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One of the most common hypotheses on the theory of non-smooth dynamical systems is a regular surface as switching manifold, at which case there is at least well-defined and established Filippov dynamics. However, systems with singular switching manifolds still lack such well-established dynamics, although present in many relevant models of phenomena where multiple switches or multiple abrupt changes occur. At this work, we leverage a methodology that, through blow-ups and singular perturbation, allows the extension of Filippov dynamics to the singular case. Specifically, tridimensional systems whose switching manifold consists of an algebraic manifold with transversal self-intersection are considered. This configuration, known as double discontinuity, represents systems with two switches and whose singular part consists of a straight line, where ordinary Filippov dynamics is not directly applicable. For the general, non-linear case, beyond defining the so-called fundamental dynamics over the singular part, general theorems on its qualitative behavior are provided. For the affine case, however, theorems fully describing the fundamental dynamics are obtained. Finally, this fine-grained control over the dynamics is leveraged to derive Peixoto like theorems characterizing semi-local structural stability., 46 pages, 16 figures; new title, abstract, introduction and conclusion; improved statements and argumentation

In this work, the structure of uninorms not locally internal on the boundary is discussed. We first study a conjunctive uninorm U with neutral element e, where the boundary function U ( 1 , ⋅ ) of U only has two points of discontinuity and one of them is 0 or e. We partially obtain the structure of the conjunctive uninorm U. Its complete structure, under some additional conditions, is achieved as well. For the structure of a disjunctive uninorm U, whose boundary function U ( 0 , ⋅ ) only has two points of discontinuity and one of them is 1 or e, we obtain similar results.

Method of asymptotic partial decomposition of a domain (MAPDD) proposed and justified earlier for thin domains (rod structures, tube structures consisting of a set of thin cylinders) generates some specific interface conditions between three-dimensional and one-dimensional parts. In the case of the heat equation these conditions ensure the continuity of the solution and continuity in average of the normal flux. However for some computational reasons the strong continuity condition for the solution may be undesirable. It may be preferable to replace it by more flexible condition of continuity in average over the interface cross section. In the present paper we introduce and justify this alternative junction condition allowing such discontinuity of the solution of both stationary and non-stationary problems. The closeness of solutions of these hybrid dimension problems to the solution of the fully three-dimensional setting is proved. At the discrete level, finite volume schemes are considered, an error estimate is established. The new version of the MAPDD is compared to the classical one via numerical tests. It shows better stability of the new scheme.

In this paper, we consider the inverse scattering problem for Sturm-Liouville operator with discontinuity conditions at some point on the positive half line. The scattering data of this boundary value problem is examined. The resolvent operator is constructed and the expansion formula with respect to the eigenfunctions of this boundary value problem is obtained. The main equation or modified Marchenko equation of the inverse scattering problem is derived and an algorithm of the construction of the potential function according to scattering data of this boundary value problem is given.

We study the ill-posedness issue for the compressible viscous heat-conductive flows in two dimensions. In the scaling invariant spaces, negative regularity of the temperature causes an essential problem for the well-posedness, and the ill-posedness is obtained for all integrability indices except for the $$L^1$$ case.

This paper is contributed to the structural stability of multi-wave configurations to Cauchy problem for the compressible non-isentropic Euler system with adiabatic exponent γ ∈ (1, 3]. Given some small BV perturbations of the initial state, the author employs a modified wave front tracking method, constructs a new Glimm functional, and proves its monotone decreasing based on the possible local wave interaction estimates, then establishes the global stability of the multi-wave configurations, consisting of a strong 1-shock wave, a strong 2-contact discontinuity, and a strong 3-shock wave, without restrictions on their strengths.

The interface crack problems in the two-dimensional (2D) decagonal quasi-crystal (QC) coating are theoretically and numerically investigated with a displacement discontinuity method. The 2D general solution is obtained based on the potential theory. An analogy method is proposed based on the relationship between the general solutions for 2D decagonal and one-dimensional (1D) hexagonal QCs. According to the analogy method, the fundamental solutions of concentrated point phonon displacement discontinuities are obtained on the interface. By using the superposition principle, the hypersingular boundary integral-differential equations in terms of displacement discontinuities are determined for a line interface crack. Further, Green’s functions are found for uniform displacement discontinuities on a line element. The oscillatory singularity near a crack tip is eliminated by adopting the Gaussian distribution to approximate the delta function. The stress intensity factors (SIFs) with ordinary singularity and the energy release rate (ERR) are derived. Finally, a boundary element method is put forward to investigate the effects of different factors on the fracture.

