Cauchy Type Research Articles

Cauchy Type Nonlinear Inverse Problem in a Two-Layer Cylindrical Area

Various types of protective coatings, such as ceramics, are used to increase the efficiency of metal parts that are exposed to high heat loads. Maintaining mechanical properties and acceptable thermal stresses requires temperature control for components such as turbine blades, gun barrels or combustion chambers. This paper presents a new method for controlling thermal parameters by solving a non-stationary Cauchy problem for the heat conduction equation. The solution is presented in a cylindrical two-layer area considering the variation of the thermophysical parameters of the metal and ceramic. The non-linear equation was solved by successive approximation method using differential quotients for the space and time variable. In order to regularize the ill-conditioned inverse problem, a quasi-regularization method was applied by performing an energy balance for the ceramic. Numerical calculations were carried out for different thicknesses of the ceramic layer. The temperature, heat flux density and heat transfer coefficient for the protective layer were calculated. For the same values of the heat transfer coefficient for 0.1 and 0.3 mm thick protective layers, the difference in permissible gas temperature differs by 10.5 °C.

On Some Cauchy Type Mean-Value Theorems with Applications

Some Cauchy type mean-value theorems for Chebychev’s inequality, Steffensen’s inequality, midpoint rule and Simpson’s rule are presented. Furthermore, we give some applications for the obtained results using the exponential and logarithmic functions, their Taylor polynomials and for some trigonometric functions. Further, we obtain some exponential, logarithmic and trigonometric equations and give two inequalities for midpoint and Simpson’s rules.

Viscosity solutions to a Cauchy type problem for timelike Lorentzian eikonal equation

In this paper, we propose a Cauchy type problem to the timelike Lorentzian eikonal equation on a globally hyperbolic space-time. For this equation, as the value of the solution on a Cauchy surface is known, we prove the existence of viscosity solutions on the past set (future set) of the Cauchy surface. Furthermore, when the time orientation of viscosity solution is consistent, the uniqueness and stability of viscosity solutions are also obtained.

Analytical solution of two non‐identical edge cracks in an infinite strip under anti‐plane shear wave

AbstractThis article presents an extensive analytical solution addressing the interaction between two non‐identical edge cracks in an infinite orthotropic strip under anti‐plane shear waves. Most studies assume identical cracks or single edge crack in a strip, but this research breaks new ground by considering cracks of different sizes. By incorporating mixed‐type boundary conditions, the study derives dual integral equations. These equations are then transformed into a singular integral equation of Cauchy type with the aid of a trial solution and contour integration technique. The singular integral equation is further converted into a system of integral equations, which are solved numerically utilizing Jacobi polynomials. The obtained solutions are utilized to derive expressions for the stress intensity factor (SIF) and crack opening displacement (COD) at the crack tip using Krenk's interpolation formulae. The derived results are presented graphically and compared against existing solutions for single edge crack and symmetric edge cracks in static scenario.

A contact model for the functionally graded coated elastic structures: Extension of the Hertz theory to the contact of beam structures

The Hertzian displacement assumption is widely used in analyzing the contact problems of non-uniform elastic bodies. It is essential to account for the support conditions of the elastic body as they significantly influence the contact stiffness and distribution of contact pressure. To address this, the deformation of beam structures is integrated into the Hertzian displacement assumption, which leads to the development of an extended contact mechanics model suitable for the elastic bodies of a beam structure which can be non-uniform and with functionally graded coatings. The problem is solved by using a numerical method based on the Gauss–Chebyshev quadrature for the singular integral equation of Cauchy type. In the contact problems of the doubly simply supported (SS-SS) and cantilever beams, the contact pressure and contact stiffness in conjunction with the interactions between the indentation and contact bodies are discussed. An in-depth study on the coupling effects between the structural deformation and functionally graded coatings is presented.

