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This article investigates a time-fractional space-nonlocal diffusion equation in a bounded domain. The fractional operators are defined rigorously, using the Caputo fractional derivative of order β and the Riemann-Liouville fractional integral of order α, where 0<α <β ᾘ ≤-1. The solution is expressed as a series involving the two-parameter Mittag-Leffler function and orthonormal eigenfunctions of the Sturm-Liouville operator. The convergence of the series is investigated, and conditions for the solution to belong to a specific function space are established. The uniqueness of the solution is demonstrated and the continuity of the solution in the specified domain is confirmed through the uniform convergence of the series. [ABSTRACT FROM AUTHOR]

This paper considers the inverse problem of determining the time‐dependent coefficient in the time‐fractional diffusion‐wave equation. In this case, an initial boundary value problem was set for the fractional diffusion‐wave equation, and an additional condition was given for the inverse problem of determining the coefficient from this equation. First of all, it was considered the initial boundary value problem. 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 a priori 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. The stability estimate is also obtained. [ABSTRACT FROM AUTHOR]

A fundamental solution matrix for elliptic systems of the second order with constant leading coefficients is constructed. It is used to obtain an integral representation of functions belonging to the Hölder class in a closed domain with a Lyapunov boundary. In the case of an infinite domain, these functions have power-law asymptotics at infinity. The representation is used to study a mixed-contact boundary value problem for a second-order elliptic system with piecewise constant leading coefficients. The problem is reduced to a system of integral equations that are Fredholm in the domain and singular at its boundary. [ABSTRACT FROM AUTHOR]

In the present paper, we consider the initial boundary value direct problem and zeroorder determination inverse problem for the time-fractional wave equation. First, we study the direct problem (using Robin-type boundary condition), and then using its properties, we investigate the inverse problem. We prove the local solvability of the inverse problem’s solution and will show its stability. [ABSTRACT FROM AUTHOR]

The existence and uniqueness theorems of the solution to the boundary-value problem for a parabolic-hyperbolic fractional-order equation with the gluing condition are proved. [ABSTRACT FROM AUTHOR]

In this work, we investigate a generalization of the Tricomi problem for a loaded mixed‐type equation with the Riemann–Liouville fractional differential operator. By using the method of integral equations, a unique solvability of the formulated problem given in piecewise non‐parallel characteristics is proven. [ABSTRACT FROM AUTHOR]

Surface tension plays a central role in the mechanical behavior of soft materials such as gels. Elastocapillary deformation of elastic graded substrates is ubiquitous in soft materials. In this work, the effect of a partially wetting sessile liquid droplet on the elastocapillary deformation of a soft elastic graded substrate is studied. The modulus is assumed to have an exponential form along the thickness direction. By applying the Fourier transformation, a mixed boundary-value problem is reduced into a dual integral equation. The numerical results show that the surface displacement is strongly affected by the inhomogeneity of the material. The study of the wetting properties of gel substrates is essential for both understanding the wetting phenomena of gels and developing gels for applications as soft actuators and sensors that can be used in wearable electronics and soft robotics. [ABSTRACT FROM AUTHOR]

In this paper we consider a boundary value problem for a loaded fractional diffusion equation. The loaded term has the form of the Riemann-Liouville fractional derivative or integral. The BVP is considered in the open right upper quadrant. The problem is reduced to an integral equation that, in some cases, belongs to the pseudo-Volterra type, and its solvability depends on the order of differentiation in the loaded term and the behavior of the support line of the load in a neighborhood of the origin. All these cases are considered. In particular, we establish sufficient conditions for the unique solvability of the problem. Moreover, we give an example showing that violation of these conditions can lead to nonuniqueness of the solution. [ABSTRACT FROM AUTHOR]

