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Two exponential wave integrator Fourier pseudospectral (EWI-FP) methods are presented and analyzed for the long-time dynamics of the Dirac equation with small potentials characterized by e ∈ ( 0 , 1 ] a dimensionless parameter. Based on the (symmetric) exponential wave integrator for temporal derivatives in phase space followed by applying the Fourier pseudospectral discretization for spatial derivatives, the EWI-FP methods are explicit and of spectral accuracy in space and second-order accuracy in time for any fixed e = e 0 . Uniform error bounds are rigorously carried out at O ( h m 0 + τ 2 ) up to the time at O ( 1 / e ) with the mesh size h, time step τ and m 0 an integer depending on the regularity of the solution. Extensive numerical results are reported to confirm our error bounds and comparisons of two methods are shown. Finally, dynamics of the Dirac equation in 2D are presented to validate the numerical schemes.

On this work, contrast between two analytical and numerical solutions of the advection-diffusion equation has been completed. We use the method of separation of variables, Hankel transform and Adomian numerical method. Also, Fourier rework, and square complement methods has been used to clear up the combination. The existing version is validated with the information sets acquired at the Egyptian Atomic Energy Authority test of radioactive Iodine-135 (I135) at Inshas in unstable conditions. On this model the wind speed and vertical eddy diffusivity are taken as characteristic of vertical height in the techniques and crosswind eddy diffusivity as function in wind speed. These values of predicted and numerical concentrations are comparing with the observed data graphically and statistically.

This paper is concerned with the numerical approximation of the time fractional Burgers equation with nonsmooth solutions. A nonlinear fully discrete scheme is presented based on the nonuniform Alikhanov formula of the Caputo time fractional derivative and Fourier spectral approximation in space. The solvability of the scheme is proved by the fixed point theorem and a priori estimate. Using the error estimation method, the proposed scheme is stable and convergent with order O ( τ min ⁡ { γ σ , 2 } + N 1 − m ) , where τ , N , γ , m and σ are the maximum time step size, polynomial degree, grading parameter, spatial regularity and temporal regularity parameter of the exact solution, respectively. A numerical example is performed to support the theoretical results.

We study a Sturm–Liouville problem in the cross-section of the ocean waveguide covered by pack ice. We prove the basis properties of the eigenfunctions, the convergence of the corresponding Fourier...

We study the homogenization of the evolutionary compressible Navier–Stokes–Fourier system in a bounded three-dimensional domain perforated with a large number of very tiny holes. We show that under suitable assumptions on the smallness and distribution of the holes the limit system remains the same in the unperforated domain. One of the main novelty in the paper consists in the treatment of the entropy inequality and thus the paper also improves the related result in the steady case from Lu and Pokorný (J Differ Equ 278:463–492, 2021).

Nonlinear vibration arises everywhere in engineering. So far there is no method to track the exact trajectory of a space fractional nonlinear oscillator; therefore, a sophisticated numerical method is much needed to elucidate its basic properties. For this purpose, a numerical method that combines the Fourier spectral method with the Runge–Kutta method is proposed. Its accuracy and efficiency have been demonstrated numerically. This approach has full physical understanding and numerical access; thus, it can be used to solve many types of nonlinear space fractional partial differential equations with periodic boundary conditions.

International audience; In this article, we develop an extension of the Fourier transform solution method in order to solve conduction equation with nonperiodic boundary conditions (BC). The periodic Lippmann-Schwinger equation for porous materials is extended to the case of non-periodicity with relevant source terms on the boundary. The method is formulated in Fourier space based on the temperature as unknown, using the exact periodic Green function and form factors to describe the boundaries. Different types of BC: flux, temperature, mixed and combined with periodicity can be treated by the method. Numerical simulations show that the method does not encounter convergence issues due to the infinite contrast and yields accurate results for both local fields and effective conductivity.

In this work one discusses a uniform observability of a semi-discrete Timoshenko beam model. We established an observability inequality to a particular class of solutions given by Fourier’s development and we prove that there exists a lack of numerical observability to the spectral problem in the setting of the spatial finite difference, i.e., the observability constant blows-up as the mesh-size h tends to zero. The semi-discrete system in finite difference avoids a numerical anomaly known as locking phenomenon on shear force and, in addition, such system raises an important problem in theoretical numerical analysis consisting in the determination of the Fourier’s solution that takes into account the parity of a sequence of the vibration modes.

