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In this article, by the similarity method, self-similar solutions of higher-order equations with constant and variable coefficients are constructed. Self-similar solutions are expressed in terms of generalized hypergeometric functions. The examples show how the fundamental solutions of known equations can be expressed through the particular solutions we have constructed. [ABSTRACT FROM AUTHOR]

The Fokker–Planck equation with one spatial variable without the lowest term is considered. The diffusion coefficient is assumed to be measurable, bounded, and separated from zero. The existence of a weak fundamental solution of the Fokker–Planck equation is proved and some properties of this solution are established. Under the additional assumption that the leading coefficient is a Hölder function, we consider the first boundary-value problem in a semi-bounded domain. We assume that the right-hand side of the equation and the initial function are zero and the boundary function is continuous. We prove the solvability of this problem in the class of bounded functions. [ABSTRACT FROM AUTHOR]

An integro‐differential equation for the probability density of the generalized stochastic Ornstein–Uhlenbeck process with jump diffusion is considered for a special case of the Laplacian distribution of jumps. It is shown that for a certain ratio between the intensity of jumps and the speed of reversion, the fundamental solution can be found explicitly, as a finite sum. Alternatively, the fundamental solution can be represented as converging power series. The properties of this solution are investigated. The fundamental solution makes it possible to obtain explicit formulas for the density at each instant of time, which is important, for example, for testing numerical methods. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Semi-Analytical method for Barrier Option (SABO) applied to continuously monitored Arithmetic Asian Options with barrier. The effectiveness and utility of this approach, based on collocation Boundary Element Method (BEM), has already been tested for European style barrier options in other frameworks and in particular with reference to the similar but unusual Geometric Asian Options. European style Arithmetic Asian Options are more common among practitioners, but much more difficult to treat. Hence a deep investigation of the strategies applied to face the numerical challenges involved in their evaluation by SABO is here presented, together with several results that show the performance of the considered approach. [ABSTRACT FROM AUTHOR]

In this work we further develop a nonlocal calculus theory (initially introduced in Bellido et al. (Adv Nonlinear Anal 12:20220316, 2023)) associated with singular fractional-type operators which exhibit kernels with finite support of interactions. The applicability of the framework to nonlocal elasticity and the theory of peridynamics has attracted increased interest and motivation to study it and find connections with its classical counterpart. In particular, a critical contribution of this paper is producing vector identities, integration by part type theorems (such as the Divergence Theorem, Green identities), as well as a Helmholtz–Hodge decomposition. The estimates, together with the analysis performed along the way provide stepping stones for proving additional results in the framework, as well as pathways for numerical implementations. [ABSTRACT FROM AUTHOR]

The paper examines the problem of unique determination of the fundamental solution of a mesh analogue of Laplace's equation within the theory of discrete gravitational potential. The mesh fundamental solution of the finite-difference analogue of Laplace's equation plays a key role in reconstructing a continuously distributed source of gravitational or magnetic field from heterogeneous and different-precision data obtained at points of a certain mesh set. [ABSTRACT FROM AUTHOR]

This article considers the biharmonic equation Δ 2 u = K (x) f (u) in R N. Under suitable assumptions, the existence of positive solutions is obtained. The methods used here contain the integral operator and the Schauder fixed point theory. Since the form of fundamental solution of Δ 2 u = 0 in R N depends on N, we divide our discussions into three cases as (a) N = 2 ; (b) N = 4 ; (c) N > 2 but N ≠ 4 . The fundamental solution of Δ 2 plays an essential role in our results. [ABSTRACT FROM AUTHOR]

Let \mathcal L=-\Delta _{\mathbb {G}}+\Upsilon be a Schrödinger operator with a nonnegative potential \Upsilon belonging to the reverse Hölder class B_{Q/2}, where Q is the homogeneous dimension of the stratified Lie group \mathbb {G}. Inspired by Shen's pioneer work and Li's work, we study fundamental solutions of the Schrödinger operator \mathcal L on the stratified Lie group \mathbb {G} in this paper. By proving an exponential decreasing variant of mean value inequality, we obtain the exponential decreasing upper estimates, the local Hölder estimates and the gradient estimates of the fundamental solutions of the Schrödinger operator \mathcal L on the stratified Lie group. As two applications, we obtain the De Giorgi-Nash-Moser theory on the improved Hölder estimate for the weak solutions of the Schrödinger equation and a Liouville-type lemma for \mathcal {L}-harmonic functions on \mathbb {G}. [ABSTRACT FROM AUTHOR]

