scholarly journals ON THE SOLVABILITY OF THE MONGE – AMPERE EQUATION ON , RELATED TO THE PROBLEM OF DIFFERENTIAL GEOMETRY

2021 ◽  
pp. 3-6
Author(s):  
Tatyana Aleksandrovna Yuryeva ◽  
Keyword(s):  
Differential Equation ◽  
Differential Geometry ◽  
Unique Solvability ◽  
Monge Ampere

The paper provides a proof of the closed nature of the solution of a family of Monge –Ampere differential one-parameter equations. The obtained result is used in the study of the unique solvability of the differential equation under study.

scholarly journals PARAMETER-UNIFORM ESTIMATES IN THE METRIC OF A FAMILY OF CLOSED CONVEX SURFACES WITH A GIVEN GAUSSIAN CURVATURE

2021 ◽  
pp. 24-26
Author(s):  
Tatyana Aleksandrovna Yuryeva ◽  
◽  
Natalya Nikolaevna Dvoeryadkina ◽  
Keyword(s):  
Differential Equation ◽  
Unique Solvability ◽  
Gaussian Curvature ◽  
Convex Surfaces ◽  
Uniform Estimates ◽  
Monge Ampere

The paper provides a proof of the boundedness of the solution of a family of Monge-Ampere differential one-parameter equations. The obtained result is used in the study of the unique solvability of the differential equation under study.

On a Singular Two-Point Boundary Value Problem for the Nonlinear mth-Order Differential Equation with Deviating Arguments

1997 ◽  
Vol 4 (6) ◽  
pp. 557-566
Author(s):  
B. Půža
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Vector Function ◽  
Unique Solvability ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Point Boundary ◽  
Deviating Arguments ◽  
Measurable Functions

Abstract Sufficient conditions of solvability and unique solvability of the boundary value problem u (m)(t) = f(t, u(τ 11(t)), . . . , u(τ 1k (t)), . . . , u (m–1)(τ m1(t)), . . . . . . , u (m–1)(τ mk (t))), u(t) = 0, for t ∉ [a, b], u (i–1)(a) = 0 (i = 1, . . . , m – 1), u (m–1)(b) = 0, are established, where τ ij : [a, b] → R (i = 1, . . . , m; j = 1, . . . , k) are measurable functions and the vector function f : ]a, b[×Rkmn → Rn is measurable in the first and continuous in the last kmn arguments; moreover, this function may have nonintegrable singularities with respect to the first argument.

scholarly journals Mixed type integro-differential equation with fractional order Caputo operators and spectral parameters

2021 ◽  
Vol 57 ◽  
pp. 190-205
Author(s):  
T.K. Yuldashev ◽  
E.T. Karimov
Keyword(s):  
Differential Equation ◽  
Fourier Series ◽  
Fractional Order ◽  
Unique Solvability ◽  
Mixed Type ◽  
Problem Solution ◽  
Algebraic Equations ◽  
Negative Part ◽  
Spectral Parameters ◽  
Integro Differential Equation

The issues of unique solvability of a boundary value problem for a mixed type integro-differential equation with two Caputo time-fractional operators and spectral parameters are considered. A mixed type integro-differential equation is a partial integro-differential equation of fractional order in both positive and negative parts of multidimensional rectangular domain under consideration. The fractional Caputo operator's order is less in the positive part of the domain, than the order of Caputo operator in the negative part of the domain. Using the method of Fourier series, two systems of countable systems of ordinary fractional integro-differential equations with degenerate kernels are obtained. Further, a method of degenerate kernels is used. To determine arbitrary integration constants, a system of algebraic equations is obtained. From this system, regular and irregular values of spectral parameters are calculated. The solution of the problem under consideration is obtained in the form of Fourier series. The unique solvability of the problem for regular values of spectral parameters is proved. To prove the convergence of Fourier series, the properties of the Mittag-Leffler function, Cauchy-Schwarz inequality and Bessel inequality are used. The continuous dependence of the problem solution on a small parameter for regular values of spectral parameters is also studied. The results are formulated as a theorem.

scholarly journals COMPOSITION METHODS FOR THE SIMULATION OF ARRAYS OF CHUA'S CIRCUITS

1999 ◽  
Vol 09 (04) ◽  
pp. 723-733
Author(s):  
YVES MOREAU ◽  
JOOS VANDEWALLE
Keyword(s):  
Differential Equation ◽  
Differential Geometry ◽  
Linear Part ◽  
Fixed Time ◽  
Time Step ◽  
Nonlinear Part ◽  
Composition Methods ◽  
Algebra Theory ◽  
Simple Integration ◽  
Integration Rules

Composition methods are methods for the integration of ordinary differential equations arising from differential geometry, or more precisely, Lie algebra theory. We apply them here to the simulation of arrays of Chua's circuits. In these methods, we split the vector field of the array of Chua's circuits into its linear part and its nonlinear part. We then solve the elementary differential equation for each part separately — which is easy since the equations for the nonlinear part are all decoupled — and recombine these contributions into a sequence of compositions. This splitting gives rise to simple integration rules for arrays of Chua's circuits, which we compare to more classical approaches: fixed time-step explicit Euler and adaptive fourth-order Runge–Kutta.

scholarly journals A Note on the Stability of the Integral-Differential Equation of the Parabolic Type in a Banach Space

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Maksat Ashyraliyev
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Difference Scheme ◽  
Unique Solvability ◽  
Parabolic Type ◽  
Stability Estimates ◽  
The Difference ◽  
The Stability ◽  
Integral Differential Equation

The integral-differential equation of the parabolic type in a Banach space is considered. The unique solvability of this equation is established. The stability estimates for the solution of this equation are obtained. The difference scheme approximately solving this equation is presented. The stability estimates for the solution of this difference scheme are obtained.

On the Prandtl Equation

1999 ◽  
Vol 6 (6) ◽  
pp. 525-536
Author(s):  
R. Duduchava ◽  
D. Kapanadze
Keyword(s):  
Differential Equation ◽  
Sobolev Space ◽  
Unique Solvability ◽  
Bessel Potential ◽  
Integro Differential Equation ◽  
Prandtl Equation ◽  
Bessel Potential Spaces

Abstract The unique solvability of the airfoil (Prandtl) integro-differential equation on the semi-axis is proved in the Sobolev space and Bessel potential spaces under certain restrictions on 𝑝 and 𝑠.

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