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The interior inverse elastic scattering problem is considered in this paper. The scattering in the cavity is illuminated by an time-harmonic elastic point source at fixed frequency. The boundary of the cavity is assumed to be a small and smooth perturbation of some circle. In this paper, we are interested in the reconstruction of the boundary according to the measured data on a closed curve inside the cavity. In implementation, we split the displacement field into compressional and shear waves by the Helmholtz decomposition. Then we use coordinate transformation, power series expansion based on perturbed parameters and Fourier expansion of corresponding power series coefficients to simplify the original boundary value problem to a successive sequence of one-dimensional boundary value problem. We linearize the inverse problem by eliminating the higher-order terms in the power series expansion; and design a reconstruction algorithm by the explicit relationship between the Fourier coefficients of the boundary function and the total field. In order to obtain better resolution, we also use a nonlinear correction method in the numerical experiments. The numerical results have shown the effectiveness and the stableness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we propose a decomposition formula for the fluid-solid interaction scattering problem. By the decomposition of the displacement, the corresponding problem is transformed into a coupled boundary value problem. For the two dimensional case, the coupled boundary value problem is about two scalar Helmholtz equations, and the other one is a scalar Helmholtz equation and a vector Helmholtz equation for the three dimensional. Then the coupled boundary value problem is solved by the method of fundamental solution. At last we give some examples to show the effectiveness of the present method in two and three dimensional case. A benchmark example is available for checking the feasibility and accuracy of the presented method. In the numerical examples, a part of exact solution for the boundary value problem is including for checking the accuracy. [ABSTRACT FROM AUTHOR]

An efficient finite element method using the sub-parametric transformations of the curved boundaries for the 45-node octic order triangular element is presented in this paper. The one-sided curved and two-sided straight triangle in the universal coordinate system (x , y) is mapped onto an isosceles triangle (ξ , η) ∈ [ 0 , 1 ] and ξ + η ≤ 1 in the (ξ , η)-coordinate system by using an isoparametric coordinate transformation. Using such a transformation, the curved edge of the triangular element is altered implicitly by a newly defined octic curve. Such an octic curve is obtained by using the topology of the curved edge. With an apparent condition of arc being a parabola, a new transformation is developed to find a unique curve for each edge that passes through the points on the original curved arc. Overall, the new sub-parametric transformations are developed, these include the transformation for the curved edge on the original boundary, in addition to the parameters from the curved triangle's interior. Several numerical experiments are presented to show the efficacy of the proposed method. The computational results obtained for a boundary value problem using octic order basis functions outperform the results obtained using the existing higher order basis functions such as quadratic (6-node), cubic (10-node), quartic (16-node), quintic (21-node), sextic (28-node), and septic (36-node) order triangular elements (TEs) with curves. The method implemented in this paper can be easily extendable to quasi-static and dynamic crack evolution problems within linear and nonlinear elasticity. [ABSTRACT FROM AUTHOR]

Numerical studies of flow in square enclosures attain great interest due to their wide applications in engineering, such as in the meteorology, navigation, mechanology and mining, etc. In this paper, the solutions are obtained to the square lid driven cavity describing classical mixed convection in the presence of an inclined magnetic field, chemical radiation and heat absorption/generation. The problem is mathematically reduced into Navier-Stokes equations. The Coiflet wavelet homotopy approach, a newly suggested computational technique, is then used to solve these nonlinear partial differential equations with their non-homogeneous boundary conditions. The numerical solutions for streamlines, isotherms, iso-concentrations contours, Nusselt number are carefully determined. The solutions so obtained are analyzed through various plots to demonstrate the effects of several physical parameters such as Reynolds numbers (Re), Richardson number (Ri), Grashof number (Gr), Hartman number (Ha), thermal radiation parameter (N R), chemical reaction coefficient (K), heat generation/absorption coefficient (Q h), the angle of inclination of magnetic field (γ), slipping parameter (λ) and Schmidt number (Sc) and their physical insights are also reported. Rigid comparison of with previous studies shows excellent computational effectiveness of newly proposed method. Additionally, according to the results, our recently proposed approach has superior characteristics and strong nonlinear processing capabilities, which render it more appropriate than the Coiflet wavelet analysis method and the traditional homotopy analysis method for tackling complex nonlinear problems. [ABSTRACT FROM AUTHOR]

