An equation with Riesz fractional space derivatives and a functional delay effect is considered. The problem is discretized. Constructions of an analog of the Crank-Nicolson difference method with piecewise linear interpolation and extrapolation by continuation are presented. The method has the second order of smallness with respect to the time and space sampling steps O and h. The basic Crank-Nicolson method with piecewise parabolic interpolation and extrapolation by continuation is constructed. The order of the residual without interpolation of the basic method is studied. The expansion coefficients of the residual with respect to O and h are written. An equation is derived for the main term of the asymptotic expansion of the global error. Under certain assumptions, the legality of using the Richardson extrapolation procedure is substantiated and the corresponding method is constructed. The main of these assumptions is the consistency of the orders of smallness Delta and h. It is proved that the method has order O(Delta(3) + h(3)).

For a space-fractional diffusion equation with a nonlinear superdiffusion coefficient and with the presence of a delay effect, the grid numerical method is constructed. Interpolation and extrapolation procedures are used to account for the functional delay. At each time step, the algorithm reduces to solving a linear system with a main matrix that has diagonal dominance. The convergence of the method in the maximum norm is proved. The results of numerical experiments with constant and variable delays are presented.

A diffusion equation with a functional delay effect is considered. The problem is discretized. Constructions of the Crank–Nicolson difference method with piecewise linear interpolation and extrapolation by continuation are given; the method here has the second order of smallness with respect to the sampling steps in time Δ and space h . The basic Crank–Nicolson method with piecewise cubic interpolation and extrapolation by continuation is constructed. The order of the residual without interpolation of the basic method is studied, and the expansion coefficients of the residual with respect to Δ and h are written. An equation for the leading term of the asymptotic expansion of the global error is written. Under certain assumptions, the validity of the application of the Richardson extrapolation procedure is substantiated and an appropriate method is constructed. The main of these assumptions is the consistency of the orders of smallness of Δ and h . It is proved that the method has order O(Δ^4+h^4) . The results of numerical experiments on test examples are presented.

A wave equation with functional delay is considered. The problem is discretized. Constructions of the difference method with weights with piecewise linear interpolation are given. A basic method with weights with piecewise cubic interpolation is constructed. The order of the residual is studied without interpolation of the basic method, and the expansion coefficients of the residual with respect to time-steps and space-steps are written out. It is proved that the weighted method with piecewise cubic interpolation converges with order 2 in the energy norm. An equation is written for the main term of the asymptotic expansion of the global error of the basic method. Under certain assumptions, the validity of the application of the Richardson extrapolation procedure is substantiated, and the corresponding numerical method is constructed, that has the fourth order of convergence with respect to time-steps and space-steps. The validity of Runge's formulas for practical estimation of the error is proved. The results of numerical experiments on a test example are presented.

We consider a system of two space-fractional superdiffusion equations with functional general delay and Neumann boundary conditions. For this problem, an analogue of the Crank-Nicolson method is constructed, based on the shifted Grünwald-Letnikov formulas for approximating fractional Riesz derivatives with respect to a spatial variable and using piecewise linear interpolation of discrete prehistory with extrapolation by continuation to take into account the delay effect. With the help of the Gershgorin theorem, the solvability of the difference scheme and its stability are proved. The order of convergence of the method is obtained. The results of numerical experiments are presented.

In this paper, a second-order method in time and space steps is constructed for a fractional diffusion equation in the presence of drift and functional delay. The basis of the algorithm is Alikhanov's method. To take into account the effect of functional delay, the interpolation and extrapolation constructions are used. The local error of the method is investigated. Using the discrete Gronwall inequality and some additional estimates, the convergence of the method is proved and the orders of convergence with respect to the partitioning steps in time and space are obtained. The results of numerical experiments on test examples are presented.

In this paper, we construct and analyze a linearized finite difference/Galerkin–Legendre spectral scheme for the nonlinear Riesz-space and Caputo-time fractional reaction–diffusion equation with prehistory. The problem is first approximated by the L1 difference method in the temporal direction, and then the Galerkin–Legendre spectral method is applied for the spatial discretization. The key advantage of the proposed method is that the implementation of the iterative approach is linear. The stability and the convergence of the semi-discrete approximation are proved by invoking the discrete fractional Halanay inequality. The stability and convergence of the fully discrete scheme are also investigated utilizing discrete fractional Grönwall inequalities, which show that the proposed method is stable and convergent. Furthermore, to verify the efficiency of our method, we provide numerical results that show a satisfactory agreement with the theoretical analysis.

Due to the lack of a discrete fractional Grönwall-type inequality, the techniques of analyzing the L2−1σ difference schemes would not be correct to apply directly to the nonlinear multi-term fractional subdiffusion equations with time delay, especially when the maximum order of the fractional derivatives is not an integer. The purpose of this paper is twofold. First, we introduce a discrete form of fractional Grönwall-type inequality, which in turn fills a gap in the proofs of convergence and stability analyses of such difference schemes. Second, some examples of improper apply of classical convergence and stability techniques are introduced. Moreover, detailed proofs for the convergence and stability theorems are provided departing from the proposed discrete fractional Grönwall-type inequalities.

For a fractional diffusion-wave equation with a nonlinear effect of functional delay, an implicit numerical method is constructed. The scheme is based on the L2-method of approximation of the fractional derivative of the order from 1 to 2, interpolation and extrapolation with the given properties of discrete prehistory and an analogue of the Crank-Nicolson method. The order of convergence of the method is investigated using the ideas of the general theory of difference schemes with heredity. The order of convergence of the method is more significant than in previously known methods, depending on the order of the starting values. The main point of the proof is the use of the stability of the L2-method. The results of comparing numerical experiments with other schemes are presented: a purely implicit method and a purely explicit method, these results showed, in general, the advantages of the proposed scheme.