A piecewise-smooth second-order singularly perturbed differential equation whose right-hand side is a nonlinear function with a discontinuity on some curve is investigated. This is a new class of problems in the case where the degenerate equation has a multiple root on the left-hand side of the curve which separates the domain and an isolated root on the right-hand side of that curve. The asymptotics of a solution with an internal layer near a point on the discontinuous curve and the transition point is constructed. The method to construct the internal layer function is proposed. The behavior of the solution in the internal layer consisting of four zones essentially differs from the case of isolated roots. For sufficiently small parameter values, the existence of a smooth solution with an internal layer from the multiple root of the degenerate equation to the isolated root in the neighborhood of a point on the discontinuous curve is proved. The method can be shown to be effective in the given example.

This paper deals with the discontinuous dynamic behaviors of a class of three-degree-of-freedom friction and collision system with symmetric bilateral rigid constraints by using the flow switching theory of discontinuous dynamical systems. The model takes into account the inequality of static and dynamic friction forces, which is more common in practice. The external excitation of the object contains a negative feedback, which can change with the variation of the velocity of the object, so that the system can adapt to different external environment. And the connection between the objects adopts nonlinear springs and viscous dampers in order to achieve a better vibration reduction effect. The model can be applied to mechanical equipment such as shock absorbers etc. According to the discontinuity caused by friction and collision, the dynamic domain and boundary of the object’s motion are defined in phase space, which requires the introduction of absolute coordinates and relative coordinates respectively to discuss the motion between objects. The vector field for the respective domain in the system is given, which can control the motion of object in each domain. The corresponding normal vector on the discontinuous boundary is introduced to determine the positive direction of the flow. The sufficient and necessary conditions for the switching motion of such dynamical system are given based on $$\mathrm{G}$$ -function and its higher order. These switching conditions can be used to predict or even control the motion of the object on the separation boundary. In addition, several typical motions, such as sliding, stick, grazing, impact motions and periodic motion, are numerically simulated to better explain the complexity of the switching motion in the discontinuous dynamic system. Finally, the stick bifurcation scenarios for excitation frequency or amplitude are presented. The research results obtained can provide a theoretical basis for better use or control of friction and collision in practical applications.

Dynamics of nonlinear oscillators with discontinuous nonlinearities subjected to harmonic and random excitations is investigated. Impact, dry friction and Hertzian type compliant contact nonlinearities are considered. Stochastic bifurcations like the P-bifurcation and D-bifurcation are discussed. P and D bifurcations are characterized respectively by the joint probability density functions (jpdf) of the response and the largest Lyapunov exponent. The jpdf is obtained by the solution of the corresponding Fokker–Planck equation by the finite element and path integral methods. The results are verified by Monte Carlo simulation methods. Adaptive time step integration procedure (ATSP) is adopted which accurately determines the point of discontinuity. A bisection method and a Brownian tree approach are used in this process and direct the solution along the correct Brownian path. Numerical results are also obtained using non-smooth coordinate transformations like the Zhuvarlev and Ivanov transformations converting the discontinuous systems to equivalent smooth systems and compared with the results of the ATSP. The Filippov convex transformation is used in the case of the dry friction nonlinearity in the integration near the discontinuity. The results are discussed with respect to some examples like the Duffing and Van der Pol oscillators with impact and dry friction.

We investigate the emergence of nonphysical solutions to the steady-state Wigner transport equation on finite-sized simulation domains with inflow boundary conditions. We find that inflow boundary conditions are generally valid, but the wave number uncertainty of injected wave packets has a lower bound that can be significantly higher than usually assumed. Large values of the so-called quantum evolution term (which captures spatial nonlocality in the Wigner transport equation) near simulation-domain boundaries are the cause of spurious reflections, the often-reported discontinuity of the Wigner function at zero wave vector, and negative probabilities. We offer a simple relationship between the lower bound of the wave number uncertainty and simulation parameters that will ensure physical results with inflow boundary conditions.

Abstract The determination of 3D particle shapes from the one-dimensional data on small-angle scattering from macromolecule solutions is ambiguous. In addition, due to the poor conditionality of the inverse problem, the determination is unstable and depends on the search algorithm parameters. Consequently, in practice, it is necessary not only to estimate the degree of stability of the solution of the problem but also to select the parameters of both the search method and the model itself. A search tactic is considered which consists in sequential determination of a set of models of particle shape represented in the form of a structure consisting of small close-packed spherical beads. The set of solutions is obtained by varying the relative contributions of objective function terms: the criterion for the deviation of the model scattering curve from the experimental one and the penalties for the looseness and discontinuity of the body structure, and the deviation of the average number of bead contacts from a specified value. Examples of the solution of model problems and the determination of molecular shape from the measurements available in the SASBDB bank of structures are presented.

The dynamics of a hysteretic relay oscillator with harmonic forcing is investigated. Periodic excitation of the system results in periodic, quasi-periodic, chaotic and unbounded behavior. An explicit Poincaré map is constructed with an implicit constraint on the switching time. The stability of the fixed points of the Poincaré map corresponding to period-one solutions is investigated. By varying the forcing parameters, we observed a saddle-center and a pitchfork bifurcation of two centers and a saddle-type fixed point. The global dynamics of the system exhibits discontinuity induced bifurcations of the fixed points.