We present an application of stable finite element (FE) approximations of convection-diffusion initial boundary value problems (IBVPs) using a weighted least squares FE method, the automatic variationally stable finite element (AVS-FE) method [1]. The transient convection-diffusion problem leads to issues in classical FE methods as the differential operator can be considered a singular perturbation in both space and time. The stability property of the AVS-FE method, allows us significant flexibility in the construction of FE approximations in both space and time. Thus, in this paper, we take two distinct approaches to the FE discretization of the convection-diffusion problem: i) considering a space-time approach in which the temporal discretization is established using finite elements, and i i) a method of lines approach in which we employ the AVS-FE method in space whereas the temporal domain is discretized using the generalized- α method. We also consider another space-time technique in which the temporal direction is partitioned, thereby leading to finite space-time "slices" in an attempt to reduce the computational cost of the space-time discretizations. We present numerical verifications for these approaches, including numerical asymptotic convergence studies highlighting optimal convergence properties. Furthermore, in the spirit of the discontinuous Petrov-Galerkin (DPG) method by Demkowicz and Gopalakrishnan [2–6] , the AVS-FE method also leads to readily available a posteriori error estimates through a Riesz representer of the residual of the AVS-FE approximations. Hence, the norm of the resulting local restrictions of these estimates serves as error indicators in both space and time for which we present multiple numerical verifications in mesh adaptive strategies. [ABSTRACT FROM AUTHOR]

In this paper, the Galerkin Finite Block Method (GFBM) is proposed for the first time to solve two-dimensional elasticity problems with functionally graded materials (FGMs). The physical variables are approximated with double layer Chebyshev polynomials with unknown coefficients for two-dimensional elasticity. The main idea is to establish a set of linear algebraic equations by the Galerkin method using Chebyshev polynomials in the weak form of the partial differential equations and the boundary conditions. By introducing the mapping technique, a block with irregular shape of the boundary is transformed from the Cartesian coordinates (x , y) to the normalized coordinates (ξ , η), | ξ | ⩽ 1 , | η | ⩽ 1. The continuity conditions of the displacement and traction on the interface between two blocks are introduced both in the weak form and strong form respectively. Several numerical examples are presented and comparisons are made with the Finite Element Method (FEM) including a cracked FGM sheet, in order to demonstrate the accuracy and efficiency of the GFBM. • True meshless method is developed using Galerkin method to two dimensional elasticity. • System matrix can be obtained analytically for homogeneous rectangular sheets. • Discontinuous boundary value problem can be solved with one block. • All integral functions in domain and boundary integrals are regular and can be obtained easily. • Numerical procedure is convergent with very high accuracy. [ABSTRACT FROM AUTHOR]

This paper mainly analyzes the heat transfer of Carreau fluid over an infinite rotating disk, and two models with variable thermal conductivity are considered. Firstly, the governing partial differential equations (PDEs) are transformed into ordinary differential equations (ODEs) by similarity transformation. Then, the boundary value problem is solved by improved bvp4c method, which reduced sensitivity to the initial values. For shear-thinning and shear-thickening fluids, the effects of Carreau fluid index n and Prandtl number Pr on velocity and temperature fields are shown and analyzed when 0.5 ≤ n ≤ 1.5 , 1 ≤ P r ≤ 10. Furthermore, the thermal conductivity is computed under two cases. [ABSTRACT FROM AUTHOR]

Transcorneal drug delivery has been used to treat eye diseases of both segments of the eye: anterior and posterior segments. Due to the low corneal permeability, this drug administration route has been shown to be very ineffective. Ultrasound has been used to increase the corneal drug transport. In this paper we consider a system of partial differential equations (PDEs) defined by a wave equation to describe the propagation of the pressure wave generated by a transducer, a convection-diffusion-reaction equation for the drug transport in each layer of the cornea - epithelium, stroma and endothelium. The last equations are linked with the acoustic pressure through the diffusion coefficient and the convective velocity. The system of PDEs is completed with convenient interface conditions, initial and boundary conditions. The initial boundary value problem is studied from analytical point of view and the qualitative behavior is numerically illustrated. The results confirm the effectiveness of ultrasound as enhancer of the drug delivery through the cornea. [ABSTRACT FROM AUTHOR]