This paper aims to study the well-posedness and the stability of two thermoelastic systems. The derivation of the first system is based on a classical coupling between the mechanical equations of Timoshenko and the thermal effects which are based on the conductivity of Fourier’s law. Whereas, the second system is derivable through a thermal coupling on the shear force. Furthermore, the damping of Kelvin–Voigt type is simultaneously presented in both the shear stress and the bending moment for the two systems.

In This paper we prove the existence of a weak solution of the complete compressible Navier-Stokes system. We follow an previous work where we added an artificial viscosity in the continuity equation and then rewrite the system in hyperbolic and symmetric form. Our study is based on the symmetric hyperbolic theory. We use for this aim a successive approximation in time to show the existence of the hyperbolic system solution and by the fixed point theorem the compacity property of some appropriate sobolev spaces and some established a priori estimates we can pass to several limits to prove our result. As state law, we use the Stiffened gas law.

We consider linear difference schemes with several degrees of freedom (DOFs) per cell for a transport equation with a constant coefficient. The Fourier transform decomposes the scheme into a number of finite systems of ODEs, the number of equations in each system being equal to the number of DOFs. The matrix of these systems is an analytical function of the wave vector. Generally this matrix is not diagonalizable and, if it is, the diagonal form may be non-smooth. We show that for L2-stable schemes in the 1D case the matrix can be locally transformed to a block-diagonal form preserving the analytical dependence on the wave number.

In this paper we shall characterize the space of Fourier ultra-hyperfunctions by the Mehler kernel method. That is, we show that any Fourier ultra-hyperfunctions are characterized as initial values...

In this paper stability of numerical solution of Fredholm integral equation of the first kind is studied for radial basis kernels which possess positive Fourier transform. As a result, the equivalence relation between strong and weak forms of partial differential equations (PDEs) is proved for some special radial test functions. Also the stability of boundary elements method (BEM) is proved analytically for Laplace and Helmholtz equations by obtaining Fourier transform of singular fundamental solutions applied in BEM. Analytical result presented in this paper is an extension of stability idea of radial basis functions (RBFs) used to interpolate scattered data described by Wendland in [51] . Similar to the interpolation, it is proved here mathematically that integral operators which have radial kernels with positive Fourier transform are strictly positive definite. Thanks to the stability idea presented in [51] , a positive lower bound for eigenvalues of these integral operators is found here, explicitly.

In this paper, we prove the existence and uniqueness results of entropy solutions to a class of nonlinear parabolic p(u)-Laplacian problem with Fourier-type boundary conditions and $$L^1$$ -data. The main tool used here is the Rothe method combined with the theory of variable exponent Sobolev spaces.

For comparison and testing purposes, an analytical and numerical method has been developed for solving the boundary value problem of convection-diffusion-decay for a homogeneous layer. A model stationary boundary value problem of the third kind is described by the equations of turbulent diffusion. In the analytical model, the Fourier symbol of the Green's function for the boundary value problem is constructed, the calculation of the Fourier integral is based on the theory of residues. In the numerical model, the Fourier integral is calculated using an algorithm based on the Gauss-Kronrod formulas.In the two-dimensional case, comparative calculations for the near and middle zones showed good agreement between the results. The accuracy of calculations within each model can be changed by several orders of magnitude. Calculations of the substance concentration function for two plane problems of convection-diffusion-decay are given as examples. The implementation of the analytical method in the flat case is relatively straightforward. Numerical calculation is much simpler and more convenient for engineering and serial scientific calculations. Numerical integration is realized just as easily in the spatial case as in the two-dimensional case, but the counting time increases significantly. Therefore, a balance is required between the required accuracy and the calculation time. For the spatial case, an analytical approach based on the theory of residues was fundamentally developed in the works of other authors. However, the method is rather cumbersome and better suited for theoretical research.