An explicit formula for the zero of the Szegö kernel for an annulus region is well-known. There exists a transformation formula for the Szegö kernel from a doubly connected region onto an annulus. Based on conformal mapping, we derive an analytical formula for the zeros of the Szegö kernel for a general doubly connected region with smooth boundaries. Special cases are the explicit formulas for the zeros of the Szegö kernel for doubly connected regions bounded by circles, limacons, ellipses, and ovals of Cassini. For a general doubly connected region with smooth boundaries, the zero of the Szegö kernel must be computed numerically. This paper describes the application of conformal mapping via integral equation with the generalized Neumann kernel for computing the zeros of the Szegö kernel for smooth doubly connected regions. Some numerical examples and comparisons are also presented. It is shown that the conformal mapping approach also yields good accuracy for a narrow region or region with boundaries that are close to each other. [ABSTRACT FROM AUTHOR]

In this paper, we consider an inverse problem of determining u(x, t) and k(t) functions in the one-dimensional integro-differential heat equation with the nonlocal initial-boundary and over-determination conditions. The unique solvability of the direct problem is rigorously proved using the Fourier method and Schauder principle. To investigate the solvability of the inverse problem, we first consider an auxiliary inverse boundary value problem, which is equivalent to the original one. Then using the Fourier method, the problem is reduced by an equivalent closed system of integral equations with respect to unknown functions. The existence and uniqueness theorem for this system of integral equations is proved by contraction mappings principle. [ABSTRACT FROM AUTHOR]

We consider a perforated domain Ω(ε) of R² with a small hole of size ε and we study the behavior of the solution of a mixed Neumann-Robin problem in Ω(ε) as the size ε of the small hole tends to 0. In addition to the geometric degeneracy of the problem, the nonlinear ε-dependent Robin condition may degenerate into a Neumann condition for ε = 0 and the Robin datum may diverge to infinity. Our goal is to analyze the asymptotic behavior of the solutions to the problem as ε tends to 0 and to understand how the boundary condition affects the behavior of the solutions when ε is close to 0. The present paper extends to the planar case the results of [36] dealing with the case of dimension n ≥ 3. [ABSTRACT FROM AUTHOR]

In this article, initial-boundary value problems for the fourth-order equation with three lines of degeneracy have been formulated in rectangular domain and one of them has been studied in detail. Applying the method of separation variables to the considered problem, the spectral problem was obtained for an ordinary differential equation. The Green’s function of the spectral problem was constructed, with the help of which the considered spectral problem is equivalently reduced to the Fredholm integral equation of the second kind with a symmetric kernel. The solution of the considered problem is represented by the Fourier series with respect to the system of eigenfunctions of the spectral problem. An estimate for solution the problem was obtained, from which follows continuous dependence on the given functions. [ABSTRACT FROM AUTHOR]

The work was carried out within the context of a competitive ideology of creating the element base of digital optical computers (transphasors, optical switches, memory elements) built on a basis other than the Fabry–Perot interferometer. Mathematical models of stationary (problem I) and non-stationary (problem II) four-beam laser interaction in optically nonlinear media are considered in detail. Problem (I) is the system of ordinary differential equations with specified boundary conditions. Problem (II) is the system of integro-differential equations with boundary conditions. We introduced the original sought-for functions z(x) and u(z, t), υ(z, t) (II). As a result, the problem (I) is reduced to solving a simple transcendental equation (canonical equation of optical hysteresis). The problem (II) is reduced to solving a system of two nonlinear integral equations for the amplitudes of interference patterns (the canonical system of equations of non-stationary optical hysteresis). [ABSTRACT FROM AUTHOR]

The content of the study is in line with the competitive ideology of creating the element base of digital optical computers (transphasors, optical switches, memory devices) on a basis other than the Fabry–Perot interferometer. Mathematical models of stationary (variant I) and nonstationary (variant II) four-beam laser interaction in optically nonlinear media are considered in detail in the form of a system of ordinary differential equations with given boundary conditions (I) and a system of integro-differential equations with boundary conditions (II). The original desired functions z (x) (I) and u (z,t), v(z,t) (II) are introduced. As a result, the solution of problem (I) is reduced to solving a simple transcendental equation (canonical equation of optical hysteresis), and the solution of problem (II) is reduced to a system of two nonlinear integral equations with respect to the amplitudes of interference patterns (canonical system of equations of nonstationary optical hysteresis). [ABSTRACT FROM AUTHOR]

