Problem for an ordinary differential equation with a general boundary condition

2021 ◽  
Vol 21 (2) ◽  
pp. 9-14
Author(s):  
L.Kh. Gadzova ◽  
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Ordinary Differential Equation ◽  
Fractional Order ◽  
Unique Solvability ◽  
Explicit Representation ◽  
General Boundary ◽  
General Boundary Condition ◽  
General Representation

For an ordinary differential equation of fractional order with a general boundary condition, a general representation of the solution of the equation is found, a condition of unique solvability is found, and an explicit representation of the solution is constructed.

BOUNDARY VALUE PROBLEM WITH SHIFT FOR A FRACTIONAL ORDER DELAY DIFFERENTIAL EQUATION

2019 ◽  
pp. 16-25
Author(s):  
М.Г. Мажгихова
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Fractional Order ◽  
Delay Differential Equation ◽  
Solvability Condition ◽  
Boundary Value ◽  
Explicit Representation ◽  
Existence And Uniqueness Theorem ◽  
The Green Function ◽  
Delay Differential

В работе доказана теорема существования и единственности решения краевой задачи со смещением для дифференциального уравнения дробного порядка с запаздывающим аргументом. Решение задачи выписано в терминах функции Грина. Получено условие однозначной разрешимости и показано, что оно может нарушаться только конечное число раз. In this paper we prove existence and uniqueness theorem to a boundary value problem with shift for a fractional order ordinary delay differential equation. The solution of the problem is written out in terms of the Green function. We find an explicit representation for solvability condition and show that it may only be violated a finite number of times

scholarly journals A Truncation Method for Solving the Time-Fractional Benjamin-Ono Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-7 ◽  
Author(s):  
Mohamed R. Ali
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Power Series ◽  
Fractional Order ◽  
Nonlinear Ordinary Differential Equation ◽  
Lie Symmetry ◽  
Series Method ◽  
Lie Symmetry Analysis ◽  
Power Series Method ◽  
Bo Equation

We deem the time-fractional Benjamin-Ono (BO) equation out of the Riemann–Liouville (RL) derivative by applying the Lie symmetry analysis (LSA). By first using prolongation theorem to investigate its similarity vectors and then using these generators to transform the time-fractional BO equation to a nonlinear ordinary differential equation (NLODE) of fractional order, we complete the solutions by utilizing the power series method (PSM).

The Mixing of Two Parallel Streams of Dissimilar Fluids—Part I: Analytical Development

1971 ◽  
Vol 38 (2) ◽  
pp. 301-309 ◽  
Author(s):  
R. L. Baker ◽  
L. N. Tao ◽  
H. Weinstein
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Ordinary Differential Equation ◽  
Schmidt Number ◽  
Similarity Transformation ◽  
Velocity Ratio ◽  
Density Ratio ◽  
Large Density ◽  
Parallel Streams ◽  
Analytical Development

The mixing region between dissimilar fluids is investigated in the region where the similarity transformation is valid. The treatment is complete in that laminar and turbulent cases both with and without large density differences are considered. The Schmidt number is an arbitrary input to the problem and may be varied. The ordinary differential equation resulting from the similarity transformation is integrated numerically and some solutions are presented. The three boundary conditions are proper; the so-called arbitrary third boundary condition is treated as originally suggested by von Karman and extended to the case of large density difference. Illustrations of the effects of varying velocity ratio, density ratio, and Schmidt number are presented.

General boundary-condition recalculation algorithm for temperature fields of different geometry

1983 ◽  
Vol 45 (5) ◽  
pp. 1294-1297 ◽  
Author(s):  
G. T. Aldoshin ◽  
A. S. Golosov ◽  
V. I. Zhuk ◽  
D. N. Chubarov
Keyword(s):  
Boundary Condition ◽  
Temperature Fields ◽  
General Boundary ◽  
General Boundary Condition

Bending of a circular plate with an eccentric circular patch symmetrically loaded with respect to its centre

1956 ◽  
Vol 52 (3) ◽  
pp. 584-598 ◽  
Author(s):  
W. A. Bassali ◽  
R. H. Dawoud
Keyword(s):  
Boundary Condition ◽  
Circular Plate ◽  
Complex Variable ◽  
Poisson’S Ratio ◽  
Complex Variable Method ◽  
Variable Method ◽  
Poisson's Ratio ◽  
General Boundary ◽  
General Boundary Condition ◽  
Circular Patch

ABSTRACTThe complex variable method is applied to obtain solutions for the deflexion of a supported circular plate with uniform line loading along an eccentric circle under a general boundary condition including the clamped boundary , a boundary with zero peripheral couple , a boundary with equal boundary cross-couples , a hinged boundary and a boundary for which , η being Poisson's ratio. These solutions are used to obtain the deflexion at any point of a circular plate having an eccentric circular patch symmetrically loaded with respect to its centre. Expressions for the slope and cross-couples over the boundary and the deflexions at the centres of the plate and the loaded patch are obtained.

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.

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