In this paper, an energy stable time-stepping method using the finite difference approximation in time and Fourier pseudo-spectral method in space is developed for the symmetric regularized long wave equation with damping mechanism. Based upon a careful treatment of nonlinear term, the suggested numerical scheme is proved to preserve energy dissipation at discrete time levels. The energy stable property implies the priori estimates of the numerical solution. The maximum norm error estimate shows that the proposed numerical scheme is of second-order accuracy in time and spectral accuracy in space. Several numerical experiments are presented to show the effectiveness of our numerical method and to confirm our theoretical analysis.

We present a method for fast computation of highly oscillatory Fourier-type integrals with monomial phase functions and Jacobi-type singularities. It is shown that on an appropriately chosen contour in the complex plane the original integral can be expressed as the sum of two nonoscillatory and rapidly decaying integrals, each of which can be efficiently computed by the generalized Gauss-Laguerre rule or by the double exponential method. For moderate and large frequencies, the proposed method is stable and more efficient than the existing methods. We also give a Mathematica program for the implementation of the method on a computer. The effectiveness and accuracy is tested by means of numerical examples.

This paper presents research on the fractional boundary layer flow and heat transfer over a stretching sheet with variable thickness. Based on the Caputo operators, the double fractional Maxwell model and generalized Fourier's law are introduced to the constitutive relationships. The governing equations are solved numerically by utilizing the finite difference method. The effects of fractional parameters on the velocity and temperature field are analyzed. The results indicate that the larger is the fractional stress parameter, the stronger is the elastic characteristic. However, fluids show viscous fluid-like behavior for a larger value of fractional strain parameter. Moreover, the numerical solutions are in good agreement with the exact solution and the convergence order can achieve the expected first order. The numerical method in this study is reliable and can be extended to other fractional boundary layer problems over a variable thickness sheet.

The main objective of this paper is to study and give basic properties of the fractional Fourier–Jacobi type transform of rapidly decreasing functions and of tempered distributions. An application in solving a generalized heat equation is given.

This study aimed at presenting a steady-state analytical solution for the two-dimensional heat conduction in a cylindrical segment made of functionally graded materials. It is acquired by taking advantage of the Fourier transform and separation of variables rather than numerical methods. Sturm–Liouville theory is employed to find the proper and adequate Fourier transformation. Continuous variations along the radial and circumferential directions based on the power-law function are taken into account, and non-homogeneous boundary conditions are applied to the problem. The obtained formulation is verified by the available solutions. Through solving an illustrative example, the temperature distribution is deliberated for a combination of boundary conditions. It is to be emphasized that mathematical robustness and generality of the solution are its primary advantage which is not often seen in the previously published literature.

In this paper, we propose a splitting Fourier pseudospectral method for Vlasov–Poisson–Fokker–Planck (VPFP) system, which describes the motion of charged particles in plasma. The numerical integration for the system is performed by employing the splitting method in time, Fourier Galerkin method in space direction, and Fourier collocation method in phase direction, respectively. The algorithm has spectral accuracy in both space and phase directions and can be implemented efficiently with the fast Fourier transform and technique of diagonalization, respectively. Extensive numerical results in one-dimensional phase space (or 1 x × 1 v ) from the proposed numerical method are shown and have proven the good agreement with the theory and previous studies. Numerical algorithm for the VPFP system in two-dimensional phase space (or 2 x × 2 v ) has been summarized in Appendix.

In this paper, a Crank–Nicolson Fourier spectral Galerkin method is proposed for solving the cubic fractional Schrodinger equation. Firstly, we discuss the mass and energy conservation laws for the nonlinear system and its corresponding fully discrete scheme. Secondly, the convergence with the spectral order accuracy in space and the second order of accuracy in time is exhibited. We perform one-dimensional calculation of the fractional derivative differential equation to verify our theoretical findings. Moreover, the proposed scheme is successfully applied to study two- and three-dimensional fractional quantum mechanics. Numerical results clearly exhibit that the fractional order can affect the shapes of soliton and rogue waves. The evolution of ground state solution can be clearly seen to be non-symmetrically configured when the fractional order becomes smaller.