This paper considers the error estimates of a family of Lagrange cubic finite volume methods for solving two dimensional elliptic boundary value problems over triangular meshes. By introducing a novel map of finite volume method, we propose a family of cubic finite volume method schemes with four independent map coefficients. It is detailedly proved that the proposed finite volume method schemes is exactly equivalent to the existing traditional finite volume method schemes of the same boundary value problems. A geometric constraint requirement on the triangular meshes of primary partitions is imposed to guarantee the uniform ellipticity of discrete elliptic bilinear forms resulted from the proposed schemes. The geometric requirements of the primary partitions could be greatly attenuated when the map coefficients are appropriately designed. Once the uniform ellipticity of the discrete elliptic bilinear forms is established, the proposed schemes have the unique solvability, and the optimal H 1 -norm error estimates without any restriction on dual elements of dual partitions of the finite volume methods, which is consistent with the results of literatures. With the help of the introduced map, the proposed schemes have the optimal L 2 -norm error estimates with a mesh restriction for the dual elements of their dual partitions significantly weaker than the existing theoretical results, in which the Aubin-Nitsche duality argument and the orthogonal conditions based on the dual partitions are employed. Finally, we validate our theoretical results by numerical experiments. [ABSTRACT FROM AUTHOR]

The method of fundamental solutions is a more and more popular meshless method for solving boundary or initial–boundary value problems. The most important issue in this method is the determination of the positions of the source points. The accuracy of the method depends strongly on the distribution of the source points. In this paper placement of these points for transient heat conduction problems is studied. The problems are initial–boundary value problems and they are considered in a time-space domain. Because of that, the placement of the source points differs from the classical distribution of the source points for boundary values problems. In the paper, four different possible sources distributions are considered for 1D, 2D and 3D transient heat conduction problems. The results show very good accuracy in case of the source points placed in a space much bigger than the considered region, additionally with the negative time coordinate. [ABSTRACT FROM AUTHOR]

The paper focuses on the new Multipoint meshless finite difference method, following the original Collatz higher order multipoint concept and the essential ideas of the Meshless FDM. The method was formulated, developed, and tested for various boundary value problems. Generalization of the multipoint method application to nonlinear analysis is the purpose of this research. The first attempt of the multipoint technique application to the geometrically nonlinear problems was successfully done recently. The case of physically nonlinear problem is considered in this paper. Several benefits of the proposed approach are highlighted, numerical algorithm and selected results are presented, and application of the multipoint method to nonlinear analysis is summarized. [ABSTRACT FROM AUTHOR]

In this paper, we consider a polynomial-augmented radial basis function collocation method (RBFCM) for solving fourth-order multi-dimensional elliptic partial differential equations (PDEs). Instead of choosing the RBF centres inside the problem domain, we use fictitious centres that can be located outside it to deal with the double boundary conditions of the fourth-order PDEs, and to also improve the computational accuracy. Furthermore, additional polynomial constraints applied to the approximation matrix of the RBFCM using fictitious centres, significantly alleviate the sensitivity of the method to the variation of the shape parameter, whereby it makes the RBFCM more practical for solving partial differential equations. We also apply the proposed method to a fourth-order elliptic PDE with random forcing terms (input data of the model). The input data are assumed to depend on a finite number of random variables at the collocation points. Numerical experiments are presented to demonstrate the effectiveness of the proposed method in terms of the selection of the shape parameter. In addition, the numerical results show that the proposed method achieves a higher accuracy than the conventional RBFCM. [ABSTRACT FROM AUTHOR]

The generalized alternating method of Schwarz was applied to boundary value problems for a multiply connected domain in the prior work Part I [62]. Approximate analytical formulas for the effective properties of dispersed composites with the exactly derived precision of their validity in concentration and in contrast parameter were derived. The present paper is devoted to comparison study of the formulas from Part I and the previously used engineering approximations. It is demonstrated that self-consistent methods (effective medium approximation, mean field, Mori-Tanaka methods etc) can be considered as the first order approximation of Schwarz's method. Implementation of the reiterated homogenization by means of Schwarz's method is also analyzed. This investigation explains a plenty of illusory different formulas which are actually reduced to the same lower order estimations for dilute composites. Some self-consistent methods violate the principles of homogenization and lead to methodologically misleading approaches. • A self-consistent method for composites is reduced to Schwarz's alternating method. • A methodology to study dispersed random composites is developed. • Limitations of various self-consistent methods are analyzed. [ABSTRACT FROM AUTHOR]