Asymptotic behavior at infinity is investigated for fundamental solutions of a hypoelliptic partial differential operator 퐏 ⁢ (i ⁢ ∂ x) = (P 1 ⁢ (i ⁢ ∂ x)) m 1 ⁢ ⋯ ⁢ (P l ⁢ (i ⁢ ∂ x)) m l with the characteristic polynomial that has real multiple zeros. Based on asymptotic expansions of fundamental solutions, asymptotic classes of functions are introduced and existence and uniqueness of solutions in those classes are established for the equation 퐏 ⁢ (i ⁢ ∂ x) ⁢ u = f in ℝ n . The obtained results imply, in particular, a new uniqueness theorem for the classical Helmholtz equation. [ABSTRACT FROM AUTHOR]

Deep operator networks (DeepONets) have demonstrated their capability of approximating nonlinear operators for initial- and boundary-value problems. One attractive feature of DeepONets is their versatility since they do not rely on prior knowledge about the solution structure of a problem and can thus be directly applied to a large class of problems. However, training the parameters of the networks may sometimes be slow. In order to improve on DeepONets for approximating the wave equation, we introduce the Green operator networks (GreenONets), which use the representation of the exact solution to the homogeneous wave equation in term of the Green's function. The performance of GreenONets and DeepONets is compared on a series of numerical experiments for homogeneous and heterogeneous media in one and two dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we present a technique for constructing generalized solutions of the Cauchy problem for abstract integro-differential equations with degeneration in Banach spaces. A generalized solution is constructed as the convolution of the fundamental operator function (fundamental solution, influence function) of the integro-differential operator of the equation with a generalized function of a special form, which involves all input data of the original problem. Based on the analysis of the representation for the generalized solution, we obtain sufficient solvability conditions for the original Cauchy problem in the class of functions of finite smoothness. Under these sufficient conditions, the generalized solution constructed turns out to be a classical solution with the required smoothness. The abstract results obtained in the paper are applied to the study of applied initial-boundary-value problems from the theory of oscillations in viscoelastic media. [ABSTRACT FROM AUTHOR]

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 present several methods for constructing the asymptotics of the fundamental solution of Fokker–Planck–Kolmogorov-type parabolic equations with a small parameter both for small and finite positive times. [ABSTRACT FROM AUTHOR]

The fundamental solution of generalized magneto thermo viscoelasticity (MTVE) with two relaxation times for perfect isotropic conduction is obtained. An infinitely long circular cylinder application has been studied. The solution is obtained by the potential method. In the transform domain Laplace and Hankel are used to obtain the solution. The inverse process is performed by asymptotic expansions appropriate for limited time values. Numerical results are displayed and graphically explained for the temperature and stress distributions. A comparison is made with the results obtained in the presence and absence of a magnetic field, for different values of time and for two theories (MTE and MTVE). [ABSTRACT FROM AUTHOR]

In [6, 7], it was shown that mixed problems can be both ill-posed for Petrovsky well-conditioned equations and well-posed for ill-conditioned equations. In the present paper we study the existence and uniqueness of the solution of a mixed problem for a class of equations with complex-valued coefficients that behave as parabolic ones, despite the fact that they can change over "time" from parabolic type to Schrödinger type, or even to antiparabolic type. [ABSTRACT FROM AUTHOR]

In this article, we study the asymptotic behavior and decay of the solution of the fully discrete heat problem. We show basic properties of its solutions, such as the mass conservation principle and their moments, and we compare them to the known ones for the continuous analogue problems. We present the fundamental solution, which is given in terms of spherical harmonics, and we state pointwise and ℓ p estimates for that. Such considerations allow to prove decay and large-time behavior results for the solutions of the fully discrete heat problem, giving the corresponding rates of convergence on ℓ p spaces. [ABSTRACT FROM AUTHOR]