We extend the applicability of the popular interior penalty discontinuous Galerkin method discretizing advection-diffusion-reaction problems to meshes comprising extremely general, essentially arbitrarily-shaped element shapes. In particular, our analysis allows for curved element shapes, without the use of non-linear elemental maps. The feasibility of the method relies on the definition of a suitable choice of the discontinuity penalization, which turns out to be explicitly dependent on the particular element shape, but essentially independent on small shape variations. This is achieved upon proving extensions of classical trace and Markov-type inverse estimates to arbitrary element shapes. A further new H 1 − L 2 H^1-L_2 -type inverse estimate on essentially arbitrary element shapes enables the proof of inf-sup stability of the method in a streamline-diffusion-like norm. These inverse estimates may be of independent interest. A priori error bounds for the resulting method are given under very mild structural assumptions restricting the magnitude of the local curvature of element boundaries. Numerical experiments are also presented, indicating the practicality of the proposed approach.

In layered media, the solution of the Maxwell equations has discontinuity of the derivative or the function itself at media interfaces. For the first time, finite-difference schemes providing convergence for discontinuous solutions across straight media interfaces are proposed for the one-dimensional formulation of the Maxwell equations. These are bicompact conservative schemes. They are two-point and layer boundaries are taken as mesh nodes. The scheme explicitly accounts for physically correct interface conditions at media interfaces. We propose an essentially new technique which accounts for medium dispersion. All these features provide the second order of accuracy even on discontinuous solutions. Calculation examples, which illustrate these results, are given. The proposed method is verified by comparison with a previously performed experiment on propagation of normally incident plane wave on one-dimensional photonic crystal. Calculated spectrum agrees well with the measured one within experimental error of 2-5%.

In this paper, we extend the classical atmospheric Ekman equations to discontinuous differential equations of Ekman flows by considering the eddy viscosity with piecewise form. Local and global analysis for the corresponding discontinuous ODEs are studied respectively. In the section of local analysis, all the possible crossings of solutions are presented over the certain discontinuity linear subspaces associated with a curve. In the section of global analysis, crossings of solutions, bounded and unbounded solutions on dynamics are obtained.

Since proposed, the self-similarity variables based genuinely multidimensional Riemann solver is attracting more attentions due to its high resolution in multidimensional complex flows. However, it needs numerous logical operations in supersonic cases, which limit the method’s applicability in engineering problems greatly. In order to overcome this defect, a hybrid multidimensional Riemann solver, called HMTHS (Hybrid of MulTv and multidimensional HLL scheme based on Self-similar structures), is proposed. It simulates the strongly interacting zone by adopting the MHLLES (Multidimensional Harten-Lax-van Leer-Eifeldt scheme based on Self-similar structures) scheme at subsonic speeds, which is with a high resolution by considering the second moment in the similarity variables. Also, it adopts the MULTV (Multidimensional Toro and Vasquez) scheme, which is with a high resolution in capturing discontinuities, to simulate the flux at supersonic speeds. Systematic numerical experiments, including both one-dimensional cases and two-dimensional cases, are conducted. One-dimensional moving contact discontinuity case and sod shock tube case suggest that HMTHS can accurately capture one-dimensional expansion waves, shock waves, and linear contact discontinuities. Two-dimensional cases, such as the double Mach reflection case, the supersonic shock / boundary layer interaction case, the hypersonic flow over the cylinder case, and the hypersonic viscous flow over the double-ellipsoid case, indicate that the HMTHS scheme is with a high resolution in simulating multidimensional complex flows. Therefore, it is promising to be widely applied in both scholar and engineering areas.

A new hybrid numerical method for the solution of nonlinear magnetostatic problems in accelerator magnets is proposed. The methodology combines the Fragile Points Method (FPM) and the Boundary Element Method (BEM). The FPM is a Galerkin-type meshless method employing for field approximation simple, local, and discontinuous point – based interpolation functions. Because of the discontinuity of these functions, the FPM can be considered as a meshless discontinuous Galerkin formulation where numerical flux corrections are employed for the treatment of local discontinuity inconsistencies. The FPM possesses the advantages of a mesh-free method, evaluates with high accuracy field gradients, and treats nonlinear magnetostatic problems by utilizing, for the same problem, fewer nodal points than the Finite Element Method (FEM). In the present work, the nonlinear ferromagnetic material of an accelerator magnet is treated by the FPM, while the BEM is employed for the infinitely extended, surrounding air space. The proposed hybrid scheme is based on the scalar potential formulation of Mayergouz et.al (1987), also used in the BEM/BEM and FEM/BEM schemes of Rodopoulos et al. (2019, 2020). The applicability of the method is demonstrated with the solution of representative 2-D magnetostatic problems and the obtained numerical results are compared to those provided by the FEM/BEM scheme of Rodopoulos et al. (2020), as well as by the commercial FEM package ANSYS. Finally, the magnetic field utilized for stable bending of the particles’ trajectory in a 16 Tesla dipole magnet design for the Future Circular Collider (FCC) project of CERN is accurately evaluated with the aid of the proposed here FPM/BEM scheme.