We propose a high order numerical method for computing time dependent 4-D Wigner equation with unbounded potentials and study a canonical quantum double-slit interference problem. To address the difficulties of 4-D phase space computations and higher derivatives from the Moyal expansion of the nonlocal pseudo-differential operator for unbounded potentials, an operator splitting technique is adopted to decompose the 4-D Wigner equation into two sub-equations, which can be computed either analytically or numerically with high efficiency. The first sub-equation contains only a linear convection term in ( x , t ) -space and can be solved with an upwinding characteristic method, while the second involves the pseudo-differential term and can be approximated by a plane wave expansion in k-space. By exploiting properties of Fourier transformations, the expansion coefficients for the second sub-equation have explicit forms and the resulting scheme is shown to be unconditionally stable for any high order derivatives in the Moyal expansion, ensuring the feasibility of 4-D Wigner numerical simulations for quantum double-slit interferences. Numerical experiments demonstrate the spectral convergence in ( x , k ) -space and provide highly accurate information on the number, position, and intensity of the interference fringes for different types of slits, quantum particle masses, and initial states (pure and mixed).

We consider a new formulation of the Fourier filtering problem based on the use of matrix filters instead of conventional filters-multiplicators in the framework of the periodic boundary value problem for a quasilinear diffusion equation of nonlinear optics. Our attention is mostly focused on studying the capabilities of matrix Fourier filtering as an efficient tool for constructing periodic solutions with prescribed spatial-temporal dynamics. To this end, we analyze the spectrum of the linearized operator and the possibility to localize it by choosing a matrix Fourier filter that provides the existence of the Hopf bifurcation. We also describe the structure of bifurcation solution for two important classes of filters: finite filters-multiplicators and matrix filters. We theoretically and numerically show that along with rotating waves, typical for the Hopf bifurcation with conventional filters-multiplicators, there exist waves with new nontrivial dynamics (rotating waves with nodes and pulsating structures). These waves excite as a result of the Hopf bifurcation under an appropriate choice of nondiagonal matrix Fourier filters.

This paper studies the analytical solution for the vibration of simply supported beams with arbitrarily and continuously varying thickness based on the two-dimensional elasticity theory. The general expression of stress function, which exactly satisfies the governing differential equations and the boundary conditions, is derived. Frequency equation governing the free vibration of beams with variable thickness can be obtained by using the Fourier sinusoidal series expansion on the upper and lower surfaces of the beam. The present solution method ensures a rapid convergence and meets the need of high accuracy in modern precise instruments. Several examples are provided to show the application of the proposed solution method which can be used to assess the validity of various approximate solutions and numerical methods for the beams with arbitrarily and continuously varying thickness.

We are concerned in this paper with the numerical computation of the spectra of highly oscillatory integrals that arise in laser simulations. Discretised using the modified Fourier basis, the spectral problem for the integral equation is converted into two independent infinite systems of linear equations whose unknowns are the coefficients of the modified Fourier functions, namely the cosine and shifted sine functions, respectively. Each ( m , n ) entry of the resulting coefficient matrices can be represented exactly by expressions involving the error function with an argument that involves the oscillatory parameter ω and the numbers m and n . Moreover, considering the behaviour of the error function for a large argument, the asymptotics for each entry are analysed for large ω or for large m and n and this enables efficient truncation of the infinite systems. Numerical experiments are provided to illustrate the effectiveness of this method.

In this work, we provide a result on the derivation of the incompressible Navier-Stokes-Fourier system from the Landau equation for hard, Maxwellian and moderately soft potentials. To this end, we first investigate the Cauchy theory associated to the rescaled Landau equation for small initial data. Our approach is based on proving estimates of some adapted Sobolev norms of the solution that are uniform in the Knudsen number. These uniform estimates also allow us to obtain a result of weak convergence towards the fluid limit system.

