Interpolating Stabilized Element Free Galerkin Method for Neutral Delay Fractional Damped Diffusion-Wave Equation

Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology, No. 424, Hafez Ave., 15914 Tehran, Iran College of Education, King Saud University, Saudi Arabia Department of Mathematics, Nazarbayev University, NurSultan, Kazakhstan Department of Computational Mathematics and Computer Science, Institute of Natural Sciences and Mathematics, Ural Federal University, ul. Mira. 19, Yekaterinburg 620002, Russia


Introduction
Partial differential equations (PDEs) with time delay play an important role in the mathematical modeling of complex phenomena and processes whose states depend not only on a given moment in time but also on one or more previous moments. We can mention a simple scenario involving the hemodynamic behavior of a person suffering from low or high glucose decompensation. This person can then be given intravenous insulin to compensate for the low level. Because the drug must be introduced into the bloodstream for it to take effect, the preceding scenario can be interpreted as a delay problem. As a result, there is a growing interest in studying biological and physical models with delay. The solutions of delay PDEs may represent voltage, concentrations, temperature, or various particle densities such as bacteria, cells, animals, and chemicals [1][2][3].
Delay PDEs with fractional derivatives have recently been studied using various numerical and analytical techniques such as [4][5][6][7][8]. It was pointed out in [9] that the derivatives of the dependent variable in the neutral type delay differential equations are both with and without time delays. Delay differential equations of neutral type appear in a variety of new phenomena, and its theory is even more complicated than the theory of nonneutral delay differential equations. From both a theoretical and practical standpoint, the oscillatory behavior of neutral system solutions is important. For some applications, such as the population growth, motion of radiating electrons, and the spread of epidemics in networks with lossless transmission lines, we refer the interested reader to [9][10][11][12][13][14].
A consideration of the following fractional PDE with a constant delay is the goal of this paper. For that end, we introduce where A novel interpolating element-free Galerkin approach to approximate the solution of the two-dimensional elastoplasticity problems was constructed in [15] using the interpolating moving least squares scheme for obtaining the shape function. Moreover, an improved element-free Galerkin scheme to solve nonlinear elastic large deformation problems was considered in [16]. The interpolating moving least squares approach using a nonsingular weight function is employed in [17] to approximate the solution of the problem of inhomogeneous swelling of polymer gels, and also the penalty scheme is used to enforce the displacement boundary condition; thus, an improved element-free Galerkin approach was constructed.
The interpolating element free Galerkin method has been developed to solve a variety of problems, including two-dimensional elastoplasticity problems [15,18], twodimensional potential problems [19], two-and threedimensional Stokes flow problems [20], two-dimensional large deformation problems [21], incompressible Navier-Stokes equation [22], steady heat conduction problems [23], two-dimensional transient heat conduction problems [24], three-dimensional wave equations [25], twodimensional Schrödinger equation [26], two-dimensional large deformation of inhomogeneous swelling of gels [27], biological populations [28], two-dimensional elastodynamics problems [29], and two-dimensional unsteady state heat conduction problems [30]. The theoretical analysis for the complex moving least squares approximation, the properties of its shape function, and its stability was analyzed in [31]. In [32], a variational multiscale interpolating element-free Galerkin scheme was established for solving the Darcy flow. For the numerical solution of generalised Oseen problems, a novel variational multiscale interpolating element-free Galerkin scheme was developed in [33] based on moving Kriging interpolation for obtaining shape functions using the Kronecker delta function. Zaky and Hendy [34] constructed a finite difference/Galerkin spectral approach for solving the Higgs boson equation in the de Sitter spacetime universe, which can inherit the discrete energy dissipation property. A high-order efficient difference/Galerkin spectral approach was proposed in [35] for solving the time-space fractional Ginzburg-Landau equation. Hendy and Zaky [36] proposed a finite difference/spectral method based on the L1 formula on nonuniform meshes for time stepping and the Legendre-Galerkin spectral approach for solving a coupled system of nonlinear multiterm timespace fractional diffusion equations.
This paper is built up as follows. In Section 2, the temporal discretization is discussed. The analysis of the temporal discretization scheme is constructed in Section 3. The moving Kriging technique and its implementation are demonstrated in Section 4. Finally, numerical experiments are presented in Section 5 to illustrate the analysis of the obtained scheme.