Convolution and indirect meshless techniques are presented to solve the free vibration problem of irregular composite plates surrounded by a nonlinear elastic matrix with three parameters. Linear and parabolic foundations are examined. The displacement field is derived based on a first order transverse shear theory. In vibration study, the governing equation is reduced for the solution of the nonlinear eigenvalue problem and is solved by a direct iterative method. The influences of variable thickness, foundation stiffness, and supporting conditions on frequency values and mode shapes are investigated. The accuracy and efficiency of the two methods are carried out in two ways, firstly by comparison with exact, classical quadrature, and Rayleigh-Ritz approaches, and secondly by calculated the CPU time. Furthermore, some detailed results are presented to explore the effects of elastic and geometric properties of the vibrated irregular plate. From the results, it is found that the indirect moving least squares quadrature technique is an efficient scheme for vibration plates containing material discontinuity.

This work is devoted to the construction of weakly nonlinear, highly oscillating, current vortex sheet solutions to the system of ideal incompressible magnetohydrodynamics. Current vortex sheets are piecewise smooth solutions that satisfy suitable jump conditions on the (free) discontinuity surface. In this article, we complete an earlier work by Alì and Hunter (Quart. Appl. Math. 61(3) (2003) 451–474) and construct approximate solutions at any arbitrarily large order of accuracy to the three-dimensional free boundary problem when the initial discontinuity displays high frequency oscillations. As evidenced in earlier works, high frequency oscillations of the current vortex sheet give rise to ‘surface waves’ on either side of the sheet. Such waves decay exponentially in the normal direction to the current vortex sheet and, in the weakly nonlinear regime which we consider here, their leading amplitude is governed by a nonlocal Hamilton–Jacobi type equation known as the ‘HIZ equation’ (standing for Hamilton–Il’insky–Zabolotskaya (J. Acoust. Soc. Amer. 97(2) (1995) 891–897)) in the context of Rayleigh waves in elastodynamics. The main achievement of our work is to develop a systematic approach for constructing arbitrarily many correctors to the leading amplitude. We exhibit necessary and sufficient solvability conditions for the corrector equations that need to be solved iteratively. The verification of these solvability conditions is based on mere algebra and arguments of combinatorial analysis, namely a Leibniz type formula which we have not been able to find in the literature. The construction of arbitrarily many correctors enables us to produce infinitely accurate approximate solutions to the current vortex sheet equations. Eventually, we show that the rectification phenomenon exhibited by Marcou in the context of Rayleigh waves (C. R. Math. Acad. Sci. Paris 349(23–24) (2011) 1239–1244) does not arise in the same way for the current vortex sheet problem.

A novel method for obtaining complete interaction diagrams of arbitrarily shaped reinforced concrete and composite sections is presented. A mathematical optimization is used to apply the force criterion that maximizes the biaxial bending capacity of a section. A strain criterion for limiting the ultimate strain can be selectively applied through the bound constraint of the optimization problem. Three of the strain parameters for defining the deformation of the section are not fixed, to determine the true ultimate strength at the specified locations in the interaction diagram. A sequential quadratic programming algorithm is used to find the solution. A rotated atan2 function is introduced to overcome the discontinuity in the objective and constraint functions. A tilted axis is used for unsymmetrical sections. Interaction diagrams are obtained for various examples and demonstrate the robustness and versatility of the proposed method. The force criterion provides accurate interaction diagrams for the section, for both normal and high-strength materials.

The boundary integral equations (BIE) and displacement discontinuity methods (DDM) are formulated for solution of smooth curved crack problems within 2D elasticity and Reissner's plate bending theory. The equivalence between the direct boundary element method and the indirect boundary element method is shown here. The Chebyshev polynomials of the second kind are employed to solve the integral equations numerically. This enables determination of the stress intensity factors at the crack tips directly5 by the coefficients of Chebyshev polynomials. In the DDM, the coefficients of stress influence for the constant displacement discontinuity element are derived in closed form for the Reissner's plate bending problem. The degree of convergence of BIE is demonstrated with increasing numbers of the collocation points. Comparison is made with the analytical solutions from the stress intensity factor handbook for an embedded circular crack in a flat plate under tensile force. High accuracy of the presented numerical solutions makes possible to use them as benchmarks to evaluate the degree of accuracy for any numerical algorithms.

