Boundary Value Problems For Functions Research Articles

The Solvability and Explicit Solutions of Singular Integral–Differential Equations with Reflection

This article deals with a classes of singular integral–differential equations with convolution kernel and reflection. By means of the theory of boundary value problems of analytic functions and the theory of Fourier analysis, such equations can be transformed into Riemann boundary value problems (i.e., Riemann–Hilbert problems) with nodes and reflection. For such problems, we propose a novel method different from classical one, by which the explicit solutions and the conditions of solvability are obtained.

The Hilbert problem in a half-plane for generalized analytic functions with a singular point on the real axis

This article analyzes the inhomogeneous Hilbert boundary value problem for an upper half-plane with the finite index and boundary condition on the real axis for one generalized Cauchy–Riemann equation with a singular point on the real axis. A structural formula was obtained for the general solution of this equation under restrictions leading to an infinite index of the logarithmic order of the accompanying Hilbert boundary value problem for analytic functions. This formula and the solvability results of the Hilbert problem in the theory of analytic functions were applied to solve the set boundary value problem.

Tricomi problem and integral equations

Formulas for inverting integral equations that arise when studying the Tricomi problem for the Lavrentyev–Bitsadze equation were derived. Solvability conditions of an auxiliary overdetermined problem in the elliptic part of the mixed domain were found using the Green function method. A connection was established between the Green functions of the Dirichlet problem and problem N for the Laplace equation in the form of integral equations mutually inverting each other. Various integral equations were considered, including explicitly solvable ones, to which the Tricomi problem can be reduced. An explicit solution of the characteristic singular equation with a Cauchy kernel was obtained without involving the theory of boundary value problems for analytic functions.

Applications of new smart algorithm based on kernel method for variable fractional functional boundary value problems

Applications of new smart algorithm based on kernel method for variable fractional functional boundary value problems

Harmonic wave scattered by an inclusion in an elastic plane: The complete Gurtin-Murdoch model

We consider the propagation of a harmonic elastic wave in a composite inclusion–matrix structure subjected to plane deformation. The interface between the inclusion and matrix is described by the complete Gurtin-Murdoch model with non-vanishing interface tension and interface stretching rigidity. We consider an inclusion of general shape and formulate the corresponding boundary value problem for the wave functions in the inclusion and matrix when a harmonic compressional or shear wave is incident on the edge of the matrix. The problem is then solved by series expansion methods for the case of a circular inclusion embedded in an infinite matrix. The series solution obtained is validated by checking its convergence and via comparisons with existing static and dynamic solutions for certain reduced cases. Numerical examples are presented for the case of a small-sized circular hole embedded in a soft matrix demonstrating the influence of surface tension on the incident wave-induced dynamic stress concentration in the matrix. We find that the presence of surface tension relieves the peak stress around the circular hole when the frequency of the incident wave is below a certain critical value, while it tends to intensify the peak stress for a high-frequency incident wave.

The inhomogeneous complex partial differential equations for bi-polyanalytic functions

<abstract><p>In this paper, we study a Riemann-Hilbert problem related to inhomogeneous complex partial differential operators of higher order on the unit disk. Applying the Cauchy-Pompeiu formula, we find out the solvable conditions and obtain the representation of the solutions. Then, we investigate the boundary value problems for bi-polyanalytic functions with the Dirichlet and Riemann-Hilbert boundary conditions, obtain the specific solution and the solvable conditions, and extend the conclusion to the corresponding higher-order problems. Therefore, we obtain the solution to the half-Neumann problem of higher order for bi-polyanalytic functions.</p></abstract>