The boundary domain integral equation (BDIE) method provides an alternative formulation to a boundary value problem (BVP) with variable coefficient in terms of integral operators defined on the boundary and the domain. In this paper, we apply two variants of the boundary domain integral equation, the united approach and the segregated approach, to the Dirichlet BVP for the steady diffusion equation with variable coefficient in two dimensions. Details on the derivation of such systems as well as equivalence and well-posedness results are provided. Moreover, we present the discretisation of the two integral equation systems and a comparison of the numerical behaviour of the approximated solutions obtained with the segregated approach and the united approach. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a new exact solution based on the classical shell theory (CST) for free vibration of cross-ply laminated composite cylindrical shells with elastic restraint ends is proposed. The present exact solution can be summed up in the following steps: Firstly, the displacement functions are constructed by the governing differential equations with the exact closed form solutions; Then, the artificial spring technology is introduced to simulate the general boundary conditions of the two end edges of shell; Thirdly, the equation for natural frequencies is obtained by means of the method of reverberation-ray matrix (MRRM); Lastly, the vibration results are presented by the modified golden section search (MGSS) algorithm. By comparing the present method with published papers, the accuracy of present method is verified. On the basis of that, some new exact nature frequencies and mode shapes of the cross-ply laminated composite cylindrical shells with various classical boundary conditions and elastic restraints are performed and they can be served as the benchmark data for the future. [ABSTRACT FROM AUTHOR]

We investigate an iterative solver for the reduced eddy currents model set in open domains. Its formulation is the result of the coupling of a variational problem restricted to a truncated domain with a transparent condition at the fictitious boundary. The non-local boundary condition is derived using simple/double layer potentials. Applying a splitting technique yields an iterative algorithm that requires to solve, at each iteration, a local Neumann boundary value problem on the current potential. A re-interpretation of the algorithm as a Schwarz method enables a tractable convergence analysis which is the core subject of the paper. Using suitable variational tools, we establish an exponential convergence of the iterative solver. Some analytical and numerical examples are presented to check the theoretical findings and the relevance of our procedure. [ABSTRACT FROM AUTHOR]

• Alternating method of Schwarz is developed for dispersed composites. • A symbolic-numerical algorithm is analyzed. • New analytical formulas for the effective properties are derived. The generalized alternating method of Schwarz can be presented as an infinite sequence of all the mutual interactions between inclusions in the boundary value problem stated for a composite. Schwarz's method for overlapping domains has been assigned to decomposition methods widely used in pure numerical computations. In the same time, the generalized alternating method for non-overlapping inclusions is convenient in symbolic-numerical computations described in this paper. Implementation of the explicit and implicit schemes yields new approximate analytical formulas for the effective properties of dispersed composites. Moreover, the exactly derived precision of their validity in concentration and in contrast parameter is established. The first order iteration corresponds to the famous Maxwell's self-consistent approach. [ABSTRACT FROM AUTHOR]

In this paper, we consider local multiscale model reduction for problems with multiple scales in space and time. We developed our approaches within the framework of the Generalized Multiscale Finite Element Method (GMsFEM) using space–time coarse cells. The main idea of GMsFEM is to construct a local snapshot space and a local spectral decomposition in the snapshot space. Previous research in developing multiscale spaces within GMsFEM focused on constructing multiscale spaces and relevant ingredients in space only. In this paper, our main objective is to develop a multiscale model reduction framework within GMsFEM that uses space–time coarse cells. We construct space–time snapshot and offline spaces. We compute these snapshot solutions by solving local problems. A complete snapshot space will use all possible boundary conditions; however, this can be very expensive. We propose using randomized boundary conditions and oversampling (cf. Calo et al., 2016). We construct the local spectral decomposition based on our analysis, as presented in the paper. We present numerical results to confirm our theoretical findings and to show that using our proposed approaches, we can obtain an accurate solution with low dimensional coarse spaces. We discuss using online basis functions constructed in the online stage and using the residual information. Online basis functions use global information via the residual and provide fast convergence to the exact solution provided a sufficient number of offline basis functions. We present numerical studies for our proposed online procedures. We remark that the proposed method is a significant extension compared to existing methods, which use coarse cells in space only because of (1) the parabolic nature of cell solutions, (2) extra degrees of freedom associated with space–time cells, and (3) local boundary conditions in space–time cells. [ABSTRACT FROM AUTHOR]