We prove the stability of a planar contact discontinuity without shear, a family of special discontinuous solutions for the three-dimensional full Euler system, in the class of vanishing dissipation limits of the corresponding Navier–Stokes–Fourier system. We also show that solutions of the Navier–Stokes–Fourier system converge to the planar contact discontinuity when the initial datum converges to the contact discontinuity itself. This implies the uniqueness of the planar contact discontinuity in the class that we are considering. Our results give an answer to the open question, whether the planar contact discontinuity is unique for the multi-D compressible Euler system. Our proof is based on the relative entropy method, together with the theory of a-contraction up to a shift and our new observations on the planar contact discontinuity.

In composite materials, the stress may be arbitrarily large in the narrow region between two close-to-touching hard inclusions. The stress is represented by the gradient of a solution to the Lame system of linear elasticity. The aim of this paper is to establish lower bounds of the gradients of solutions of the Lame system with partially infinite coefficients as the distance between the surfaces of discontinuity of the coefficients of the system tends to zero. Combining it with the pointwise upper bounds obtained in our previous work, the optimality of the blow-up rate of gradients is proved for inclusions with arbitrary shape in dimensions two and three. The key to show this is that we find a blow-up factor, a linear functional of the boundary data, to determine whether the blow-up will occur or not.

Some direct numerical method is presented for solving the inverse coefficient problem for an elliptic equation with piecewise constant coefficients. The discontinuity points of the coefficients are assumed known. The algorithm is based on the theory of spectral problems of linear algebra and the application of finite-difference methods for solving elliptic equations. The values (measurements) of the solution at the discontinuity points of the coefficients are used as additional information. In the case of unperturbed additional information, the coefficients are reconstructed precisely.

In this paper, we present a deep learning framework for solving two-dimensional elliptic equations with singular forces on arbitrary domains. This work follows the ideas of the physical-inform neural networks to approximate the solutions and the immersed boundary method to deal with the singularity on an interface. Numerical simulations of elliptic equations with regular solutions are initially analyzed in order to deeply investigate the performance of such methods on rectangular and irregular domains. We study the deep neural network solutions for different number of training and collocation points as well as different neural network architectures. The accuracy is also compared with standard schemes based on finite differences. In the case of singular forces, the analytical solution is continuous but the normal derivative on the interface has a discontinuity. This discontinuity is incorporated into the equations as a source term with a delta function which is approximated using a Peskin's approach. The performance of the proposed method is analyzed for different interface shapes and domains. Results demonstrate that the immersed boundary neural network can approximate accurately the analytical solution for elliptic problems with and without singularity.

A singularly perturbed parabolic problem of convection-diffusion type with a discontinuous initial condition is examined. A particular complimentary error function is identified which matches the discontinuity in the initial condition. The difference between this analytical function and the solution of the parabolic problem is approximated numerically. A coordinate transformation is used so that a layer-adapted mesh can be aligned to the interior layer present in the solution. Numerical analysis is presented for the associated numerical method, which establishes that the numerical method is a parameter-uniform numerical method. Numerical results are presented to illustrate the pointwise error bounds established in the paper.

In this paper, we consider inverse problems for the Dirac operator with complex-valued weight and the jump conditions inside the interval. We prove that the potential on the whole interval can be uniquely determined by the Weyl-type function or two spectra.

An inverse problem is considered for a system of semilinear elliptic equations that simulate radiative heat transfer with Fresnel matching conditions on the surfaces of discontinuity of the refractive index. The problem consists in finding the right-hand side of the heat equation, which is a linear combination of given functionals from their specified values on the solution. The solvability of the inverse problem is proved without restrictions on smallness. A sufficient condition for the uniqueness of the solution is presented.

There are some special phenomena like longitudinal end effect, saturation effect, skin effect, edge effect, and half-filled slots which make the single-sided linear induction motor (SSLIM) analysis more complicated compared to its rotary counterparts. Finite element method (FEM) can be the best choice for complete modeling of SSLIM considering all phenomena. But it is not possible to include the non-linear magnetization curve of iron into the time-harmonic FEM analysis. Thus, this article proposes a 2-D time-harmonic FEM considering all phenomena as well as the non-linear magnetization characteristics of iron using appropriate approximation. In order to overcome the Lorentz method calculation error of vertical force caused by inherent discontinuity of magnetic field intensity at the boundaries of elements, a combined Lorentz–Maxwell method is presented. Validity of the methods and final results is proved by comparing the results with analytical results and experimental measurements.

This paper discusses the well posedness of an initial value problem describing the motion of a Bingham fluid in a basin with a degenerate bottom topography. A physical interpretation of such motion is discussed. The system governing such motion is obtained from the Shallow Water-Bingham models in the regime where the Froude number degenerates, i.e taking the limit of such equations as the Froude number tends to zero. Since we are considering equations with degenerate coefficients, then we shall work with weighted Sobolev spaces in order to establish the existence of a weak solution. In order to overcome the difficulty of the discontinuity in Bingham's constitutive law, we follow a similar approach to that introduced in [G. DUVAUT and J.-L. LIONS, Springer-Verlag, 1976]. We study also the behavior of this solution when the yield limit vanishes. Finally, a numerical scheme for the system in 1D is furnished.