ФУНКЦИОНАЛЬНЫЕ УРАВНЕНИЯ КАК МАТЕМАТИЧЕСКИЕ МОДЕЛИ ЗАДАЧ СОПРЯЖЕНИЯ С ЦИКЛИЧЕСКИМ СДВИГОМ НА СЛОЖНЫХ КРИВЫХ

The paper considers linear functional equations with a shift function having a nonzero derivative satisfying the Helder condition on an arbitrary piecewise smooth curve. Such equations are studied in connection with the theory of boundary value problems for analytical functions, which are a mathematical tool in the study of mathematical models of elasticity theory in which the conjugation conditions contain a boundary shift. The shift function acts cyclically on a set of simple curves forming a given curve, and except the ends of simple curves, there are no periodic points relative to the shift function. The purpose of the study is to find conditions for the existence and uniqueness of a solution (and in the case of non–uniqueness of the cardinality of the set of solutions) of such equations in the classes of Helder and primitive Lebesgue functions with a coefficient and the right part of the same classes.

A Class of Inverse Boundary Value Problems for (λ, 1) Bi-Analytic Functions

In this paper, a class of inverse boundary value problems for (λ, 1) bi-analytic functions is given. Using the method of Riemann boundary value problem for analytic functions, the conditions of solvability and the expression of the solutions for the inverse problems are obtained.

Positive solutions for fractional iterative functional differential equation with a convection term

<abstract><p>In this paper, we deal with the fractional iterative functional differential equation nonlocal boundary value problem with a convection term. By using the fixed point theorems, some results about existence, uniqueness, continuous dependence and multiplicity of positive solutions are derived.</p></abstract>

Краевая задача Гильберта для одного класса обобщенных аналитических функций с сингулярной линией

In this paper, we study an inhomogeneous Hilbert boundary value problem with a finite index and a boundary condition on the real axis for one generalized equation Cauchy — Riemann with singular coefficient. A structural formula for the general solution of this equation is obtained under constraints leading to an infinite index of the accompanying Hilbert boundary value problem for analytic functions. The study of the solvability of the latter is the basis for solving the boundary value problem for generalized analytical functions. Note that here we have obtained a boundary value problem with an infinite index and two swirl points. Such a situation is not studied in the case of an unbounded domain, even for analytic functions. We have obtained a formula for the general solution of the indicated Hilbert problem in the theory of analytic functions. We carried out a complete study of the solvability of this problem in terms of the characteristics of the coefficient of the boundary condition. Note that here we have obtained a boundary value problem with an infinite index and two swirl points. Such a situation is not studied in the case of an unbounded domain, even for analytic functions. We have obtained a formula for the general solution of the indicated Hilbert problem in the theory of analytic functions. We carried out a complete study of the solvability of this problem in the class of bounded analytic functions. This study made it possible to obtain a formula for solving the Hilbert problem already for generalized analytic functions. We also formulated the conditions under which the problem has no solution, has a single solution or an infinite number of solutions.

Boundary Value Problems for Analytic Functions with Boundary Data from Grand Lebesgue Spaces

Boundary Value Problems for Analytic Functions with Boundary Data from Grand Lebesgue Spaces

Solvability of mixed Hilfer fractional functional boundary value problems with p-Laplacian at resonance

This article investigates the existence of solutions of mixed Hilfer fractional differential equations with p-Laplacian under the functional boundary conditions at resonance. By defining Banach spaces with appropriate norms, constructing suitable operators, and using the extension of the continuity theorem, some of the current results are extended to the nonlinear situation, and some new existence results of the problem are obtained. Finally, an example is given to verify our main results.

Existence of solutions for $n$-dimensional fractional order bvp with $\infty$–point boundary conditions via the concept of measure of noncompactness

In this paper, using the concept of measure of noncompactness we present some results on the existence of $n$-fixed points for a class of operators. Also as an application, we derive the sufficient conditions for the existence of solutions for $n$-dimensional fractional order functional boundary value problems.