Temporal Discretization
Assume that τ = s/m such that m is a positive integer. Take N = dT/τe and t n = nτ, ∀n ∈ ℕ + ∪ f0g. Also, to make t = s, 2s, ⋯ being grid points, the time-variable step size should be surrounded by s = mτ instead of τ = T/N 1 for N 1 ∈ ℤ + . Thus, t n = nτ for n = −m, −m + 1, ⋯, 0. Here, we present a time-discrete scheme for Equation (1). For any function ξ n = ξðx, y, t n Þ, we set Lemma 1 (see [37]). Assume ϕðtÞ ∈ C 2 ½0, t n and ν ∈ ð1, 2Þ. Then in which Let Ψ be the exact solution of (1) and where vðt, x, y, zÞ = ∂Ψðt, x, y, zÞ/∂t. Thus, Equation (1) at ðt n , x, y, zÞ can be rewritten as Making use of Taylor expansion yields   2 Journal of Function Spaces Employing Lemma 1 and putting v 0 = vðx, 0Þ = φðxÞ = φ give Furthermore, there is a constant c > 0 that Substituting the above result into (10) arrives at in which there exists C ∈ ℝ + such that In the current paper, U n is an approximation of exact solution Ψ n .

Analysis of the Temporal Discretization
In the current section, we check the stability of the numerical procedure.
Multiplying relation (19) by τδ t W s−ð1/2Þ , integrating over Λ and then summing from s = 1 to M give Recalling the left hand side of the above relation, invoking Schwartz inequality and Lemma 3 yields

Journal of Function Spaces
Moreover, for the first term in the right hand side of Equation (20), we have On the other hand, according to some simple mathematical actions, we have Also, for the delay term, we arrive at Replacing the above relations in Equation (20) yields or Now, Equation (26) can be simplified as Changing index from M to s arrives at The use of Equation (29) and Lemma 2 yields Thus, there exists C ∈ ℝ + that

Moving Kriging Interpolation and Its Implementation
Following [39,40], we will invoke the technique of moving Kriging. Up to our knowledge and armed by the fact of the advantage of less CPU time consuming needed for the element free Galerkin approach based on the shape functions of moving Kriging than what needed for the element free Galerkin approach based on the shape functions of moving least squares approximation. In the meantime, the shape functions of moving Kriging interpolation can be deduced, which is analogous to moving least squares approximation over subdomain Λ 1 ⊂ Λ. Let Ψ h ðxÞ is the approximate solution of ΨðxÞ on Λ. The local approximation is formulated for any subdomain as 4

Journal of Function Spaces
such that q r and a r are monomial basis functions and monomial coefficients, respectively. Also, SðxÞ be the realization of a stochastic process. The covariance matrix of SðxÞ is given as in which (i) E½Eðx i , x r Þ is the correlation matrix (ii) Eðx i , x r Þ is the correlation function between any pair of nodes located at x i and x r The correlation function is defined as [39,40] such that θ > 0 is a value of the correlation parameter [39,40]. Using the best linear unbiased (BLUP) [39], we can write Equation (31) as follows [39,40] in which We will introduce some notations. The vector of known m functions can be written as follows [39,40] and the matrix of defined function values at the nodes x 1 , x 2 , ⋯, x n has the following representation [39,40] The correlation matrix is given as [39,40] The correlation vector at the nodes x 1 , x 2 , ⋯, x n has the following form : ð39Þ The matrices A and B are given as where I is the n × n identity matrix. Accordingly, Equation (34) can be written as follows [39,40] or where the moving Kriging approach's shape functions are as follows [39,40]: Now, we are ready to implement that kind of interpolation to the problem under consideration. Let the approximation solution of this equation be in which ϕ j ðxÞ are shape functions of moving Kriging approximation. Substituting Equation (44) in relation (15) gives 5 Journal of Function Spaces By collocating a set of arbitrary distributed nodes After doing some simplifications, we have where μ = 1/Γð2 − νÞτ. Now, the above formulation yields the following system of equations in which

Conclusion
The current paper presented a new numerical procedure for solving fractional damped diffusion-wave equations with delay. In this process, the time derivative is discretized by a finite difference scheme, and we constructed a time-discrete scheme. The stability and convergence of the proposed numerical formulation are studied, analytically and numerically. Then, the moving Kriging interpolation technique, as a meshless method, is used to get a fully discrete scheme. The proposed numerical method is flexible to simulate a wide range of PDEs including delay PDEs on irregular computational domains. Finally, an example is

Data Availability
The data used to support the findings of this study are included within the article.