Heat equations can estimate the thermal distribution and phase transformation in real-time based on the operating conditions and material properties. Such wonderful features have enabled heat equations in various fields, including laser and electron beam processing. The integral transform technique (ITT) is a powerful general-purpose semi-analytical/numerical method that transforms partial differential equations into a coupled system of ordinary differential equations. Under this category, Fourier and non-Fourier heat equations can be implemented on both equilibrium and non-equilibrium thermo-dynamical processes, including a wide range of processes such as the Two-Temperature Model, ultra-fast laser irradiation, and biological processes. This review article focuses on heat equation models, including Fourier and non-Fourier heat equations. A comparison between Fourier and non-Fourier heat equations and their generalized solutions have been discussed. Various components of heat equations and their implementation in multiple processes have been illustrated. Besides, literature has been collected based on ITT implementation in various materials. Furthermore, a future outlook has been provided for Fourier and non-Fourier heat equations. It was found that the Fourier heat equation is simple to use but involves infinite speed heat propagation in comparison to the non-Fourier heat equation and can be linked with the Two-Temperature Model in a natural way. On the other hand, the non-Fourier heat equation is complex and involves various unknowns compared to the Fourier heat equation. Fourier and Non-Fourier heat equations have proved their reliability in the case of laser–metallic materials, electron beam–biological and –inorganic materials, laser–semiconducting materials, and laser–graphene material interactions. It has been identified that the material properties, electron–phonon relaxation time, and Eigen Values play an essential role in defining the precise results of Fourier and non-Fourier heat equations. In the case of laser–graphene interaction, a restriction has been identified from ITT. When computations are carried out for attosecond pulse durations, the laser wavelength approaches the nucleus-first electron separation distance, resulting in meaningless results.

The time spectral method is a very popular reduced order frequency method for analyzing unsteady flow due to its advantage of being easily extended from an existing steady flow solver. Condition number of the inverse Fourier transform matrix used in the method can affect the solution convergence and stability of the time spectral equation system. This paper aims at evaluating the effect of the condition number of the inverse Fourier transform matrix on the solution stability and convergence of the time spectral method from two aspects. The first aspect is to assess the impact of condition number using a matrix stability analysis based upon the time spectral form of the scalar advection equation. The relationship between the maximum allowable Courant number and the condition number will be derived. Different time instant groups which lead to the same condition number are also considered. Three numerical discretization schemes are provided for the stability analysis. The second aspect is to assess the impact of condition number for real life applications. Two case studies will be provided: one is a flutter case, NASA rotor 67, and the other is a blade row interaction case, NASA stage 35. A series of numerical analyses will be performed for each case using different time instant groups corresponding to different condition numbers. The conclusion drawn from the two real life case studies will corroborate the relationship derived from the matrix stability analysis.

This paper designs a general type time–frequency transform whose kernel function involves a nonlinear modulation and a frequency-varying dilation. The corresponding inversion formula is established.

In this paper we propose and analyze a (temporally) third order accurate exponential time differencing (ETD) numerical scheme for the no-slope-selection (NSS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectral discretization in space. A linear splitting is applied to the physical model, and an ETD-based multistep approximation is used for time integration of the corresponding equation. In addition, a third order accurate Douglas-Dupont regularization term, in the form of $$-A {\Delta t}^2 \phi _0 (L_N) \Delta _N^2 ( u^{n+1} - u^n)$$ , is added in the numerical scheme. A careful Fourier eigenvalue analysis results in the energy stability in a modified version, and a theoretical justification of the coefficient A becomes available. As a result of this energy stability analysis, a uniform in time bound of the numerical energy is obtained. And also, the optimal rate convergence analysis and error estimate are derived in details, in the $$\ell ^\infty (0,T; H_h^1) \cap \ell ^2 (0,T; H_h^3)$$ norm, with the help of a careful eigenvalue bound estimate, combined with the nonlinear analysis for the NSS model. This convergence estimate is the first such result for a third order accurate scheme for a gradient flow. Some numerical simulation results are presented to demonstrate the efficiency of the numerical scheme and the third order convergence. The long time simulation results for $$\varepsilon =0.02$$ (up to $$T=3 \times 10^5$$ ) have indicated a logarithm law for the energy decay, as well as the power laws for growth of the surface roughness and the mound width. In particular, the power index for the surface roughness and the mound width growth, created by the third order numerical scheme, is more accurate than those produced by certain second order energy stable schemes in the existing literature.

