scholarly journals Classical solution to the mixed problems for the Klein–Gordon–Fock-type equation with curve derivatives in boundary conditions

2018 ◽  
Vol 62 (5) ◽  
pp. 531-539
Author(s):  
V. I. Korzyuk ◽  
I. I. Stolyarchuk
Keyword(s):  
Boundary Conditions ◽  
Type Equation ◽  
Approximate Solutions ◽  
Analytical Form ◽  
Problem Analysis ◽  
Matching Conditions ◽  
Differentiable Functions ◽  
Uniqueness Of The Solution ◽  
Klein Gordon ◽  
Necessary And Sufficient

The mixed problem for one-dimensional Klein–Gordon–Fock-type equation with curve derivatives in boundary conditions is considered in half-strip. The solution of this problem is reduced to solving the second type Volterra integral equations. Theorems of existence and uniqueness of the solution in the class of the twice continuously differentiable functions were proven for these equations when initial functions are smooth enough. It is proven that fulfillment of the matching conditions on the given functions is necessary and sufficient for the existence of the unique smooth solution when initial functions are smooth enough. The method of characteristics is used for the problem analysis.This method is reduced to the splitting the original area of the definition to the subdomains. The solution of the subproblem can be constructed in each subdomain with the help of the initial and boundary conditions. Then obtained solutions are glued in common points, and received glued conditions are the matching conditions. This approach can be used in constructing as analytical solution, in case when solution of the integral equation can be found in explicit way, so for approximate solution. Moreover, approximate solutions can be constructed in numerical and analytical form. When numeric solution is constructed, then matching conditions are essential and they need to be considered while developing numerical methods.

scholarly journals Classical solution of the mixed problem for the Klein – Gordon – Fock type equation in the half-strip with curve derivatives at boundary conditions

2019 ◽  
Vol 54 (4) ◽  
pp. 391-403 ◽  
Author(s):  
V. I. Korzyuk ◽  
I. I. Stolyarchuk
Keyword(s):  
Boundary Conditions ◽  
Mixed Problem ◽  
Type Equation ◽  
Approximate Solutions ◽  
Analytical Form ◽  
Problem Analysis ◽  
Matching Conditions ◽  
Half Strip ◽  
Uniqueness Of The Solution ◽  
Klein Gordon

The mixed problem for the one-dimensional Klein – Gordon – Fock type equation with curve derivatives at boundary conditions is considered in the half-strip. The solution of this problem is reduced to solving the second-type Volterra integral equations. Theorems of existence and uniqueness of the solution in the class of twice continuously differentiable functions were proven for these equations when initial functions are smooth enough. It is proven that the fulfillment of the matching conditions on the given functions is necessary and sufficient for the existence of the unique smooth solution when initial functions are smooth enough. The method of characteristics is used for the problem analysis. This method is reduced to splitting the original area of definition to the subdomains. The solution of the subproblem can be constructed in each subdomain with the help of the initial and boundary conditions. Then, the obtained solutions are glued in common points, and the obtained glued conditions are the matching conditions. This approach can be used in constructing as an analytical solution when a solution of the integral equation can be found in an explicit way, so an approximate solution. Moreover, approximate solutions can be constructed in numerical or analytical form. When a numerical solution is built, the matching conditions are essential and they need to be considered while developing numerical methods.

scholarly journals Classical solution of the mixed problem for the Klein – Gordon – Fock type equation with characteristic oblique derivatives at boundary conditions

2019 ◽  
Vol 55 (1) ◽  
pp. 7-21
Author(s):  
V. I. Korzyuk ◽  
I. I. Stolyarchuk
Keyword(s):  
Boundary Conditions ◽  
Mixed Problem ◽  
Type Equation ◽  
Approximate Solutions ◽  
Analytical Form ◽  
Problem Analysis ◽  
Matching Conditions ◽  
Uniqueness Of The Solution ◽  
Klein Gordon ◽  
The One

The mixed problem for the one-dimensional Klein – Gordon – Fock type equation with oblique derivatives at boundary conditions in the half-strip is considered. The solution of this problem is reduced to solving the second-type Volterra integral equations. Theorems of existence and uniqueness of the solution in the class of twice continuously differentiable func tions were proven for these equations when initial functions are smooth enough. It is proven that fulfilling the matching conditions on the given functions is necessary and sufficient for existence of the unique smooth solution, when initial functions are smooth enough. The method of characteristics is used for the problem analysis. This method is reduced to splitting the ori ginal definition area into subdomains. The solution of the subproblem can be constructed in each subdomain with the help of the initial and boundary conditions. The obtained solutions are then glued in common points, and the obtained glued сonditions are the matching conditions. Intensification of smoothness requirements for source functions is proven when the di rections of the oblique derivatives at boundary conditions are matched with the directions of the characteristics. This approach can be used in constructing both the analytical solution, when the solution of the integral equation can be found explicitly, and the approximate solution. Moreover, approximate solutions can be constructed in numerical and analytical form. When a numerical solution is constructed, the matching conditions are significant and need to be considered while developing numerical methods.

scholarly journals Solving the mixed problem for the Klein–Gordon–Fock type equation with integral conditions in the case of the inhomogeneous matching conditions

2019 ◽  
Vol 63 (2) ◽  
pp. 142-149
Author(s):  
V. I. Korzyuk ◽  
I. I. Stolyarchuk
Keyword(s):  
Mixed Problem ◽  
Type Equation ◽  
Approximate Solutions ◽  
Analytical Form ◽  
Integral Conditions ◽  
Matching Conditions ◽  
Differentiable Functions ◽  
Klein Gordon ◽  
Time Argument ◽  
The Given