In this paper, we construct and analyze a linearized finite difference/Galerkin–Legendre spectral scheme for the nonlinear multiterm Caputo time fractional-order reaction-diffusion equation with time delay and Riesz space fractional derivatives. The temporal fractional orders in the considered model are taken as 0 < β 0 < β 1 < β 2 < ⋯ < β m < 1 . The problem is first approximated by the L 1 difference method on the temporal direction, and then, the Galerkin–Legendre spectral method is applied on the spatial discretization. Armed by an appropriate form of discrete fractional Grönwall inequalities, the stability and convergence of the fully discrete scheme are investigated by discrete energy estimates. We show that the proposed method is stable and has a convergent order of 2 − β m in time and an exponential rate of convergence in space. We finally provide some numerical experiments to show the efficacy of the theoretical results.

A two-dimensional in space fractional diffusion equation with functional delay of a general form is considered. For this problem, the Crank-Nicolson method is constructed, based on shifted Grunwald-Letnikov formulas for approximating fractional derivatives with respect to each spatial variable and using piecewise linear interpolation of discrete history with continuation extrapolation to take into account the delay effect. The Douglas scheme is used to reduce the emerging high-dimensional system to tridiagonal systems. The residual of the method is investigated. To obtain the order of the method, we reduce the systems to constructions of the general difference scheme with heredity. A theorem on the second order of convergence of the method in time and space steps is proved. The results of numerical experiments are presented.

We constructGorbova, T.V. Pimenov, V.G. Solodushkin, S.I. a Crank–Nicolson numerical algorithm for nonlinear initial-boundary value problem of parabolic type complicated by heredity effect. Nonlinearity is present in the partial differential operator as well as in the inhomogeneity function. Stability and convergence property of the elaborated algorithm are studied. Proposed numerical algorithm was implemented in Python 3.7. Numerical experiments have been carried through. Numerical results coincides with the theoretical ones.

A fractional diffusion equation with the presence of a drift and a delay of a general form is considered. For this problem, a family of grid schemes with weights is constructed based on the L1-method for approximating the fractional derivative and applying the piecewise constant interpolation of discrete history. The algorithm is reduced to solving linear systems with a tridiagonal matrix. The order of the residual of the method without interpolation and the order of the residual of the method with interpolation are investigated. The stability conditions of the algorithm are obtained. A theorem is proved that under stability conditions the method has the first order with respect to the time-step and the second order with respect to the space-step.

Low density foams are widely used for laser plasma interaction studies. Here, we present the results of first direct measurements of a residual level of inhomogeneity of a foam plasma by using a pump-probe technique. It is demonstrated that large scale density modulations in such a plasma can survive a time larger than 1 ns, much longer than the plasma formation time.

A class of one-dimensional time-fractional parabolic differential equations with delay effects of functional type in the time component is numerically investigated in this work. To that end, a compact difference scheme is constructed for the numerical solution of those equations based on the idea of separating the current state and the prehistory function. In these terms, the prehistory function is approximated by means of an appropriate interpolation-extrapolation operator. A discrete form of the fractional Gronwall inequality is employed to provide an optimal error estimate. The existence and uniqueness of the numerical solutions, the order of approximation error for the constructed scheme, the stability and the order of convergence are mathematically investigated in this work.

We developed a procedure for the high-precision determination of matrix elements of glasses of the Ga–Ge–As–Se system, including those doped with praseodymium, in the range of concentrations of gallium from 1 to 5 at %, germanium from 16 to 24 at %, arsenic from 14 to 18 at %, and selenium from 57 to 65 at %. A procedure for determining 0.05–0.5 wt % of praseodymium in these glasses by inductively coupled plasma–atomic emission spectrometry is also proposed. The expanded uncertainty (P = 0.95) of the results in determining the matrix elements ranged from 0.05 to 0.1 at %, and for praseodymium, it was from 0.002 from 0.02 wt %. A method for preparing standard solutions necessary to achieve the stated level of uncertainty of the analysis results is described. The minimum sample weight for determining matrix elements is approximately 1 mg, and for the determination of praseodymium, it is approximately 10 mg, which enables analyzing not only bulk glass samples but also fibers made of them.

A space-fractional two-dimensional diffusion equation with a delay of a general form is considered. For this problem, the ADI method is constructed based on the shifted Grunwald-Letnikov formula for approximating the fractional derivatives and applying the piecewise constant interpolation of discrete history. The method convergence order is obtained. The results of numerical experiments are presented.

One of the most important stages of the high-purity chalcogenide glasses’ analytical control is the determination of matrix elements’ content with the uncertainty at the levels of 0.1–0.2 mol.%. The content of the macro-components may differ from the composition of the initial charge; therefore, an important task is the macro-composition determination of the final materials. This article describes the development of the technique for determining the matrix elements of high-purity Ge-Se-Te glasses in the range of germanium content from 10 to 35 mol. %, selenium and tellurium content from 20 to 50 mol. % with the expanded uncertainty within 0.01–0.2 mol. % (P = 0.95) using the inductively coupled plasma atomic emission spectrometry (ICP-AES). A simple technique for the preparation of primary calibration solutions from pure elementary Ge, Se and Te is proposed. The correctness of the analysis results is confirmed by comparing the calculated matrix composition of model glass samples, prepared by direct synthesis from high-purity simple substances in the sealed quartz glass ampoule, with the analysis results. The main advantage of the proposed analysis technique is the absence of the need for the reference samples identical to the analyzed material, which is especially important for determination of new materials’ matrix composition. The minimum sample mass for the determination of matrix elements is about 1 mg, which makes it possible to analyze not only bulk glass samples, but also fibers and expensive materials.