This paper deals with hyperbolic conservation laws exhibiting a flux discontinuity at the origin and which does not admit a weak solution satisfying the Rankine–Hugoniot jump condition. We therefore seek unbounded solutions in the form of shadow waves supported by at the origin. The shadow waves are defined as nets of piecewise constant functions approximating a shock wave to which we add a delta function and possibly another unbounded part.

The Stefan two-phase problem with an internal source, as well as the heat conduction problem for rectangular and curvilinear regions, has been solved by the method of rapid sine-expansion, which involves the Fourier series. The problem of heat conduction in a body of rectangular form was solved in a general form with the Dirichlet conditions and an internal heat source by the method of rapid expansion, and thereafter the problem for a body of curvilinear shape. The solutions obtained were used in considering the Stefan problem with unknown curvilinear interface. Expressions are presented for matching the input data of the problem, on satisfaction of which the solution will be universally continuous everywhere in the region of the rectangle together with derivatives up to the sixth order inclusive. The nonfulfillment of these conditions leads to the discontinuity of the solution at the corner points. The algorithms of the solution of problems in rectangular and curvilinear regions, as well as of the Stefan twophase problem, are written. The solution for the rectangular region is represented in an explicit form, whereas the solution for the curvilinear region and the solution of the Stefan problem are reduced to a closed algebraic system of equations of small volume.

In this article, we study the classical solution of the mixed problem in a quarter of a plane and a half-plane for a one-dimensional wave equation. On the bottom of the boundary, Cauchy conditions are specified, and the second of them has a discontinuity of the first kind at one point. Smooth boundary condition is set at the side boundary. The solution is built using the method of characteristics in an explicit analytical form. Uniqueness is proved and conditions are established under which a piecewise-smooth solution exists. The problem with linking conditions is considered.

Method of successive approximations for the solution of the Wiener---Hopf functional equations system is offered. The method uses the presentation of matrix coefficient of the system as a sum of two matrices, where the first matrix assumes the exact factorization, and the second matrix --- matrix-perturbation --- is much smaller than the first matrix in the domain of system definition. Solution of the system is sought in the form of expansions in powers of the matrix-perturbation. At each step of the approximation the solution of the system is carried out by means of the Wiener---Hopf method using factorization of the main component of the matrix coefficient. The example of the use of method is considered for the solution of the problem about the calculation of the pre-fracture zone parameters in connecting material near the tip of the interfacial crack outcoming from the angular point of the broken interface of two different materials. The zone is modeled by the discontinuity line of displacement, on which the stresses meet the Mises---Hill failure criterion. It is shown that under certain conditions, already in the first approximation, the method allows to obtain solutions with permissible accuracy.

We consider Sturm-Liouville problems with a discontinuity in an interior point, which are motivated by the inverse problems for the torsional modes of the Earth. We assume that the potential on the right half-interval and the coefficient in the right boundary condition are given. Half-inverse problems are studied, that consist in recovering the potential on the left half-interval and the left boundary condition from the eigenvalues. If the discontinuity belongs to the left half-interval, the position and the parameters of the discontinuity also can be reconstructed. In this paper, we provide reconstructing algorithms and prove existence of solutions for the considered inverse problems. Our approach is based on interpolation of entire functions.

Periodic motions in a discontinuous dynamical system are studied. The discontinuous dynamical system consists of three distinct linear dynamical systems with two different circular boundaries. Analytical conditions for switching and sliding motions at the two circular boundaries are developed. From such analytical conditions, periodic motions in discontinuous dynamical systems can be determined through specific mapping structures. Illustrations give periodic motions in discontinuous dynamical systems, which are different from the continuous dynamical systems. In different domains, the motions are different, and the discontinuity of the periodic motions at the boundaries is observed, and the sliding motion on the boundaries is also a portion of periodic motion. Such periodic motions in discontinuous dynamical systems do not have any Fourier series. The methodology presented in this paper can be applied to other discontinuous dynamical systems. The circular boundaries can be treated different energy levels as control conditions in engineering systems.

We consider an inhomogeneous boundary value problem for a system of semilinear elliptic equations modeling radiative heat transfer with Fresnel matching conditions on the surfaces of discontinuity of the refractive index. The unique solvability of the boundary value problem is proved without requiring the initial data to be small.

This paper investigates the variational finite element formulation and its numerical implementation of the damage evolution in solids, using a new discrete embedded discontinuity approach. For this purpose, the kinematically optimal symmetric (KOS) formulation, which guarantees kinematics, is consistently derived. In this formulation, rigid body motion of the parts in which the element is divided is obtained. To guarantee equilibrium at the discontinuity surfaces, the length of the discontinuity is introduced in the numerical implementation at elemental level. To illustrate and validate this approach, two examples, involving mode-I failure, are presented. Numerical results are compared with those reported from experimental tests. The presented discontinuity formulation shows a robust finite element method to simulate the damage evolution processes in quasi-brittle materials, without modifying the mesh topology when cohesive cracks propagate.