A multipole expansion of the fundamental solution of the fractional power of the Laplace operator is constructed in terms of the Gegenbauer polynomials. Based on the decomposition constructed and the idea of the fast multipole method, we propose a numerical algorithm for solving the fractional differential generalization of the Poisson equation in the two-dimensional and three-dimensional spaces. [ABSTRACT FROM AUTHOR]

This paper is concerned with the approximate controllability of a class of semi-linear neutral integro-differential equation with nonlocal condition. The basic tools of this study are the theory of resolvent operators of linear neutral integro-differential equation and fractional powers. The fundamental solution of linear neutral integro-differential equations is also constructed to deal with the non-uniform boundedness of the extra linear term. Sufficient conditions of approximate controllability are then obtained through the resolvent condition. Finally, an example is provided to illustrate the applications of the obtained results. [ABSTRACT FROM AUTHOR]

An analytical solution of unsaturated poroelastic transversely isotropic (TI) half‐space under surface time‐harmonic load with consideration of compressibility, viscous and inertial coupling of soil skeleton, pore‐water and pore‐air, as well as the capillary pressure is investigated in this paper. Firstly, the governing equations of unsaturated TI soil are established in cylindrical coordinate system. Then, the displacement function and operator theory are introduced to decouple the general solution. Secondly, Fourier expansion and Hankel transform are applied to the circumferential and radial coordinates, and the integral general solutions of soil skeleton displacement, pore‐water displacement and pressure, pore‐air displacement and pressure, and the total stress components are derived in the transformed domain. Combined with boundary conditions and continuity conditions, the fundamental solution of unsaturated TI half‐space is obtained. Finally, the proposed solution is verified by benchmarking with two existing solutions. The influence of the TI parameters, vibration frequency and the degree of saturation on the dynamic response of unsaturated poroelastic half‐space is investigated. [ABSTRACT FROM AUTHOR]

We consider the linear field-road system, a model for fast diffusion channels in population dynamics and ecology. This system takes the form of a system of PDEs set on domains of different dimensions, with exchange boundary conditions. Despite the intricate geometry of the problem, we provide an explicit expression for its fundamental solution and for the solution to the associated Cauchy problem. The main tool is a Fourier (on the road variable)/Laplace (on time) transform. In addition, we derive estimates for the decay rate of the L ∞ norm of these solutions. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of anisotropic parabolic equation related to the p i -Laplacian with a source term f (u) . If f (u) = 0 , then the fundamental solution of the equation is constructed. If there are some restrictions on the growth order of u in the source term, the initial energy E (0) is positive and has a super boundedness, which depends on the Sobolev imbedding index, then the local solution may blow up in finite time. [ABSTRACT FROM AUTHOR]

This study considers the wave phenomena and fundamental solution in swelling porous elastic solids consisting of a mixture of solid, liquid, and gas. There exist three dilatational and two transversal waves which vibrate at distinct speeds. The basic characteristics of waves (phase velocity and attenuation coefficient) are computed numerically and displayed in the form of graphs. Also, a fundamental solution is explored for steady oscillation and some preliminary features are explored in this model. The obtained results for both waves are analyzed with and without swelling porous elastic solid. This new model enables us to improve the efficacy of swelling porous elastic solid which involves solid and fluid and find it applicable in exploration industries and to investigate various wave propagation problems. [ABSTRACT FROM AUTHOR]

In this article, we study the existence of mild solutions and the approximate controllability for a class of systems governed by neutral equations of second-order with infinite delay in infinite-dimensional Hilbert spaces. The mild solution and approximate controllability are achieved by constructing the fundamental solution for the associated linear equation and assuming that the linear system is approximately controllable. The discussion is based on the fundamental solution theory and Rothe's fixed point theorem. In addition, an example is given to illustrate our main conclusion. [ABSTRACT FROM AUTHOR]

The Laplace equation in three-dimensional Euclidean space is R -separable in bi-cyclide coordinates leading to harmonic functions expressed in terms of Lamé–Wangerin functions called internal and external bi-cyclide harmonics. An expansion for a fundamental solution of Laplace's equation in products of internal and external bi-cyclide harmonics is derived. In limiting cases this expansion reduces to known expansions in bi-spherical and prolate spheroidal coordinates. [ABSTRACT FROM AUTHOR]

