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We introduce Besov space $ \dot {B}_{p}^{\alpha,q}(\partial \Omega _k) $ B ˙ p α , q (∂ Ω k) and Triebel-Lizorkin space $ \dot {F}^{\alpha,q}_{p}(\partial \Omega _k) $ F ˙ p α , q (∂ Ω k) on a family of model domains $ \partial \Omega _k=\left \{(\mathbf{z},z_{n+1})=(z_1,z_2,\ldots, z_{n+1}):\right. \left. \mbox {Im} (z_{n+1})=\phi (|\mathbf{z}|^2)\right \} $ ∂ Ω k = { (z , z n + 1) = (z 1 , z 2 , ... , z n + 1) : Im (z n + 1) = ϕ (| z | 2) } with $ \phi (x)=x^{k} $ ϕ (x) = x k in $ \mathbf {C}^{n+1} $ C n + 1 which can be considered as a space of homogeneous type in the sense of Coifman RR, Weiss G.[Analyse Harmonique Non-commutative and Certains Espaces Homogǹes. Berlin: Springer; 1971. (Lecture Notes in Math.; 242).], Coifman R, Weiss G. [Extensions of Hardy spaces and their use in analysis. Bull Amer Math Soc. 1977;83:569–645.]. We study the sharp estimates on the fundamental solution for the Kohn Laplacian and Cauchy-Szegö projection on $ \partial \Omega _k $ ∂ Ω k in these spaces. [ABSTRACT FROM AUTHOR]

We consider a pseudodifferential equation of parabolic type with a fractional power of the Laplace operator of order $\alpha\in(0;1)$ acting with respect to the spatial variable. This equation naturally generalizes the well-known fractal diffusion equation. It describes the local interaction of moving objects in the Riesz gravitational field. A simple example of such system of objects is stellar galaxies, in which interaction occurs according to Newton's gravitational law. The Cauchy problem for this equation is solved in the class of continuous bounded initial functions. The fundamental solution of this problem is the Polya distribution of probabilities $\mathcal{P}_\alpha(F)$ of the force $F$ of local interaction between these objects. With the help of obtained solution estimates the correct solvability of the Cauchy problem on the local field fluctuation coefficient under certain conditions is determined. In this case, the form of its classical solution is found and the properties of its smoothness and behavior at the infinity are studied. Also, it is studied the possibility of local strengthening of convergence in the initial condition. The obtained results are illustrated on the $\alpha$-wandering model of the Lévy particle in the Euclidean space $\mathbb{R}^3$ in the case when the particle starts its motion from the origin. The probability of this particle returning to its starting position is investigated. In particular, it established that this probability is a descending to zero function, and the particle "leaves" the space $\mathbb{R}^3$.

We establish the Caccioppoli inequality, a reverse Hölder inequality in the spirit of the classic estimate of Meyers, and construct the fundamental solution for linear elliptic differential equations of order 2m2m with certain lower order terms.

In this paper, layer potential techniques are investigated for solving the thermal diffusion problem. We construct the Green function to get the analytic solution. Moreover, by combining Fourier transform some attractive relation between initial heat distribution and the final observation is obtained. Finally iteration scheme is developed to solve the inverse heat conduction problem and convergence results are presented.

‎In this paper‎, ‎for a singular perturbation problem consist of the Cauchy-Euler equation with local and non-local boundary conditions‎. ‎We investigate the condition of the self-adjoint and the non-self-adjoint‎, ‎also look for the formation or non-formation of boundary layers for local boundary conditions using the Frequent uniform limit method‎. ‎Also‎, ‎for the state of non-local conditions‎, ‎we convert the non-local boundary conditions into local conditions by finding the fundamental solution and then obtaining the necessary conditions with the help of the 4-step method‎. ‎Finally‎, ‎we determine the formation or non-formation of a boundary layer for non-local conditions such as local conditions‎.