The classical solution of the mixed problem with integral conditions for the Klein–Gordon–Fock type equation in the half strip is considered when inhomogeneous matching conditions are fulfilled. An equivalent conjugation problem is formulated where conjugation conditions are set on characteristics. Constructed inhomogeneous conditions uniquely define gaps of the solution or its derivatives on characteristics and given gaps can be either remained or smoothed while the time argument increases depending on the kernel of the integral operator in unlocal conditions. The solution of this problem is reduced to solving the second-type Volterra integral equations and their systems. The unique solution of these equations in the class of the twice continuously differentiable functions exists when the initial functions are smooth enough. While considering the given problem the method of characteristics is used to construct both an analytical solution, when the solution of the integral equation can be found explicitly, and an approximate solution. Moreover, approximate solutions can be constructed in numerical and analytical form. When the numerical solution is constructed, matching conditions are significant and need to be considered while developing numerical methods.

scholarly journals Classical solution of the mixed problem in the quarter of the plane for the wave equation

2019 ◽  
Vol 62 (6) ◽  
pp. 647-651
Author(s):  
V. I. Korzyuk ◽  
I. S. Kozlovskaya ◽  
V. Yu. Sokolovich
Keyword(s):  
Wave Equation ◽  
Boundary Conditions ◽  
Classical Solution ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Analytical Form ◽  
Partial Solution ◽  
Differentiable Functions ◽  
Straight Line ◽  
Straight Lines

This article presents the classical solution with mixed boundary conditions in the quarter of the plane for the wave equation in the analytical form. The boundary of the region consists of two perpendicular half-straight lines. On one of them, Cauchy’s boundary conditions are assigned. The second half-straight line is divided into two parts. Dirichlet’s condition is assigned on the straight line and Neumann’s conditions – on the half-straight line. The classical solution of the considered problem is defined in the class of double continuous differentiable functions in the quarter of the plane. To build this solution, the partial solution of the initial wave equation is written. For the assigned functions of the problem, the matching conditions are written, which are necessary and enough so that the solution of the problem would be classical and unique.

scholarly journals DEZIN NONLOCAL PROBLEM FOR A MIXED-TYPE EQUATION WITH POWER DEGENERATION

2017 ◽  
Vol 22 (3-4) ◽  
pp. 24-31
Author(s):  
V. A. Gushchina
Keyword(s):  
Nonlocal Problem ◽  
Sufficient Conditions ◽  
Type Equation ◽  
Normal Derivative ◽  
Transition Line ◽  
Lower Base ◽  
Rectangular Area ◽  
Uniqueness Of The Solution ◽  
Non Local ◽  
Necessary And Sufficient

In this article, for the equation of mixed elliptic - hyperbolic type with a power degeneracy on the transition line in a rectangular area are studied the problem Dezin with periodicity conditions and non-local condition, binding values of the normal derivative on the lower base of the rectangle with the value of the target solution on the line type of study. Necessary and sufficient conditions for the uniqueness of the solution were settled, and the uniqueness of the solution was proved problem on the based on completeness of the system of peculiar functions of one-problem or the peculiar.

scholarly journals Lidstone–Euler Second-Type Boundary Value Problems: Theoretical and Computational Tools

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Francesco Aldo Costabile ◽  
Maria Italia Gualtieri ◽  
Anna Napoli
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Interpolation Problem ◽  
Boundary Value ◽  
Computational Tools ◽  
Numerical Examples ◽  
Odd Order ◽  
Uniqueness Of The Solution ◽  
Type Boundary

AbstractGeneral nonlinear high odd-order differential equations with Lidstone–Euler boundary conditions of second type are treated both theoretically and computationally. First, the associated interpolation problem is considered. Then, a theorem of existence and uniqueness of the solution to the Lidstone–Euler second-type boundary value problem is given. Finally, for a numerical solution, two different approaches are illustrated and some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms.

Mechanics of a charged particle on the Kaluza–Klein background

1995 ◽  
Vol 73 (9-10) ◽  
pp. 602-607 ◽  
Author(s):  
S. R. Vatsya
Keyword(s):  
Electromagnetic Field ◽  
Charged Particle ◽  
Integral Method ◽  
Gordon Equation ◽  
Type Equation ◽  
Fifth Dimension ◽  
Klein Gordon ◽  
Path Integral Method ◽  
Klein Gordon Equation ◽  
Kaluza Klein

The path-integral method is used to derive a generalized Schrödinger-type equation from the Kaluza–Klein Lagrangian for a charged particle in an electromagnetic field. The compactness of the fifth dimension and the properties of the physical paths are used to decompose this equation into its infinite components, one of them being similar to the Klein–Gordon equation.

NEW MODELS IN MICROPOLAR FLUID AND THEIR APPLICATION TO LUBRICATION

2005 ◽  
Vol 15 (03) ◽  
pp. 343-374 ◽  
Author(s):  
GUY BAYADA ◽  
NADIA BENHABOUCHA ◽  
MICHÈLE CHAMBAT
Keyword(s):  
Boundary Condition ◽  
Friction Coefficient ◽  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Micropolar Fluid ◽  
Reynolds Equation ◽  
Slip Boundary Condition ◽  
Slip Boundary ◽  
New Models ◽  
Uniqueness Of The Solution

A thin micropolar fluid with new boundary conditions at the fluid-solid interface, linking the velocity and the microrotation by introducing a so-called "boundary viscosity" is presented. The existence and uniqueness of the solution is proved and, by way of asymptotic analysis, a generalized micropolar Reynolds equation is derived. Numerical results show the influence of the new boundary conditions for the load and the friction coefficient. Comparisons are made with other works retaining a no slip boundary condition.

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