We propose a new method to solve the initial-boundary value problem for hyperbolic-dissipative partial differential equations (PDEs) based on the spirit of LY algorithm [T.-P. Liu and S.-H. Yu, Dirichlet–Neumann kernel for hyperbolic-dissipative system in half-space, Bull. Inst. Math. Acad. Sin. 7 (2012) 477–543]. The new method can handle more general domains than that of LYs'. We convert the evolutionary PDEs into the elliptic PDEs by the Laplace transformation. Using the Laplace transformation of the fundamental solutions of the evolutionary PDEs and the image method, we can construct Green's functions for the corresponding elliptic PDEs. Finally, we obtain Green's functions for the evolutionary PDEs by inverting the Laplace transformation. As a consequence, we establish Green's functions for some basic PDEs such as the heat equation, the wave equation and the damped wave equation, in a half space and a quarter plane with various boundary conditions. On the other hand, the structure of hyperbolic-dissipative PDEs means its fundamental solution is non-symestric and hence the image method does not work. We utilize the idea of Laplace wave train introduced by Liu and Yu in [Navier–Stokes equations in gas dynamics: Greens function, singularity and well-posedness, Comm. Pure Appl. Math. 75(2) (2022) 223–348] to generalize the image method. Combining this with the notions of Rayleigh surface wave operators introduced in [S. J. Deng, W. K. Wang and S.-H. Yu, Green's functions of wave equations in ℝ + n × ℝ + , Arch. Ration. Mech. Anal. 216 (2015) 881–903], we are able to obtain the complete representations of Green's functions for the convection-diffusion equation and the drifted wave equation in a half space with various boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, we study the heat equation with non-smooth perturbation caused by a delay applied to the time variable through a piecewise constant function, but keeping the time positive, which does not merit changing the phase space as usually occurs when the delay makes the time negative. To do so, we first prove the existence of solutions using Fourier's transform. Next, we prove the uniqueness of solutions by applying the maximum principle method. After that, we study the stability of these solutions. Finally, we propose some problems that can be solved with this or similar techniques. [ABSTRACT FROM AUTHOR]

This is a continuation of [5] which is concerned about the regularity property of stochastic convolutions for abstract linear stochastic retarded differential equations with unbounded operators on delay terms. In this work, we improve and generalize the main results in [5] by considering those delay operators which may have the same order as the infinitesimal generator of the system under consideration. To this end, we need restrict the weight function of distributed delay term to be Hölder continuous type in this system. [ABSTRACT FROM AUTHOR]

In the theory of partial differential equations, the fundamental solutions have undoubtedly occupied serious attention, as they have a critical role to play in a wide range of problems under mathematical physics and continuum mechanics. The present work explores the Galerkin-type representation of the field equations based on the recently developed Moore–Gibson–Thompson thermoelasticity theory. With the help of the Laplace transform technique, we then establish the short-time approximated fundamental solutions for two separate cases: concentrated body force and concentrated heat source. For the time domain, the fundamental solutions are obtained by taking inverse Laplace transforms of displacement and temperature components. Lastly, we obtain the fundamental solution for steady vibration in terms of elementary functions. [ABSTRACT FROM AUTHOR]

In this study, the singular boundary method (SBM) is employed for the simulation of nonlinear Klein-Gordon equation with initial and Dirichlet-type boundary conditions. The ?-weighted and Houbolt finite difference method is used to discretize the time derivatives. Then the original equations are split into a system of partial differential equations. A splitting scheme is applied to split the solution of the inhomogeneous governing equation into homogeneous solution and particular solution. To solve this system, the method of particular solution in combination with the singular boundary method is used for particular solution and homogeneous solution, respectively. Finally, several numerical examples are provided and compared with the exact analytical solutions to show the accuracy and efficiency of method in comparison with other existing methods. [ABSTRACT FROM AUTHOR]

The hybrid fundamental solution-based finite element method was proposed for heat conduction problems. Firstly, 2 independent fields were assumed: the intra-element temperature field approximated through the linear combination of fundamental solutions, and the auxiliary frame temperature field in the same form as that in the conventional finite element method. Then, a modified variational functional was employed to link the 2 independent fields and derive the finite element formulation. However, the accuracy of the method is strongly dependent on the distribution and the number of source points. The source points were usually placed on 2 fictitious boundaries outside the element: one is similar to the element shape, the other is a circular one. Furthermore, the dual fictitious boundary scheme was proposed for comparison with the above fictitious boundaries. With different configurations of source points, 2 typical numerical examples were given to demonstrate the validity and the insensitivity to mesh distortion of the proposed method. [ABSTRACT FROM AUTHOR]