In this paper, we consider the Cauchy problem for parabolic Shilov equations with continuous bounded coefficients. In these equations, the inhomogeneities are continuous exponentially decreasing functions, which have a certain degree of smoothness by the spatial variable. The properties of the fundamental solution of this problem are described without using the kind of equation. The corresponding volume potential, which is a partial solution of the original equation, is investigated. For this Cauchy problem the correct solvability in the class of generalized initial data, which are the Gelfand and Shilov distributions, is determined.

A potential theory for a three-dimensional elliptic equation with one singular coefficient is considered. Double- and simple-layer potentials with unknown density are introduced, which are expressed in terms of the fundamental solution of the mentioned elliptic equation. When studying these potentials, the properties of the Gaussian hypergeometric function are used. Theorems are proved on the limiting values of the introduced potentials and their conormal derivatives, which make it possible to equivalently reduce boundary value problems for singular elliptic equations to an integral equation of the second kind, to which the Fredholm theory is applicable. The Holmgren problem is solved for a three-dimensional elliptic equation with one singular coefficient in the domain bounded \(x=0\) by the coordinate plane and the Lyapunov surface for \(x0\) as an application of the stated theory. The uniqueness of the solution to the stated problem is proved by the well-known \(abc\) method, and existence is proved by the method of the Green's function, the regular part of which is sought in the form of the double-layer potential with an unknown density. The solution to the Holmgren problem is found in a form convenient for further research.

In This paper, we consider two boundary value problems for non homogenuous Cauchy- Riemann equation with local and non-local boundary conditions separatly.In the first problem boundary condition is local and for proving uniqueness of solution, we show that homogenuous boundary value problem has only non-trivial solution. In the second problem, the boundary condition is non-local. At first, the given problem changed to a second kind system of Fredholm integral equation. Then the singularities in the Kernels of integral equations are removed. Finaly, for the uniqueness of solution, we show that the regularized kernels satify in contraction mapping theorem. ./files/site1/files/%D8%B9%D8%A8%D8%A7%D8%AF%D9%BE%D9%88%D8%B1.pdf

The paper considers a new approach to constructing generalized solutions of degenerate integro-differential equations convolution type in Banach spaces. The principal idea of the method proposed implies the refusal of the condition of existence of the full Jordan set for the Fredholm operator of the higher derivative with respect to the operator bundle formed by the rest of operator coefficients of the differential part and by the operator kernel of the integral component of the equation. The conditions are superimposed upon the values of the operator function specially constructed on the basis elements of the Fredholm operator kernel. Under such an approach, the differential part of the equation may include not only the higher derivative but also any combination lower derivatives, what allows one to consider the convolution integral-differential equations from universal positions, without any special account of the structure of the structure of the operator bundle. The method proposed represents a form of generalization of the technique based on the application of Jordan sets of Fredholm operators, and in the case of existence of the latter the method coincides with this technique. The generalized solutions are constructed in the form of a convolution of the fundamental operator function, which corresponds to the equation under investigation, and a function, which includes the right-hand side of the equation and the initial data. The conditions, under which such a generalized solution does not contain any singular component, and the regular component converts the initial equation into an identity and satisfies the initial data, and the result will provide for the resolvability of the initial problem in the class of functions of characterized by the respective smoothness. In this case, the generalized solution constructed will be classical. The theorem on the form of fundamental operator function has been proved, the abstract results have been illustrated via examples of initialboundary value problems of applied character from the theory electromagnetic fields, the theory of oscillations in visco-elastic media, the theory of vibrations of thermal-elastic plates.

In the article, using the modified Levy method, a Green's function for a class of ultraparabolic equations of high order with an arbitrary number of parabolic degeneration groups is constructed. The modified Levy method is developed for high-order Kolmogorov equations with coefficients depending on all variables, while the frozen point, which is a parametrix, is chosen so that an exponential estimate of the fundamental solution and its derivatives is conveniently used.