In this paper, parallel to the ideas based on Ahmadullah et al. [4, 5, 6], and Eke et al. [18], we prove the existence and uniqueness of the common fixed point for a pair of self-mappings employing (E. A.)-property in metric-like spaces for implicit contractive mappings related to binary relation. Henceforth, results obtained will be verified with the help of illustrative examples. As an application of the results, we solve two boundary value problems of the second-order differential equation. [ABSTRACT FROM AUTHOR]

A Rayleigh–Ritz Method is suggested for solving linear Fredholm integral equations of the second kind numerically in a desired accuracy. To test the performance of the present approach, the classical one-dimensional Schrödinger equation - y ″ (x) + v (x) y (x) = λ y (x) , x ∈ (- ∞ , ∞) has been converted into an integral equation. For a regular problem, the unbounded interval is truncated to x ∈ [ - ℓ , ℓ ] , where ℓ is regarded as a boundary parameter. Then, the resulting integral equation has been solved and the results are compared with the very well known eigenvalues of the Schrödinger equation with several types of potential functions v(x). It is shown that the eigenvalues recorded to about 15 significant figures are in excellent agreement with the results that exist in the literature. [ABSTRACT FROM AUTHOR]

The boundary value problem for the Helmholtz equation in the m-dimensional free space is considered. The problem is reduced to the Lippmann–Schwinger integral equation over the inhomogeneity domain. The operator of the integral equation is shown to be an invertible Fredholm operator. The inverse coefficient problem is considered. An application of the two-step method reduces the inverse problem to the source-type integral equation with a smooth kernel. Special classes of solutions to this equation are introduced. The uniqueness of a solution to the integral equation of the first kind is proved in the defined function classes. [ABSTRACT FROM AUTHOR]

The boundary value problem for the Helmholtz equation in the m-dimensional free space is considered. The problem is reduced to the Lippmann–Schwinger integral equation over the inhomogeneity domain. The operator of the integral equation is shown to be an invertible Fredholm operator. The inverse coefficient problem is considered. An application of the two-step method reduces the inverse problem to the source-type integral equation with a smooth kernel. Special classes of solutions to this equation are introduced. The uniqueness of a solution to the integral equation of the first kind is proved in the defined function classes. [ABSTRACT FROM AUTHOR]

Boundary-value problems with phase constraints for linear ordinary differential equations are considered. The necessary and sufficient conditions for the existence of a solution to the boundary value problems of linear ordinary differential equations with boundary conditions from given sets in the presence of phase constraints are obtained. A method is proposed for constructing a solution to a boundary value problem with phase constraints by constructing minimizing sequences in a functional space. An estimate of the convergence rate of minimizing sequences is obtained. The basis of the proposed method for solving boundary value problems with phase constraints is the ability to reduce these problems to one class of the Fredholm integral equation of the first kind. The Fredholm integral equation of the first kind is among the poorly studied problems of mathematics. Therefore, basic research on integral equations and the solution based on them of boundary value problems of differential equations is the main promising direction in mathematics. A new method is proposed for solving boundary value problems of linear ordinary differential equations with phase constraints, which has numerous applications in the theory of dynamical systems. The scientific novelty of the results is: Formalization of the general problem of dynamical systems and its reduction to boundary value problems of ordinary differential equations with phase constraints; A new criterion is found for the existence of a solution to boundary value problems in the form of the immersion principle based on the existence theorem and the construction of a solution to the integral equation; A new method has been created for solving boundary value problems of linear ordinary differential equations by constructing minimizing sequences for a special initial optimal control problem.

