scholarly journals Distributional derivatives of functions of two variables of finite variation and their application to an impulsive hyperbolic equation

1998 ◽  
Vol 48 (1) ◽  
pp. 145-171
Author(s):  
Dariusz Idczak
Keyword(s):  
Hyperbolic Equation ◽  
Finite Variation ◽  
Functions Of Two Variables ◽  
Derivatives Of

scholarly journals ON SMOOTHNESS OF SOLUTION OF ONE NONLOCAL PROBLEM FOR HYPERBOLIC EQUATION

2021 ◽  
Vol 26 (2) ◽  
pp. 15-22
Author(s):  
V. A. Kirichek
Keyword(s):  
Hyperbolic Equation ◽  
Nonlocal Problem ◽  
Generalized Solution ◽  
Integral Boundary ◽  
Apriori Estimates ◽  
The Space L2 ◽  
Elastic Fixation ◽  
The Right ◽  
Derivatives Of ◽  
Smoothness Of Solution

In this paper we consider a nonlocal problem with integral boundary condition for hyperbolic equation. The conditions of the problem contain derivatives of the first order with respect to both x and t,, which can be interpreted as an elastic fixation of the right end rod in the presence of a certain damper, and since the conditions also contain integral of the desired solution, this condition is nonlocal. It is known that problems with nonlocal integral conditions are non-self-adjoint and, therefore, the study of solvability encounters difficulties that are not characteristic of self-adjoint problems. Additional difficulties arise also due to the fact that one of the conditions is dynamic. The attention of the article is focused on studying thesmoothness of the solution of the nonlocal problem. The concept of a generalized solution is introduced, and the existence of second-order derivatives and their belonging to the space L2 are proved. The proof is basedon apriori estimates obtained in this work.

scholarly journals New Hermite-Hadamard type inequalities for m and (α, m)-convex functions on the coordinates via generalized fractional integrals

2021 ◽  
Vol 40 (6) ◽  
pp. 1449-1472
Author(s):  
Seth Kermausuor
Keyword(s):  
Convex Functions ◽  
Type Inequality ◽  
Second Order ◽  
Fractional Integrals ◽  
Absolute Value ◽  
Partial Derivatives ◽  
Independent Variables ◽  
Functions Of Two Variables ◽  
Derivatives Of ◽  
Two Independent Variables

In this paper, we obtained a new Hermite-Hadamard type inequality for functions of two independent variables that are m-convex on the coordinates via some generalized Katugampola type fractional integrals. We also established a new identity involving the second order mixed partial derivatives of functions of two independent variables via the generalized Katugampola fractional integrals. Using the identity, we established some new Hermite-Hadamard type inequalities for functions whose second order mixed partial derivatives in absolute value at some powers are (α, m)-convex on the coordinates. Our results are extensions of some earlier results in the literature for functions of two variables.

scholarly journals Reconstructed Interpolating Differential Operator Method with Arbitrary Order of Accuracy for the Hyperbolic Equation

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 295
Author(s):  
Shijian Lin ◽  
Qi Luo ◽  
Hongze Leng ◽  
Junqiang Song
Keyword(s):  
Differential Operator ◽  
Hyperbolic Equation ◽  
Arbitrary Order ◽  
Operator Method ◽  
High Order Accuracy ◽  
Order Of Accuracy ◽  
Moment Methods ◽  
Linear Transform ◽  
Spatial Derivatives ◽  
Derivatives Of

We propose a family of multi-moment methods with arbitrary orders of accuracy for the hyperbolic equation via the reconstructed interpolating differential operator (RDO) approach. Reconstruction up to arbitrary order can be achieved on a single cell from properly allocated model variables including spatial derivatives of varying orders. Then we calculate the temporal derivatives of coefficients of the reconstructed polynomial and transform them into the temporal derivatives of the model variables. Unlike the conventional multi-moment methods which evolve different types of moments by deriving different equations, RDO can update all derivatives uniformly via a simple linear transform more efficiently. Based on difference in introducing interaction from adjacent cells, the central RDO and the upwind RDO are proposed. Both schemes enjoy high-order accuracy which is verified by Fourier analysis and numerical experiments.

The Design of Linkages to Generate Functions of Two Variables

1959 ◽  
Vol 81 (1) ◽  
pp. 23-28 ◽  
Author(s):  
C. W. Allen
Keyword(s):  
Single Point ◽  
Graphical Methods ◽  
Functions Of Two Variables ◽  
Basic Approaches ◽  
Derivatives Of

Graphical methods are presented for designing linkages that have two inputs and one output. Two basic approaches are developed. The desired function is matched at a limited number of positions with the maximum number being seven. Designs are also developed which match the derivatives of the desired function at a single point. The designs were developed with ease of solution and broadness of application as primary considerations.

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