We construct the fundamental solution of second order parabolic equations in non-divergence form under the assumption that the coefficients are of Dini mean oscillation in the spatial variables. We also prove that the fundamental solution satisfies a sub-Gaussian estimate. In the case when the coefficients are Dini continuous in the spatial variables and measurable in the time variable, we establish the Gaussian bounds for the fundamental solutions. We present a method that works equally for second order parabolic systems in non-divergence form. [ABSTRACT FROM AUTHOR]

We derive a fundamental solution E to a space-fractional diffusion problem on the half-line. The equation involves the Caputo derivative. We establish properties of E as well as formulas for solutions to the Dirichlet and fixed slope problems in terms of convolution of E with data. We also study integrability of derivatives of solutions given in this way. We present conditions, which are sufficient for uniqueness of solutions. Finally, we show the infinite speed of signal propagation. [ABSTRACT FROM AUTHOR]

In this work, we establish the theory of fundamental solution, including the existence and some essential properties, for neutral linear integro-differential equations with finite delay. In this way, we are able to represent the mild solutions of semi-linear neutral retarded integro-differential equations by fundamental solutions through the Laplace transformation method. Then, as applications, we investigate the problems of approximate controllability, optimal control and time optimal control for this kind of integro-differential evolution systems based on the theory of fundamental solution, resolvent operators and fractional powers, and sufficient conditions for these control problems are obtained, respectively. Finally, an example is presented to illustrate the achieved results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper deals with the approximate controllability problem for a class of linear neutral evolution systems with infinite delay in Banach spaces. Since the concerned system involves additional linear terms, the fundamental solution theory is used to describe the mild solutions. Based on the spectral analysis and the adjoint system, sufficient conditions for the approximate controllability of the concerned system are established. The rank condition is also shown when the control space is finite-dimensional. A practical example is given to show the application of the obtained results. [ABSTRACT FROM AUTHOR]

Three nonlinear equations for ion acoustic and drift waves in a plasma are derived. The fundamental solution of the common linear part of the resulting nonlinear equations is constructed, and its smoothness properties are studied. Next, the second Green formula in a bounded domain is constructed, which is then used to derive the third Green formula in a bounded domain. Finally, two variants of the third Green formula in the entire space are constructed in a certain class of functions. [ABSTRACT FROM AUTHOR]

In this paper, a novel Burton-Miller-type singular boundary method (BM-SBM) formulation is proposed for the acoustic design sensitivity analysis, with the help of the direct differentiation method. The Burton-Miller formulation is employed to overcome the fictitious frequency problem in the numerical solutions of exterior acoustic problems. The simple empirical formulas are used to estimate the origin intensity factors in the BM-SBM. Compared with the traditional methods, the present scheme is an accurate and semi-analytical approach with the merits of truly meshless, integration-free, mathematical simplicity, and easy-to-program. As a boundary-type method based on the fundamental solution, furthermore, the BM-SBM is straightforward for addressing the exterior Helmholtz problems encountered in the acoustic design sensitivity analysis. Numerical experiments, including 2D and 3D geometries with Neumann boundary conditions, indicate that the proposed methodology is an accurate, stable, and convergent numerical approach for the acoustic design sensitivity analysis. [ABSTRACT FROM AUTHOR]

The Flamant problem of a cubic quasicrystal half-plane is solved when its surface is loaded by normal and tangential concentrated forces. Its solution serves as a fundamental solution since its superposition forms other solutions. The Flamant problem is converted to an associated boundary value problem. The Fourier transform method is applied to solve the Flamant problem. Explicit expressions for the fundamental solution of the phonon and phason stresses are obtained. A comparison between the fundamental solutions of the Flamant problem for cubic quasicrystals and conventional cubic crystals is made. The influence of the presence of phason field on the phonon stress and deformation is shown in graph. [ABSTRACT FROM AUTHOR]