By introducing a scalar auxiliary variable (SAV), we propose a class of semi-implicit time-stepping, unconditionally energy stable numerical schemes for simulating wrinkling phenomena of an elastic thin films on a compliant substrate. These SAV schemes only need to solve, twice, a decoupled linear, fourth-order integro-differential equations with constant coefficients at each time step, and these linear equations can be efficiently implemented by using Fourier spectral method to discretize spatial derivatives. Numerical results have demonstrated that these SAV schemes (including first-order SAV/BDF1, second-order SAV/BDF2 and SAV/CN2) are highly efficient and accurate. Furthermore, many interesting wrinkling phenomena, such as pattern formations (e.g., stripes, checkerboards, labyrinths, herringbones) and pattern transitions under different loading processes, are investigated by using the proposed schemes.

In this paper, we propose a simple fast Fourier spectral method for the inelastic Boltzmann collision operator, with its application to one of the widely used models of granular gases, the inelastic Boltzmann equation with a heating source. Compared to the direct Fourier spectral method, our fast algorithm reduces the computational complexity from O ( N 6 ) to O ( M N 4 log ⁡ N ) per evaluation of the collision operator in three dimensions, where N is the number of discretization points in each velocity dimension and M ≪ N 2 is the number of quadrature points used on the unit sphere. We test the numerical accuracy and efficiency of the proposed method in both two dimensional and three dimensional examples, where in the latter case the famous Haff's cooling law for granular flows is successfully recovered.

In this paper, the L1 Fourier spectral method is considered to solve the time-fractional Navier–Stokes equation with periodic boundary condition. The Fourier spectral method is employed for spatial approximation, and the L1 finite difference scheme is used to discrete the Caputo time fractional derivative. Analysis of stability and convergence are accomplished as well, leading to the conclusion that our numerical method is unconditionally stable, and the solution converges to the exact one with order O ( τ 2 − α + N − s ) , where τ and N are the time step size and polynomial degree, respectively. The numerical example is provided to testify the effectiveness of our scheme, from the results of which, it turns out that the L1 Fourier spectral method is effective for solving the time-fractional Navier–Stokes equation.

This paper presents stress influence functions for uniformly distributed, time-harmonic rectangular loads within a three-dimensional, viscoelastic, isotropic full-space. The coupled differential equations relating displacements and stresses in the full-space are solved through double Fourier integral transforms in the wave number domain, in which they can be solved algebraically. The final stress fields are expressed in terms of double indefinite integrals arising from the Fourier transforms. The paper presents numerical schemes with which to integrate these functions accurately. The article presents numerical validation of the synthesized stress kernels and their behavior for high frequencies and large distances from the excitation source. The influence of damping ratio on the dynamic results is also investigated. This article is complementary to previous results of the authors in which the corresponding displacement solutions were derived. Stress influence functions, together with their displacement counterparts, are a fundamental part of many numerical methods of discretization such the boundary element method.

The plane problem for an elastic layer under a nonstationary load is solved analytically. The Laplace and Fourier transforms are used. The solution has the form of a series of reflected waves where a finite number of terms are retained for numerical purposes. The original functions are recovered using the method of joint inversion of integral transforms. The numerical analysis is made for a load suddenly applied to the layer and propagating with constant velocity. The normal stress on the symmetry axis is calculated for the time interval that takes the head expansion wave to cross the thickness of the layer for four times. The results obtained are compared to a similar problem for mixed boundary conditions.

We propose and compare numerically spatial/temporal resolution of various efficient numerical methods for solving the Klein–Gordon–Dirac system (KGD) in the nonrelativistic limit regime. The KGD system involves a small dimensionless parameter $$0

This result sharpens the bilinear to linear deduction of Lee and Vargas for extension estimates on the hyperbolic paraboloid in $\mathbb R^3$ to the sharp line, leading to the first scale-invariant restriction estimates, beyond the Stein--Tomas range, for a hypersurface on which the principal curvatures have different signs., Comment: 8 pages

In this paper, we present an application of a Galerkin-Petrov method to the spatially one-dimensional Boltzmann equation. The three-dimensional velocity space is discretised by a spectral method. The space of the test functions is spanned by polynomials, which includes the collision invariants. This automatically insures the exact conservation of mass, momentum and energy. The resulting system of hyperbolic PDEs is solved with a finite volume method. We illustrate our method with two standard tests, namely the Fourier and the Sod shock tube problems. Our results are validated with the help of a stochastic particle method.