In this paper, the time domain Maxwell’s equations are solved using the discrete exterior calculus (DEC) formalism in the two-dimensional space. To truncate the computational domain, the complex frequency-shifted perfectly matched layer (CFS-PML) concept is applied to create a reflectionless artificial layer. The paper presents a new numerical procedure to easily implement the CFS-PML with curved inner boundary. In order to numerically realize the PML, in a simplicial mesh, this paper proposes to utilize the nearest neighbor algorithm to associate point sets to boundary points. The distance from points to the boundary curve defines the attenuation function inside the PML. The performance of the approach is assessed by measuring the reflection error for three numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, a semi analytical method is used to investigate the free vibration of doubly-curved shells of revolution with arbitrary boundary conditions. The doubly-curved shells of revolution are divided into their segments in the meridional direction, and the theoretical model for vibration analysis is formulated by applying Flügge’s thin shell theory. Regardless of the boundary conditions, the displacement functions of shell segments are composed by the Jacobi polynomials along the revolution axis direction and the standard Fourier series along the circumferential direction. The boundary conditions at the ends of the doubly-curved shells of revolution and the continuous conditions at two adjacent segments were enforced by the penalty method. Then, the natural frequencies of the doubly-curved shells are obtained by using the Rayleigh–Ritz method. For arbitrary boundary conditions, this method does not require any changes to the mathematical model or the displacement functions, and it is very effective in the analysis of free vibration for doubly-curved shells of revolution. The credibility and exactness of proposed method are compared with the results of finite element method (FEM), and some numerical results are reported for free vibration of the doubly-curved shells of revolution under classical and elastic boundary conditions. Results of this paper can provide reference data for future studies in related field. [ABSTRACT FROM AUTHOR]

This paper deals with the following quasilinear chemotaxis-growth system u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( u ∇ v ) + μ u ( 1 − u ) , x ∈ Ω , t > 0 , v t = Δ v − v + w , x ∈ Ω , t > 0 , τ w t + δ w = u , x ∈ Ω , t > 0 , in a smoothly bounded domain Ω ⊂ R n ( n ≥ 3 ) under zero-flux boundary conditions. The parameters μ , δ and τ are positive and the diffusion function D ( u ) is supposed to generalize the prototype D ( u ) ≥ D 0 u θ with D 0 > 0 and θ ∈ R . Under the assumption θ > 1 − 4 n , it is proved that whenever μ > 0 , τ > 0 and δ > 0 , for any given nonnegative and suitably smooth initial data ( u 0 , v 0 , w 0 ) satisfying u 0 ≢ 0 , the corresponding initial–boundary problem possesses a unique global solution which is uniformly-in-time bounded. The novelty of the paper is that we use the boundedness of the | | v ( ⋅ , t ) | | W 1 , s ( Ω ) with s ∈ [ 1 , 2 n n − 2 ) to estimate the boundedness of | | ∇ v ( ⋅ , t ) | | L 2 q ( Ω ) ( q > 1 ) . Moreover, the result in this paper can be regarded as an extension of a previous consequence on global existence of solutions by Hu et al. (2016) under the condition that D ( u ) ≡ 1 and n = 3 . [ABSTRACT FROM AUTHOR]

In this paper, we present a modification of the Somigliana identity for the 3D Navier–Lamé equation in order to analytically include in its mathematical formalism the boundary represented by Coons and Bézier parametric surface patches. As a result, the equations called the parametric integral equation system (PIES) with integrated boundary shape are obtained. The PIES formulation is independent from the boundary shape representation and it is always, for any shape, defined in the parametric domain and not on the physical boundary as in the traditional boundary integral equations (BIE). This feature is also helpful during numerical solving of PIES, as from a formal point of view, a separation between the approximation of the boundary and the boundary functions is obtained. In this paper, the generalized Chebyshev series are used to approximate the boundary functions. Numerical examples demonstrate the effectiveness of the presented strategy for boundary representation and indicate the high accuracy of the obtained results. [ABSTRACT FROM AUTHOR]