The paper deals with possible behaviour at infinity of solutions to the Cauchy problem for a parabolic type equation whose elliptic part is the generator of a Markov jump process , i.e. a nonlocal diffusion operator. The analysis of the behaviour of the solutions at infinity is based on the results on the asymptotics of the fundamental solutions of nonlocal parabolic problems. It is shown that such fundamental solutions might have different asymptotics and decay rates in the regions of moderate, large and super-large deviations. The asymptotic formulae for the said fundamental solutions are then used for describing classes of unbounded functions in which the studied Cauchy problem is well-posed. We also consider the question of uniqueness of a solution in these functional classes.

We introduce a stochastic partial differential equation (SPDE) with elliptic operator in divergence form, with measurable and bounded coefficients and driven by space-time white noise. Such SPDEs could be used in mathematical modelling of diffusion phenomena in medium consisting of different kinds of materials and undergoing stochastic perturbations. We characterize the solution and, using the Stein–Malliavin calculus, we prove that the sequence of its recentered and renormalized spatial quadratic variations satisfies an almost sure central limit theorem. Particular focus is given to the interesting case where the coefficients of the operator are piecewise constant.

In this paper, we construct an explicit fundamental solution for the fractional Kolmogorov–Fokker–Planck equation. To achieve this goal, the Fourier transform is applied, and the method of characteristics and the properties of positive definite matrix are also exploited.

This paper proposes a fast meshless scheme for acoustic sensitivity analysis by using the Burton–Miller-type singular boundary method (BM-SBM) and recursive skeletonization factorization (RSF). The Burton–Miller formulation was adopted to circumvent the fictitious frequency that occurs in external acoustic analysis, and then the direct differentiation method was used to obtain the sensitivity of sound pressure to design variables. More importantly, RSF was employed to solve the resultant linear system obtained by the BM-SBM. RSF is a fast direct factorization technique based on multilevel matrix compression, which allows fast factorization and application of the inverse in solving dense matrices. Firstly, the BM-SBM is a boundary-type collocation method that is a straightforward and accurate scheme owing to the use of the fundamental solution. Secondly, the introduction of the fast solver can effectively reduce the requirement of computer memory and increase the calculation scale compared to the conventional BM-SBM. Three numerical examples including two- and three-dimensional geometries indicate the precision and efficiency of the proposed fast numerical technique for acoustic design sensitivity analysis associated with large-scale and complicated structures.

We find the fundamental solution to the p-Laplace equation in a class of Hormander vector fields that generate neither a Carnot group nor a Grushin-type space. The singularity occurs at the sub-Riemannian points, which naturally corresponds to finding the fundamental solution of a generalized operator in Euclidean space. We then extend these solutions to a generalization of the p-Laplace equation and use these solutions to find infinite harmonic functions and their generalizations. We also compute the capacity of annuli centered at the singularity.

A localized virtual boundary element–meshless collocation method (LVBE-MCM) is proposed to solve Laplace and Helmholtz equations in complex two-dimensional (2D) geometries. “Localized” refers to employing the moving least square method to locally approximate the physical quantities of the computational domain after introducing the traditional virtual boundary element method. The LVBE-MCM is a semi-analytical and domain-type meshless collocation method that is based on the fundamental solution of the governing equation, which is different from the traditional virtual boundary element method. When it comes to 2D problems, the LVBE-MCM only needs to calculate the numerical integration on the circular virtual boundary. It avoids the evaluation of singular/strong singular/hypersingular integrals seen in the boundary element method. Compared to the difficulty of selecting the virtual boundary and evaluating singular integrals, the LVBE-MCM is simple and straightforward. Numerical experiments, including irregular and doubly connected domains, demonstrate that the LVBE-MCM is accurate, stable, and convergent for solving both Laplace and Helmholtz equations.