The propagation of sound waves in an infinite circular cylindrical pipe with an inserted perforated screen is investigated rigorously through the mode matching technique. The pipe walls are assumed to be rigid for $$ - \infty < z < - l$$ and coated for $$ - l < z < 0,{\text{ }}0 < z < \infty $$ with different linings. An analytical solution for the field terms are determined in form of eigenmodes which are matched across the boundary of each junction discontinuity. Numerical results are performed to show the effect of the different parameters such as waveguide radius, length of the partial lined part and acoustic absorbing lining properties on the propagation phenomenon. The use of such components is effective in reducing noise effects from various sources. The results of the mode matching technique are also compared graphically with the results of the Wiener–Hopf technique which is more difficult to implement and perfect agreement are observed.

In this paper the displacement discontinuity method of high-order accuracy and its application to the problems of fracture mechanics are considered. In common practice of applications of boundary element methods, the methods with the piecewise-constant function of boundary displacement are often used. Their advantage against other algorithms is the simplicity of calculation scheme with a rather good accuracy of the solution at the points of region distant from the boundary. In the fracture mechanics (with lines of surfaces of discontinuity of the displacement field), it is required to describe the stress behavior in the proximity of the crack edges with the highest accuracy possible, which leads to necessity of increasing the degree of accuracy of the used numerical methods. It is shown that the methods with high-order continuity of displacements at the boundary proposed in this work substantially improve the accuracy of computation of displacement and stress fields in the neighborhood of crack edges within the region.

In this paper, inverse spectral problems of discontinuous Sturm–Liouville problems contained in the discontinuous coefficient and discontinuity conditions at an interior point of the finite interval are, respectively, studied according to

In this paper the new semi-analytical solution for the moving mass problem on massless foundation, published by the author of this paper, is extended to account for inertial foundation modelled by a continuous homogeneous finite depth foundation with simplified shear resistance. Derivations are presented for infinite as well as finite homogeneous beams. Mode expansion method is used to solve the problem on finite beams, thus vibration modes, the corresponding orthogonality condition, reengagement of coupled equations to ensure significant calculation time savings are derived. Methods of integral transforms and contour integration are exploited to obtain the solution on infinite beams. Resulting vibrations are derived as a sum of the steady and unsteady harmonic vibrations and a transient contribution. The unsteady harmonic vibration is proven to be a useful indicator of unstable behaviour through the mass induced frequencies. Besides frequency lines also discontinuity lines are determined and their influence on the proximity of harmonic and full solutions is discussed. Even if the differences between these two versions are larger than for the massless foundation, it is shown that the harmonic solution provides a very good estimate of the full solution (in several cases perfect match is achieved) with the advantage to be obtainable by a simple evaluation of the derived closed-form results. Like for the massless foundation, also here, vibrations on infinite beams can be obtained on long finite beams with eliminated effect of its supports. All mentioned approaches are also validated by the finite element method.

Dynamics of discontinuous nonlinear systems subjected to random excitation is studied. Such systems occur in many mechanical and aerospace applications involving impact, friction, clearance, backlash, freeplay etc. These systems are characterized by sharp switches in dynamical behaviour described by discontinuous stochastic differential equations. An adaptive time stepping approach is developed in combination with a bisection algorithm to locate precisely the discontinuity point in the numerical integration advanced by the Milstein method. The Brownian tree approach is used to direct the integration along the correct Brownian path. The examples of a Duffing oscillator with one- and two-sided impacts and a linear oscillator with a nonlinear discontinuous dry friction-type damper (Coulomb damping) subjected to combined harmonic and white noise excitations are considered. Stable periodic motion, D-bifurcation and chaotic dynamics are exhibited in different parametric regimes. The path-wise numerical integration procedure demonstrates the accuracy and efficiency of the proposed scheme in the dynamic analysis of the noisy vibro-impact oscillator and the friction oscillator.

A novel mathematical apparatus allowing formulation and justification of a new approach towards the setting of the mathematical problem of band-pass layered dielectric filters (LDF) synthesis is developed. Аrbitrary layered dielectric systems with piecewise continuous physical media parameters given by the functions of dielectric permittivity and of magnetic permeability, both depending on the coordinate along the normal to the layer pile, with fixed discontinuity points of at least one of the mentioned functions are examined. For such systems, an important conservation law for the difference of the squares of absolute amplitude values of plane waves propagating left and right in given layered medium is stated, which further leads to the traditional energy conservation law in lossless layered media. This new identity law allows turning from synthesis problems in terms of fractional rational energy reflectivity and transmittance of layered systems to equivalent tasks for profiling functions introduced in the work, representing only the numerator or only the denominator of the expressions usually considered in the synthesis. A new concept of the feasible ideal is introduced for the energy coefficients of reflection and transmission of layered systems. It is shown that the feasibility of the energy coefficients of reflection and transmission of layered systems is equivalent to the feasibility of the profiling functions of such systems, which together with the main identity allows a significant change of the existing LDF synthesis approach. The rule for converting the ideal of the reflection or transmission coefficient into the ideal of the profiling function is given. The proposed synthesis problem statement leads to considerably less intensive computational procedures.

