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 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 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 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.

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.

scholarly journals A Modified Techniques of Fractional-Order Cauchy-Reaction Diffusion Equation via Shehu Transform

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Mounirah Areshi ◽  
A. M. Zidan ◽  
Rasool Shah ◽  
Kamsing Nonlaopon
Keyword(s):  
Fractional Order ◽  
Reaction Diffusion ◽  
Approximate Solutions ◽  
Transformation Method ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Closed Form Solutions ◽  
Homotopy Perturbation ◽  
In Series ◽  
The Given

In this article, the iterative transformation method and homotopy perturbation transformation method are applied to calculate the solution of time-fractional Cauchy-reaction diffusion equations. In this technique, Shehu transformation is combined of the iteration and the homotopy perturbation techniques. Four examples are examined to show validation and the efficacy of the present methods. The approximate solutions achieved by the suggested methods indicate that the approach is easy to apply to the given problems. Moreover, the solution in series form has the desire rate of convergence and provides closed-form solutions. It is noted that the procedure can be modified in other directions of fractional order problems. These solutions show that the current technique is very straightforward and helpful to perform in applied sciences.

scholarly journals Сlassical solution of the mixed problem for the one-dimensional wave equation with the nonsmooth second initial condition

2020 ◽  
Vol 64 (6) ◽  
pp. 657-662
Author(s):  
V. I. Korzyuk ◽  
J. V. Rudzko
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Mixed Problem ◽  
Method Of Characteristics ◽  
Smooth Boundary ◽  
Analytical Form ◽  
Initial Condition ◽  
One Dimensional ◽  
The One ◽  
Dimensional Wave

In this article, we study the classical solution of the mixed problem in a quarter of a plane and a half-plane for a one-dimensional wave equation. On the bottom of the boundary, Cauchy conditions are specified, and the second of them has a discontinuity of the first kind at one point. Smooth boundary condition is set at the side boundary. The solution is built using the method of characteristics in an explicit analytical form. Uniqueness is proved and conditions are established under which a piecewise-smooth solution exists. The problem with linking conditions is considered.

Asymptotic stepwise solutions of the Korteweg--de Vries equation with a singular perturbation and their accuracy

2021 ◽  
Vol 8 (3) ◽  
pp. 410-421
Author(s):  
S. I. Lyashko ◽  
◽  
V. H. Samoilenko ◽  
Yu. I. Samoilenko ◽  
I. V. Gapyak ◽  
...  
Keyword(s):  
Singular Perturbation ◽  
Small Parameter ◽  
Asymptotic Approximation ◽  
Vries Equation ◽  
Approximate Solutions ◽  
Variable Coefficients ◽  
Main Term ◽  
Asymptotic Approximations ◽  
De Vries ◽  
The Given

The paper deals with the Korteweg-de Vries equation with variable coefficients and a small parameter at the highest derivative. The asymptotic step-like solution to the equation is obtained by the non-linear WKB technique. An algorithm of constructing the higher terms of the asymptotic step-like solutions is presented. The theorem on the accuracy of the higher asymptotic approximations is proven. The proposed technique is demonstrated by example of the equation with given variable coefficients. The main term and the first asymptotic approximation of the given example are found, their analysis is done and statement of the approximate solutions accuracy is presented.

scholarly journals Hawking radiation from cubic and quartic black holes via tunneling of GUP corrected scalar and fermion particles

2019 ◽  
Vol 34 (09) ◽  
pp. 1950057 ◽  
Author(s):  
Wajiha Javed ◽  
Rimsha Babar ◽  
Ali Övgün
Keyword(s):  
Black Holes ◽  
Hawking Radiation ◽  
Generalized Uncertainty Principle ◽  
Hawking Temperature ◽  
Jacobi Method ◽  
Dirac Equations ◽  
Tensor Theory ◽  
Klein Gordon ◽  
The Given ◽  
Do So

We analyze the effect of the generalized uncertainty principle (GUP) on the Hawking radiation from the hairy black hole in U(1) gauge-invariant scalar–vector–tensor theory by utilizing the semiclassical Hamilton–Jacobi method. To do so, we evaluate the tunneling probabilities and Hawking temperature for scalar and fermion particles for the given spacetime of the black holes with cubic and quartic interactions. For this purpose, we utilize the modified Klein–Gordon equation for the Boson particles and then Dirac equations for the fermion particles, respectively. Next, we examine that the Hawking temperature of the black holes do not depend on the properties of tunneling particles. Moreover, we present the corrected Hawking temperature of scalar and fermion particles which look similar in both interactions, but there are different mass and momentum relationships for scalar and fermion particles in cubic and quartic interactions.

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