The goal of this work is to look at generalised contractions and show that they may be used to derive fixed point theorems in generalised metric spaces. Furthermore, we'll also use fixed point theorems to show that the outcome of the ordinary differential equation exists and is exclusive (ODE). Fixed point theorem applies to modified Lipschitz condition. First, we have proved that generalised condition of Boyd Wong theorem and using this concept we have modified Lipschitz condition by adding constant which is suitable for solving differential equation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a nonlocal boundary value problem for an equation of special type. For y > 0 it is a fractional diffusion equation, which contains the Riemann-Liouville fractional derivative. For y < 0 it is a generalized equation of moisture transfer. A unique solvability of the considered problem is proved. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of solutions for the following fractional boundary value problem c D α u (t) = f (t , u (t)) , α ∈ (n , n + 1) , 0 ≤ t < + ∞ , u (0) = 0 , u ′ ′ (0) = 0 , ... , u (n) (0) = 0 , lim t → + ∞ c D α − 1 u (t) = β u (ξ). The considerations of this paper are based on the concept of a new family of measures of noncompactness in the space of functions C n , γ (R +) satisfying the Hölder condition and a fixed point theorem of Darbo type. We also provide an illustrative example in support of our existence theorems. Finally, to credibility, we apply successive approximation and homotopy perturbation method to find solution of the above problem with high accuracy. [ABSTRACT FROM AUTHOR]

We define a minimal operator L0 generated by an integral equation with an operator measure and prove necessary and sufficient conditions for the operator L0 to be densely defined. In general, L ∗ 0 is a linear relation. We give a description of L ∗ 0 and establish that there exists a one-to-one correspondence between relations Lb with the property L0 ⊂Lb⊂L ∗ 0 and relations θ entering in boundary conditions. In this case we denote Lb = Lθ. We establish conditions under which linear relations Lθ and θ together have the following properties: a linear relation (l.r) is self-adjoint; l.r is closed; l.r is invertible, i.e., the inverse relation is an operator; l.r has the finitedimensional kernel; l.r is well-defined; the range of l.r is closed; the range of l.r is a closed subspace of the finite codimension; the range of l.r coincides with the space wholly; l.r is continuously invertible. We describe the spectrum of Lθ and prove that families of linear relations Lθ(λ) and θ(λ) are holomorphic together. [ABSTRACT FROM AUTHOR]

The standard Caputo fractional derivative is generalized for the piecewise continuous functions. A more general boundary value problem for the impulsive Langevin fractional differential equation involving the Caputo fractional derivatives is studied. New existence results for solutions of concerned problems are established. [ABSTRACT FROM AUTHOR]

In this paper, we study an analog of the Gellerstedt problem for the third‐order loaded equation with boundary conditions on nonparallel characteristics in the hyperbolic domain of the equation. [ABSTRACT FROM AUTHOR]

This paper considers two variants of problems that arise in developing optical computers. The first variant is related to the mathematical analysis of problems of optical bistability in the case of multibeam interaction of laser radiation in nonlinear media. The existence of optical bistability is confirmed by the results of solving the boundary value problem for a system of nonlinear ordinary differential equations. In the general case of an arbitrary nonstationary process, the problem is reduced to solving a system of two nonlinear integral equations with respect to complex amplitudes describing interference patterns. The second variant of problems is devoted to studying the absorption and scattering of light by nanomaterials. As a result, a multidimensional integral equation was obtained for the complex amplitude of the electric field. A fundamentally important feature of this equation is its singularity inside a nanoparticle. [ABSTRACT FROM AUTHOR]

It is proved that the family of all pairwise products of regular harmonic functions on D and of the Newtonian potentials of points on the line L ⊂ Rn is complete in L2(D), where D is a bounded domain in Rn, n ≥ 3, such that D¯ ∩ L = ∅. This result is used in the proof of uniqueness theorems for the inverse acoustic sounding problem in R3. [ABSTRACT FROM AUTHOR]