We consider a boundary value problem for an ordinary singularly perturbed second-order differential equation whose right-hand side is a nonlinear function with a discontinuity along some curve that is independent of the small parameter. For this problem, we study the existence of a smooth solution with steep gradient in a neighborhood of some point lying on this curve. The point itself and an asymptotic representation for the solution are to be determined. The existence theorem is proved by the method of matching asymptotic expansions. To this end, we use theorems on existence of solutions of boundary value problems for singularly perturbed equations and methods for constructing asymptotic approximations to these solutions.

Objective. In this paper, the authors study problems of a plane strain of elastic bodies containing internal rectilinear fractures. In each case, the margins of the considered areas are supported by thin flexible coatings. The first part of the paper is devoted to the problem of an infinite elastic wedge, the faces of which are free from the outside and reinforced with a thin flexible material, and the bisector contains a rectilinear fracture with regular forces applied to the margins, and to the study of the stress concentration at the fracture vertices. In the second part of the paper, the authors consider the problem of an equilibrium radial internal fracture in the cross-section of a round pipe. The inner surface of the pipe experiences hydrostatic pressure; the outer surface is reinforced with a thin flexible coating. The purpose of the study in each of the presented tasks is to determine the values of the influence factor.Methods. Both problems are united by a single approach, in which the presence of a coating is modeled mathematically, using special marginal conditions obtained based on an asymptotic analysis of the exact solution for a strip or ring flexible coating of small relative thickness. In the first issue, the singular integral equation is derived using the Mellin transform, which allows proceeding to the solution of a system of ordinary differential equations and obtaining a singular integral equation relative to the derivative of the discontinuity function of the first kind with a Cauchy kernel. In the second issue, discontinuous solutions are constructed using the Fourier series, resulting in a singular integral equation of a similar structure. Previously, similar ideas were successfully implemented by the authors in the study of the problem of the equilibrium state of a strip with a coating weakened by an internal transverse fracture under arbitrary conditions on the lower edge of the strip.Conclusion. Singular integral equations for the considered problems are obtained. The collocation method is used to construct solutions of singular integral equations for various combinations of geometric and physical characteristics of issues. In all the considered cases, the values of the influence factor were calculated. The analysis of changes in the influence factor depending on various combinations of geometric parameters and mechanical characteristics of problems is carried out. It is noted that with increasing rigidity of the coating and increasing its thickness, the values of the influence factor decrease; the increase in the value of the influence factor is provided by approaching the fracture to the body margin and increasing its relative length.

As a free discontnuity problem, the predicton of the path that a crack chooses while it propagates through a britle material has been a long standing problem in fracture mechanics. To circumvent the mathematcal difcultes in the modeling of fracture behaviors, this paper takes advantage of the regularizaton approach to approximate the Munford-Shah functonal with Γ-convergence. The governing equatons of physical felds are constructed based on the concept of energy minimizaton. This method is capable of capturing the crack initaton and propagaton without a specifc tracking algorithm. Additonally, a pure elastc phase can be considered by introducing an energy threshold. By the use of the concept of equivalent stress, a stress-based fracture governing criterion was established. The numerical implementaton is based on a standard fnite element discretzaton and on the staggered algorithmic. Two classic concrete fracturing tests were investgated to demonstrate the performance of the model by comparing the numerical results with experimental results.

Generally, the damage of the structure will lead to the discontinuity of the local mode shape, which can be well reflected by the modal curvature of the structure, and the local information entropy of the beam structure will also change with the discontinuity of the mode. In this paper, based on the information entropy theory and combining the advantages of modal curvature index in damage identification of beam structure, the modal curvature utility information entropy index is proposed. The modal curvature curves of nondestructive structures were obtained by fitting the modal curvature curves of damage structures with the gapped smoothing technique to avoid dependence on the baseline data of nondestructive structures. The index comprehensively reflects the damage state of the structure by calculating mutual weight change matrix and the weight-probability coefficient. The performance of the new index was verified by the finite element simulation and model test of simply supported beam, respectively. The results show that the modal curvature utility information entropy index takes advantage of the modal curvature index which is sensitive to damage and can overcome its shortcomings effectively. The index proposed can identify the damage location and damage degree accurately and has certain noise immunity, which provides an effective damage identification indicator for beam structures.