A mixed initial boundary value problem for the heat conduction equation is investigated and solved. The problem statement includes three intervals: the first one (0 → T1) describes heating of an internal combustion chamber and the second one (T1 → T2), the chamber cooling and slower cooling of its wall. The third interval describes natural cooling of the chamber wall when the chamber temperature coincides with that of the environment. The applicability of the Fourier transform in time to this problem is proved. This makes it possible to transform the governing equation to an ordinary differential equation. By using the resulting equation, an inverse boundary value problem for the heat conduction equation is solved by a nonlinear method of projection regularization, and an error estimate of the approximate solution is obtained.

The paper proposes a novel computational method for simulating diffraction of obliquely incident plane waves by a doubly periodic array of arbitrarily shaped planar delaminations in elastic layered structures. The presented approach is an extension of the boundary integral equation method (BIEM) based on the boundary integral approach. Periodically situated delaminations are simulated via spring boundary conditions (SBCs), which allows considering bridged cracks. A boundary integral equation is formulated for a reference delamination based on SBCs in terms of the Fourier transform of Green's matrix and solved using Galerkin method. Two modifications of the method are proposed: the spectral BIEM applied only for rectangular delaminations and the meshless BIEM with axisymmetric basis functions, which is applied for delaminations of arbitrary shapes. The effectiveness of regular and irregular meshes is studied for the meshless BIEM. Comparison of both modifications of the BIEM is provided for open and bridged cracks. The proposed method allows considering specific wave phenomena such as a band-gap formation or opening of an interface for wave propagation in elastic layered structures with a doubly periodic array of inhomogeneities (e.g. phononic crystals and damaged interfaces).

The solution of a nonlinear diffusion equation is numerically investigated using the generalized Fourier transform method. This equation includes fractal dimensions and power-law dependence on the radial variable and on the diffusion function. The generalized Fourier transform approach is the extension of the Fourier transform method used for the normal diffusion equation. The feasibility of the approach is validated by comparing the numerical result with the exact solution for a point-source. The merit of the numerical method is that it provides a way to calculate anomalous diffusion with an arbitrary initial condition.

Due to the operation of fractional differentiation introduced with the help of Fourier integral, the results of calculating fractional derivatives for certain types of functions are given. Using the numerical method of integration, the values of fractional derivatives for arbitrary dimensionality e , (where e is any number greater than zero) are calculated. It is proved that for {integer} values of e we obtain ordinary derivatives of the first, second and more high orders. As an example it was considered heat conduction equation of Fourier, where spatial derivation was realized with the use of fractional derivatives. Its solution {is} given {by} Fourier integral. {moreover} , it was shown that integral went into the required results in special case of the whole e obtained in n -dimensional case, where n = 1 , 2 . . . , etc.

In this paper, we develop a mixed Hermite–Fourier spectral method for the Fokker–Planck equation of periodic field. Radical to the numerical solutions of partial differential equations of the sort, results on mixed generalized Hermite–Fourier orthogonal approximation are established. The convergence of the constructed spectral scheme is proved. Numerical results show the efficiency of this approach and coincide well with theoretical analysis.

The accurate simulation of scattering of electromagnetic waves in three dimensions by a diffraction grating is crucial in many applications of engineering and scientific interest. In this contribution we present a novel High-Order Perturbation of Surfaces method for the numerical approximation of vector electromagnetic scattering by a doubly periodic layered medium. For this we restate the governing time harmonic Maxwell equations as vector Helmholtz equations which are coupled by transmission boundary conditions at the layer interface. We then apply the method of Transformed Field Expansions which delivers a Fourier collocation, Legendre–Galerkin, Boundary Perturbation approach to solve the problem in transformed coordinates. A sequence of numerical simulations demonstrate the efficient and robust spectral convergence which can be achieved with the proposed algorithm.