A Robin problem is a mixed problem with a linear combination of Dirichlet and Neumann D-N conditions. The aim of this paper are presents a new boundary integral equation BIE method for the solution of unbounded Robin boundary value problem BVP in the simply connected domain. The method show how to reformulate the Robin boundary value problem BVP as Riemann-Hilbert problem RHP which lead to the system of integral equation and the related differential equations are also created that give rise to unique solutions. Numerical results on several tests regions by the Nyström method NM with the trapezoidal rule TR are presented to clarify the solution technique for the Robin problem when the boundaries are sufficiently smooth. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of solutions for a boundary value problem involving both left Riemann-Liouville and right Caputo-type fractional derivatives. For this, we convert the posed problem to a sum of two integral operators, then we apply Krasnoselskii's fixed point theorem to conclude the existence of nontrivial solutions. [ABSTRACT FROM AUTHOR]

This paper presents our study of American floating strike lookback options written on dividend-paying assets. The valuation of these options can be mathematically formulated as a free boundary inhomogeneous Black–Scholes PDE with a Neumann boundary condition, which we, by using a Mellin transform, convert into a relatively simple ordinary differential equation with Dirichlet boundary conditions. We then use these results to derive an integral equation that can be used to calculate the price of American floating strike lookback options. In addition, we also used Mellin transforms to derive the closed-form of the perpetual case. [ABSTRACT FROM AUTHOR]

In this paper, by using the Guo-Krasnoselskii theorem, we investigate the existence and nonexistence of positive solutions of a system of integral equation with parameters which can be seen as an effective generalization of various types of systems of boundary value problems for differential equation on continuous interval and time scales or fractional differential equations. We give a general approach of positive solutions to cover various systems of boundary value problems in a unified way, which avoids treating these problems on a case-by-case basis. Under some growth conditions imposed on the nonlinear term, we obtain explicit ranges of values of parameters with which the problem has a positive solution and has no positive solution, respectively. By giving some examples, we will show how our results may be applied to consider existence of positive solutions to a variety of system of boundary value problems of differential equations, differential equations on time scales or fractional differential equations. [ABSTRACT FROM AUTHOR]

Consider the scattering of long ocean tidal waves by an island taking into account the influence of daily rotation of the Earth, which is modeled by an exterior boundary value problem for the two-dimensional Helmholtz equation with generalized oblique derivative boundary condition. In this paper, we are concerned with a corresponding inverse scattering problem which is to reconstruct the unknown obstacle (island) from the far-field data. After proving the unique solvability of the direct scattering problem in a suitable function space required for our inverse scattering problem, we establish the linear sampling method (LSM) for reconstructing the boundary of the obstacle from the far-field data. To clarify the validity of such a sampling-type method which essentially depends on the solvability of an interior boundary value problem, we show that, except a discrete set of wave numbers, such an interior problem has a unique solution. Finally, some numerical examples are presented to demonstrate the efficiency of the reconstruction scheme. [ABSTRACT FROM AUTHOR]

This paper deals with a new approach to study the nonlinear inviscid flow over arbitrary bottom topography. The problem is formulated as a nonlinear boundary value problem which is reduced to a Dirichlet problem using certain transformations. The Dirichlet problem is solved by applying Plemelj–Sokhotski formulae and it is noticed that the solution of the Dirichlet problem depends on the solution of a coupled Fredholm integral equation of the second kind. These integral equations are solved numerically by using a modified method. The free-surface profile which is unknown at the outset is determined. Different kinds of bottom topographies are considered here to study the influence of bottom topography on the free-surface profile. The effects of the Froude number and the arbitrary bottom topography on the free-surface profile are demonstrated in graphical forms for the subcritical flow. Further, the nonlinear results are validated with the results available in the literature and compared with the results obtained by using linear theory. [ABSTRACT FROM AUTHOR]

We establish the unique solvability for an inverse problem for semi- linear fractional telegraph equation Dtα u + r(t) Dtβ u - Δu = Fo(x,t,u, Dtβ u), (x,t) ∈ Ω0 x (0,T] with regularized fractional derivatives Dtαu, Dtβu of orders α ∈ (1,2), β ∈ (0,1) with respect to time on bounded cylindrical domain. This problem consists in the determination of a pair of functions: a classical solution u of the first boundary-value problem for such equation, and an unknown continuous coefficient r(t) under the over-determination condition ... with given functions φ and F. [ABSTRACT FROM AUTHOR]

