as a zero of the characteristic function $\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D706})$. This is a new criterion for determining the multiplicities of eigenvalues for concrete operators.

scholarly journals Some Janowski Type Harmonic q-Starlike Functions Associated with Symmetrical Points

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 629 ◽  
Author(s):  
Muhammad Arif ◽  
Omar Barkub ◽  
Hari Srivastava ◽  
Saleem Abdullah ◽  
Sher Khan
Keyword(s):  
Differential Operator ◽  
Harmonic Functions ◽  
Sufficient Conditions ◽  
Starlike Functions ◽  
Partial Sums ◽  
Circular Region ◽  
Necessary And Sufficient ◽  
New Criterion ◽  
Symmetrical Points ◽  
Convexity Conditions

The motive behind this article is to apply the notions of q-derivative by introducing some new families of harmonic functions associated with the symmetric circular region. We develop a new criterion for sense preserving and hence the univalency in terms of q-differential operator. The necessary and sufficient conditions are established for univalency for this newly defined class. We also discuss some other interesting properties such as distortion limits, convolution preserving, and convexity conditions. Further, by using sufficient inequality, we establish sharp bounds of the real parts of the ratios of harmonic functions to its sequences of partial sums. Some known consequences of the main results are also obtained by varying the parameters.

Approximation by Dilated Averages and K-Functionals

2010 ◽  
Vol 62 (4) ◽  
pp. 737-757 ◽  
Author(s):  
Z. Ditzian ◽  
A. Prymak
Keyword(s):  
Banach Space ◽  
Differential Operator ◽  
Characteristic Function ◽  
Lebesgue Measure ◽  
Interior Point ◽  
Convex Set ◽  
Finite Measure ◽  
Elliptic Differential Operator ◽  
Averaging Operator ◽  
Bounded Convex

AbstractFor a positive finite measure dμ(u ) on ℝd normalized to satisfy , the dilated average of f (x ) is given byIt will be shown that under some mild assumptions on dμ(u ) one has the equivalencewhere means , B is a Banach space of functions for which translations are continuous isometries and P(D) is an elliptic differential operator induced by μ. Many applications are given, notable among which is the averaging operator with where S is a bounded convex set in ℝd with an interior point, m(S) is the Lebesgue measure of S, and 𝒳S(u ) is the characteristic function of S. The rate of approximation by averages on the boundary of a convex set under more restrictive conditions is also shown to be equivalent to an appropriate K-functional.

scholarly journals Matrix Differentiation of the Characteristic Function

1931 ◽  
Vol 2 (4) ◽  
pp. 256-264 ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Differential Operator ◽  
Characteristic Function ◽  
Matrix Differential Operator ◽  
Independent Variables ◽  
Scalar Matrix ◽  
Scalar Quantity ◽  
The Matrix ◽  
Matrix Differential

The following work is a sequel to three previous communications, and more particularly to the first. The present object is to shew the effect of repeated operation with the matrix differential operator , when it acts upon a scalar matrix formed from an n rowed determinant |xij|, or sums of principal minors, the n2 elements xij being treated as independent variables. Thus when z is a scalar quantity ω z means the matrix [∂z/∂xij], whose ijth element is the derivative.

Burgers fluid flow in perspective of Buongiorno’s model with improved heat and mass flux theory for stretching cylinder

2020 ◽  
Vol 92 (3) ◽  
pp. 31101
Author(s):  
Zahoor Iqbal ◽  
Masood Khan ◽  
Awais Ahmed
Keyword(s):  
Diffusion Process ◽  
Mathematical Formulation ◽  
Thermal Relaxation ◽  
Mass Diffusion ◽  
Stretching Cylinder ◽  
Discussion Section ◽  
Buongiorno’S Model ◽  
Homotopy Analysis ◽  
Burgers Fluid ◽  
Diffusion Phenomena

In this study, an effort is made to model the thermal conduction and mass diffusion phenomena in perspective of Buongiorno’s model and Cattaneo-Christov theory for 2D flow of magnetized Burgers nanofluid due to stretching cylinder. Moreover, the impacts of Joule heating and heat source are also included to investigate the heat flow mechanism. Additionally, mass diffusion process in flow of nanofluid is examined by employing the influence of chemical reaction. Mathematical modelling of momentum, heat and mass diffusion equations is carried out in mathematical formulation section of the manuscript. Homotopy analysis method (HAM) in Wolfram Mathematica is utilized to analyze the effects of physical dimensionless constants on flow, temperature and solutal distributions of Burgers nanofluid. Graphical results are depicted and physically justified in results and discussion section. At the end of the manuscript the section of closing remarks is also included to highlight the main findings of this study. It is revealed that an escalation in thermal relaxation time constant leads to ascend the temperature curves of nanofluid. Additionally, depreciation is assessed in mass diffusion process due to escalating amount of thermophoretic force constant.

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