Fast numerical integration of highly oscillatory Bessel transforms with a Cauchy type singular point and exotic oscillators

Fast numerical integration of highly oscillatory Bessel transforms with a Cauchy type singular point and exotic oscillators

Developing Higher-Order Unconditionally Positive Finite Difference Methods for the Advection Diffusion Reaction Equations

This study introduces the higher-order unconditionally positive finite difference (HUPFD) methods to solve the linear, nonlinear, and system of advection–diffusion–reaction (ADR) equations. The stability and consistency of the developed methods are analyzed, which are necessary and sufficient for the numerical approach to converge to the exact solution. The problem under consideration is of the Cauchy type, and hence, Von Neumann stability analysis is used to analyze the stability of the proposed schemes. The HUPFD’s efficacy and efficiency are investigated by calculating the error, convergence rate, and computing time. For validation purposes, the higher-order unconditionally positive finite difference solutions are compared to analytical calculations. The numerical results demonstrate that the proposed methods produce accurate solutions to solve the advection diffusion reaction equations. The results also show that increasing the order of the unconditionally positive finite difference leads an implicit scheme that is conditionally stable and has a higher order of accuracy with respect to time and space.

Solution of a Cauchy type problem for an integral equation of Volterra type with singular kernels, when the roots of the characteristic equations are complex conjugate

Solution of a Cauchy type problem for an integral equation of Volterra type with singular kernels, when the roots of the characteristic equations are complex conjugate

On a composite obtained by a mixture of a dipolar solid with a Moore–Gibson–Thompson media

Our study is dedicated to a mixture composed of a dipolar elastic medium and a viscous Moore–Gibson–Thompson (MGT) material. The mixed problem with initial and boundary data, considered in this context, is approached from the perspective of the existence of a solution to this problem as well as the uniqueness of the solution. Considering that the mixed problem is very complex, both from the point of view of the basic equations and that of the initial conditions and the boundary data, the classical methods become difficult. That is why we preferred to transform it into a problem of Cauchy type on a conveniently constructed Hilbert space. In this way, we immediately proved both the existence and uniqueness of the solution, with techniques from the theory of semigroups of linear operators.

Solvability of nonlinear boundary value problems for non-sloping Timoshenko-type isotropic shells of zero principal curvature

We study the solvability of boundary value problem for a system of second order partial differential equations under boundary given conditions describing the equilibrium of elastic non-sloping isotropic inhomogeneous shells with free boundary in the framework of the Timoshenko shear model. The base of the study method are the integral representations for generalized motions involving arbitrary functions, which also involve arbitrary holomorphic functions. The arbitrary functions ate determined so that the generalized motions satisfy a linear system of equations and linear boundary conditions extracted from the original boundary value problem. The holomorphic functions are sought as Cauchy type integrals with real densities. The integral representations allow us to reduce the initial boundary value problem to a nonlinear operator equations for generalized motions in the Sobolev space. While studying the solvability of this operator equation, the most essential point is to invert it with respect to the linear part. As a result, the work is reduced to an equation, the solvability of which is established on the base of the contracting mapping principle.

Solution of an Initial Value Problem of Cauchy Type for One Equation

Solution of an Initial Value Problem of Cauchy Type for One Equation

Analysis of a nonlinear problem involving discrete and proportional delay with application to Houseflies model

<abstract><p>This manuscript established a comprehensive analysis of a general class of fractional order delay differential equations with Caputo-Fabrizio fractional derivative (CFFD). Functional analysis was used to examine the existence and uniqueness of the suggested class and to generate sufficient requirements for Ulam-Hyers (UH) type stability. Further, a numerical method based on Lagrange interpolation is used to compute approximate solution. Then, some applications in physical dynamics including a houseflies model and a Cauchy type problem were discussed to illustrate the established analysis with graphical illustrations.</p></abstract>

Efficient computational methods of highly oscillatory Bessel transforms with a singular point of Cauchy type and a nonlinear special oscillator

In this article, we propose two efficient combined methods to compute the integral ∫01f(x)x−cJm(ωxr)dx with a special and nonlinear oscillator xr, where 0<c<1,m≥0,2r∈N+. One is established by combining the complex integration method with the two-point Taylor interpolation method. The other is to combine two complex integration methods. The explicit formula of the integral ∫0+∞1x−cJm(ωxr)dx is expressed in terms of the Meijer G function. Importantly, the rigorous error analysis is performed by a large amount of theoretical analysis. The asymptotic error estimates in inverse powers of ω are achieved. Finally, numerical experiments show the validity of theoretical analysis and the efficiency of the proposed methods. In addition, we make numerical comparisons between the two proposed combined methods and another combined method.