The solvability issues of counterpart Holmgren’s boundary value problem with mixed conditions for a degenerate four-dimensional second-order Gellerstedt equation Hu≡ymzktluxx+xnzktluyy+xnymtluzz+xnymzkutt=0, m,n,k,l≡const>0, are studied in the finite domain R4+, where the values of normal derivatives are set on the piecewise smooth part of the boundary and the values of the desired function are set on the remaining part of the boundary. The main results of the work are the proof of the uniqueness of the considered problem solution by using an energy integral’s method and the construction of the solution of counterpart Holmgren’s boundary value problem in explicit form by means of Green’s function method, containing the hypergeometric Lauricella’s function FA4. Using the corresponding fundamental solution for the considered generalized Gellerstedt equation of elliptic type, we construct Green’s function. In addition, formulas of differentiation, some adjacent relations, decomposition formulas, and various properties of Lauricella’s hypergeometric functions were used to establish the main results for the aforementioned problem.

We construct fundamental solutions for two-multidimensional elliptic equations. The solutions are written in explicit form via hypergeometric Gauss functions for $\lambda=0$ and via confluent Horn functions for $\lambda\neq 0$. It is proved that the fundamental solutions found possess power-type singularity $\rho^{2-n}$ as $\rho\to 0$.

The expression u′′ + aδ′u, which is consisted derivative of delta-function as a coefficient, is a formal expression and doesnʼt define operator in L2(R), because a product δ′u is not defined. So according to these reasons the study investigated the family of operators, which are approximated by the following formal expression (L(ε, a, φ)u)(x) = u′′(x) + a(ε) ⋅ (∫Ψε(y)u(y)dy ⋅ φε(x) + ∫φε(y)u(y)dy ⋅ Ψε(x), where φ ∈ D(R); φ(x) ∈ R; ∫φ(x)dx = 1; φε(x) = 1/εφ(x/ε); coefficient a(ε) could be real-valued and not null. The main results of the study were finding the limit in the family in sense of resolvent convergence. As the result, the five different kinds of limits of resolvents in this family had been received which are depended on a behavior of coefficient a(ε) and function φ properties. Therefore the formal expression u′′ + aδ′u could not put in accordance to the operator in L2(R) uniquely. This is the fundamental difference with the case u′′ + aδu expression for which the limit of resolvents doesn’t depend on choosing approximated family.

Because of the parabolic instability of the Shilov systems to change their coefficients, the definition parabolicity of Shilov for systems with time-dependent $t$ coefficients, unlike the definition parabolicity of Petrovsky, is formulated by imposing conditions on the matricant of corresponding dual by Fourier system. For parabolic systems by Petrovsky with time-dependent coefficients, these conditions are the property of a matricant, which follows directly from the definition of parabolicity. In connection with this, the question of the wealth of the class Shilov systems with time-dependent coefficients is important. A new class of linear parabolic systems with partial derivatives to the first order by the time $t$ with time-dependent coefficients is considered in this work. It covers the class by Petrovsky systems with time-dependent younger coefficients. A main part of differential expression of each such system is parabolic (by Shilov) expression with constant coefficients. The fundamental solution of the Cauchy problem for systems of this class is constructed by the Fourier transform method. Also proved their parabolicity by Shilov. Only the structure of the system and the conditions on the eigenvalues of the matrix symbol were used. First of all, this class characterizes the wealth by Shilov class of systems with time-dependents coefficients. Also it is given a general method for investigating a fundamental solution of the Cauchy problem for Shilov parabolic systems with positive genus, which is the development of the well-known method of Y.I. Zhitomirskii.

In this paper, we present an explicit formula for the approximate solution of the Cauchy problem for the matrix factorizations of the Helmholtz equation in a bounded domain on the plane. Our formula for an approximate solution also includes the construction of a family of fundamental solutions for the Helmholtz operator on the plane. This family is parameterized by function K(w) which depends on the space dimension. In this paper, based on the results of previous works, the better results can be obtained by choosing the function K(w).