The incorporation of analytical kernel information is exploited in the construction of Nyström discretization schemes for integral equations modeling planar Helmholtz boundary value problems. Splittings of kernels and matrices, coarse and fine grids, high-order polynomial interpolation, product integration performed on the fly, and iterative solution are some of the numerical techniques used to seek rapid and stable convergence of computed fields in the entire computational domain. [ABSTRACT FROM AUTHOR]

This article develops a numerical-analytical method for solving the problem of scattering acoustic and electromagnetic waves by an impedance grating. This problem leads to a third-type boundary problem for the two-dimensional Helmholtz equation with additional conditions. The boundary value problem is reduced to boundary integral equations, and the discrete singularities method is modified to numerically solve them. Dependencies of integral characteristics of solutions to the problem on frequency are obtained. [ABSTRACT FROM AUTHOR]

SUMMARY A parallelized multilevel fast multipole algorithm (MLFMA) is presented for simulating electromagnetic scattering from complex targets with anisotropic impedance surfaces. By employing both surface electric and magnetic currents as unknowns and weakly enforcing the anisotropic impedance boundary condition, a combined integral equation is formulated to generate a set of well-conditioned linear systems to be solved by MLFMA. To further improve the iterative convergence of the linear systems, a parallel sparse approximate inverse preconditioner is constructed from the near-field interaction of the system matrix. The MLFMA is parallelized to enable computation on a large number of processors for large-scale problems. Several numerical examples are presented to validate the algorithm and demonstrate its accuracy, scalability, and capability in handling large complex objects with anisotropic impedance surfaces. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

New integral formulas for upward/downward continuation of gravitational gradients onto gravitational gradients are derived in this article. They provide more options for continuation of gravitational gradient combinations and extend available mathematical apparatus formulated for this purpose up to now. The starting point represents the analytical solution of the spherical gradiometric boundary value problem in the spatial domain. Applying corresponding differential operators on the analytical solution of the spherical gradiometric boundary value problem, a total of 18 integral formulas are provided. Spatial and spectral forms of isotropic kernels are given and their behaviour for parameters of a GOCE-like satellite is investigated. Correctness of the new integral formulas and the isotropic kernels is tested in a closed-loop simulation. The derived integral formulas and the isotropic kernels form a theoretical basis for validation purposes and geophysical applications of satellite gradiometric data as provided currently by the GOCE mission. They also extend the well-known Meissl scheme. [ABSTRACT FROM AUTHOR]

We prove the existence and construct the form of the asymptotic solution for an initial-boundary value problem that describes diffusive filling of thin shells with a real gas. Such problems arise in the manufacturing of laser targets, where a thin spherical shell is filled with hydrogen isotopes to a high pressure [1-3]. In this article we apply the methods of [4], but the nonlinearity of the boundary conditions requires new approaches to the problem. [ABSTRACT FROM AUTHOR]

In this paper we investigate the boundedness of the Gerber-Shiu expected discounted penalty function applied in insurance mathematics. We give a necessary and sufficient condition for its boundedness. This condition is essentially the boundedness of the conditional expectation given the ruin occurs at the first claim. It assures that the integral equation satisfied by the Gerber- Shiu function has a unique bounded solution and it is exactly the Gerber-Shiu function. By the help of the proven necessary and sufficient condition we can also simplify the theorem concerning the necessary and sufficient condition for the limit property. We do not assume independence between the inter-claim time and the claim size but the results can be applied to the usually investigated independent case as well. We present examples to show the applicability of the condition. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and uniqueness of positive solutions for a class of nonlinear operator equations on ordered Banach spaces. Various applications are also considered to illustrate our obtained results (existence of solutions to quadratic integral equations with a linear modification of the argument, positive solution of second-order Neumann boundary value problem, and positive definite solutions of a class of nonlinear matrix equations). [ABSTRACT FROM AUTHOR]