Cauchy type integrals and a boundary value problem in a complex Clifford analysis

Cauchy type integrals and a boundary value problem in a complex Clifford analysis

Inverse coefficient problems for a time-fractional wave equation with the generalized Riemann-Liouville time derivative

This paper considers the inverse problem of determining the time-dependent coeffiicient in the fractional wave equation with Hilfer derivative. In this case, the direct problem is initial-boundary value problem for this equation with Cauchy type initial and nonlocal boundary conditions. As overdetermination condition nonlocal integral condition with respect to direct problem solution is given. By the Fourier method, this problem is reduced to equivalent integral equations. Then, using the Mittag-Leffler function and the generalized singular Gronwall inequality, we get apriori estimate for solution via unknown coefficient which we will need to study of the inverse problem. The inverse problem is reduced to the equivalent integral of equation of Volterra type. The principle of contracted mapping is used to solve this equation. Local existence and global uniqueness results are proved.

Giaccardi inequality for generalized convex functions and related results

In this article, the famous Giaccardi inequality is generalized for modified (h,m)-convex functions under the condition that h is a multiplicative positive function. The results are also discussed for different values of h and m. Also, Lagrange and Cauchy type mean value theorems for the functional due to Giaccardi’s inequality are obtained.

An application of Lobatto–Chebyshev method to a nonhomogeneous plane problem with two cracks

In this manuscript, Lobatto–Chebyshev method, which is an effective collocation method, is applied to a system of singular integral equations, which leads from a nonhomogeneous plane problem. It is assumed that there are two cracks in a nonhomogeneous medium. The problem is formulated in the view of the basics of the elasticity theory and boundary conditions of the problem. By using the method of singular integral equation, the problem is converted to a system of first kind Cauchy type singular integral equations. It is aimed to determine the stress intensity factors (SIFs) of the crack problem. It is seen that Lobatto–Chebyshev quadrature has many advantages in determining SIFs. To verify the validity of the method, the obtained results corresponding to the one crack case are compared with the results in literature.

Constructing Solutions of Cauchy Type Integral Equations by Using Four Kinds of Basis

We have four kinds of solutions for Cauchy type integral equations: by expanding on known functions and using the Maclaurin series, we can convert these four kinds of solutions into linear combinations of some elements that are basis of these solutions. Using these bases gives exact solutions for polynomials and, for some other functions, a high-accuracy approximate solution.

Constructing Solutions of Cauchy Type Integral Equations by Using Four Kinds of Basis

Constructing Solutions of Cauchy Type Integral Equations by Using Four Kinds of Basis

Dvumernye zadachi fil'tratsii zhidkosti s granichnymi istochnikami v anizotropnom neodnorodnom sloe

We study the first and second boundary value problems and the transmission problem for the complex potential of a two-dimensional filtration flow in an anisotropic and inhomogeneous (variable permeability and thickness) porous layer. The flow sources are arbitrary discrete and can generally be located both on the boundaries and outside the boundaries. The boundaries are modeled by arbitrary smooth (piecewise smooth) closed lines, and the flow sources are singularities (isolated singular points) of the complex potential. The presence of a system of sources on the boundaries leads to a fundamentally new generalization (complication) of the boundary conditions, which are characterized by singular functions with isolated singular points. In the case of an anisotropic homogeneous (constant permeability and thickness) layer and rectilinear boundaries, the solutions of the problems are presented in closed form. In the general case, when an arbitrary smooth closed curve models a boundary with sources located on it, a generalized Cauchy type integral for the complex flow potential is used. This permitted reducing the second boundary value problem and the transmission problem to boundary singular integral equations. The problems studied are mathematical models of two-dimensional filtration processes in layered porous media, which are of interest, for example, for the practice of extracting fluids (oil, water) from natural anisotropically heterogeneous soil layers.