In this article we discuss a method for constructing trace formulae for the heat-volume potential of the time-fractional diffusion equation to lateral surfaces of cylindrical domains and use these conditions to construct as well as to study a nonlocal initial boundary value problem for the time-fractional diffusion equation.

We give an explicit formula for the fundamental solutions of an elliptic linear partial differential operator of the second order with analytic coefficients and simple complex characteristics in an open set Ω ⊂ Rn. We prove that those fundamental solutions can be continued at least locally as multi-valued analytic functions x → E(x, y) in Cn up to the complex bicharacteristic conoid. This extension ramifies along its singular set the bicharacteristic conoid and belongs to the Nilsson class

We consider a diffusion–wave equation with fractional derivative with respect to the time variable, defined on infinite interval, and with the starting point at minus infinity. For this equation, we solve an asympotic boundary value problem without initial conditions, construct a representation of its solution, find out sufficient conditions providing solvability and solution uniqueness, and give some applications in fractional electrodynamics.

We consider an integro-differential equation in convolutions of a special kind in Banach spaces with the Fredholm operator in the main part. The article concerns with the problem of unique solvability of the Cauchy-problem for this equation in the class of distributions with left-bounded support. The research is based on the theory of fundamental operator-functions of integro-differential operators in Banach spaces. The Fredholm operator from the differential part of the equation has the complete Jordan set. The kernel of the integral part of the equation is equal to zero at the starting point, which multiplicity is determined by a maximum length of Jordan chains elements of the Fredholm operator kernel and by the order of the equation’s differential operator. Under these assumptions, we prove the theorem on the structure of the fundamental operator-function (the fundamental solution) of the equation. Based on the fundamental operator-function the generalized solution is constructed. The dependence between the generalized solution and the classical (smooth) solution is considered. The abstract results are illustrated by an example of the initial-boundary value problem for the partial integrodifferential equation. The presented research continues the papers in the field, and can be generalized to other cases of a singular operator of the leading derivative (Noetherity, spectral, sectorial or radial boundedness). The results of these investigations make it possible to explore the mathematical models of the theory of oscillations in viscoelastic media and of the theory of electric chains.

Boundary layer problems (Singular perturbation problems) more have been applied for ordinary differential equations. While this theory for partial differential equations have many applications in several fields of physics and engineering. Because of complexity of limit and boundary behavior of the solutions of partial differential equations these problems considered less than ordinary case. In this paper, a boundary layer problem including perturbed Cauchy- Riemann equation is considered with a non local boundary condition. For the given problem , some sufficient conditions will be presented so that the problem be well posed and without any boundary layer. In this case, the approximate solutions of the problem can be written same as boundary layer problems for ordinary differential equations

Fundamental solution of the model equation of anomalous diffusion with Riemann-Liouville operator is constructed. Using the properties of the integral transformation with Wright function in kernel, we give estimates for the fundamental solution. When the considered equation transformes into the diffusion equation of fractional order, constructed fundamental solution goes into the corresponding fundamental solution of the diffusion equation of fractional order. General solution of the model equation of anomalous diffusion of fractional order is constructed.

This document focuses attention on the fundamental solution of an autonomous linear retarded functional differential equation (RFDE) along with its supporting cast of actors: kernel matrix, characteristic matrix, resolvent matrix; and the Laplace transform. The fundamental solution is presented in the form of the convolutional powers of the kernel matrix in the manner of a convolutional exponential matrix function. The fundamental solution combined with a solution representation gives an exact expression in explicit form for the solution of an RFDE. Algebraic graph theory is applied to the RFDE in the form of a weighted loop-digraph to illuminate the system structure and system dynamics and to identify the strong and weak components. Examples are provided in the document to elucidate the behavior of the fundamental solution. The paper introduces fundamental solutions of other functional differential equations.

We present necessary and sufficient conditions for the nonoscillation of the fundamental solutions to a linear autonomous differential equation with distributed delay. The conditions are proposed in both the analytic and geometric forms.