Periodic Riemann boundary value problem

In this paper, we discuss the Riemann boundary value problems for the analytic functions with period in the situation where the solution can be unbounded at . When the jump curve is a closed contour, by using the tangent transformation , we transform the singularities on the z plane to the singularities on the ζ plane. When the jump curve is the real axis, by using the transformation , we transform the singularities on the z plane to the singularities 0 and ∞ on the ω plane. Thus, the original periodic Riemann boundary value problems are reduced to the classical Riemann boundary value problems whose solutions and the corresponding solvability conditions are well known.

Plemelj formula of inframonogenic functions and their boundary value problems

In this paper, firstly, we study the continuity of Cauchy-type integral operator associated with inframonogenic functions and give the Plemelj formula. Secondly, we prove the properties of the Teodorescu operator related to the inframonogenic functions, including its boundness, continuity and differentiability. Finally, we give the related integral representation of Riemann boundary value problems for inframonogenic functions and generalized inframonogenic functions.

Nonlinear two-point iterative functional boundary value problems on time scales

Nonlinear two-point iterative functional boundary value problems on time scales

The method of fundamental solutions for analytic functions in complex analysis

<abstract><p>This paper extends the method of fundamental solutions (MFS) for solving the boundary value problems of analytic functions based on Cauchy-Riemann equations and properties of harmonic functions. The conformal mapping technique is applied to introduce the singularities of the approximate analytic functions and reconstruct the fundamental solutions. The presented method can naturally introduce the information of homogeneous boundary conditions and singularity properties, when the conformal mapping technique or the reconstructed fundamental solutions are used. The numerical examples show that the proposed method has the advantages of conciseness, reliability, efficiency, high accuracy and easy-using, respectively. The developed method can be used to solve the boundary value problems (BVPs) of analytic functions without considering single-valuedness, which simplify the numerical analysis.</p></abstract>

A NEW EFFICIENT METHOD FOR TWO CLASSES OF SINGULAR CONVOLUTION INTEGRAL EQUATIONS OF NON-NORMAL TYPE WITH CAUCHY KERNEL

In this article, our task is to study the existence and Noethericity of solution for two classes of singular convolution integral equations with Cauchy kernels in the non-normal type case. To obtain the conditions of solvability for such equations, we establish regularity theory of solvability. By means of the theory of Fourier analysis, we will transform the equations into boundary value problems for holomorphic functions. The holomorphic solutions and conditions of solvability are obtained by using the method of complex analysis in class {0}. Moreover, we also discuss the asymptotic property of solution near nodes. Therefore, our work generalizes and improves the theories of integral equations and the classical boundary value problems for holomorphic functions.

SOLVING CONVOLUTION SINGULAR INTEGRAL EQUATIONS WITH REFLECTION AND TRANSLATION SHIFTS UTILIZING RIEMANN-HILBERT APPROACH

<abstract> In this paper, method of solution for some kinds of convolution singular integral equations with reflection will be discussed in class {0}. By means of the theory of Fourier analysis and the theory of boundary value problems of analytic functions, such equations can be transformed into Riemann boundary value problems (i.e., Riemann-Hilbert problems) with nodes and reflection, or a system of linear algebraic equations. In spite of the classical method for solution, we are to give a new method, by which analytic solutions and conditions of Noether solvability are obtained respectively. At the end of this paper, we propose two kinds of convolution singular integral equations with reflections and a finite set of translation shifts. </abstract>

A SCHAUDER TYPE HYBRID FIXED POINT THEOREM IN A PARTIALLY ORDERED METRIC SPACE WITH APPLICATIONS TO NONLINEAR FUNCTIONAL INTEGRAL EQUATIONS

In this paper we prove some hybrid fixed point theorems for the monotonic nondecreasing mappings in a partially ordered metric space which includes the Schauder type fixed point theorems in an ordered Banach space proved by Dhage (2013,2014) and Dhage et al. (2022) in a partially ordered Banach space as the special cases. As an application, we discuss a nonlinear functional integral equation of Fredholm type and a nonlinear functional boundary value problem for proving the existence and approximation of solution by constructing the algorithms via Dhage monotone iteration method under some natural condtions.