The paper shows application of the method of fundamental solutions in combination with the radial basis functions for analysis of fluid flow and heat transfer in an internally corrugated tube. Cross-section of such a tube is mathematically described by a cosine function and it can potentially represent a natural duct with internal corrugations, e.g. inside arteries. The boundary value problem is described by two partial differential equations (one for fluid flow problem and one for heat transfer problem) and appropriate boundary conditions. During solving this boundary value problem the average fluid velocity and average fluid temperature are calculated numerically. In the paper the Nusselt number and the product of friction factor and Reynolds number are presented for some selected geometrical parameters (the number and amplitude of corrugations). It is shown that for a given number of corrugations a minimal value of the product of friction factor and Reynolds number can be found. As it was expected the Nusselt number increases with increasing amplitude and number of corrugations. [ABSTRACT FROM AUTHOR]

This paper presents a fast computational technique based on the wavelet collocation method for the numerical solution of an optimal control problem governed by elliptic variational inequalities of obstacle type. In this problem, the solution divides the domain into contact and noncontact sets. The boundary between the contact and noncontact sets is a free boundary, which is not known a priori and the solution is not smooth on it. Accordingly, a very fine grid is needed in order to obtain a solution with a reasonable accuracy. In this paper, our aim is to propose an adaptive scheme in order to generate an appropriate and economic irregular dyadic mesh for finding the optimal control and state functions. The irregular mesh will be generated such that its density around the free boundary is higher than in other places and high-resolution computations are focused on these zones. To this aim, we use an adaptive wavelet collocation method and take advantage of the fast wavelet transform of compact-supported interpolating wavelets to develop a multi-level algorithm, which generates an adaptive computational grid. Using this adaptive grid takes less CPU time than using a full regular mesh. At each step of the algorithm, the active set method is used for solving the optimality system of the obstacle problem on the adapted mesh. Finally, the numerical examples are presented to show the validity and efficiency of the technique. [ABSTRACT FROM AUTHOR]

A new weak Galerkin (WG) finite element method for solving the second-order elliptic problems on polygonal meshes by using polynomials of boundary continuity is introduced and analyzed. The WG method is utilizing weak functions and their weak derivatives which can be approximated by polynomials in different combination of polynomial spaces. Different combination gives rise to different weak Galerkin finite element methods, which makes WG methods highly flexible and efficient in practical computation. This paper explores the possibility of certain combination of polynomial spaces that minimize the degree of freedom in the numerical scheme, yet without losing the accuracy of the numerical approximation. Error estimates of optimal order are established for the corresponding WG approximations in both a discrete H 1 norm and the standard L 2 norm. In addition, the paper also presents some numerical experiments to demonstrate the power of the WG method. The numerical results show that the WG method achieves the optimal convergence order with a relatively low computational cost, which reveals a great promise of the flexibility and accuracy of the WG method. [ABSTRACT FROM AUTHOR]

In this paper, a fast finite volume method is proposed for the initial and boundary value problems of spatial fractional diffusion equations on nonuniform meshes. The discretizations of the Riemann-Liouville fractional derivatives lead to unstructured dense coefficient matrices, differing from the Toeplitz-like structure under the uniform mesh. The fast algorithm is proposed by using the sum-of-exponentials (SOE) technique to the spatial kernel x α − 1 , α ∈ (0 , 1). Then, the matrix-vector multiplications of the resulting coefficient matrices could be implemented in O (m log 2 ⁡ m) operations, where m denotes the size of matrices. Iterative solvers are preferably applied to obtain the numerical solution. The proposed fast scheme is proved to be unconditionally stable for sufficiently accurate SOE approximation. Meanwhile, a banded preconditioner is exploited to accelerate the Krylov subspace method. Numerical experiments are provided to demonstrate the efficiency of the proposed fast algorithm. [ABSTRACT FROM AUTHOR]

In this paper, the novel localized singular boundary method (LSBM) in conjunction with the Chebyshev collocation scheme (CCS) is developed to simulate large-scale inhomogeneous problems governed by elliptic operators. By means of the Gauss–Lobatto collocation points, the CCS is employed to obtain the particular solution. With the obtained particular solution, the inhomogeneous elliptic boundary value problem can be transformed into a classical homogeneous problem which can be solved by the LSBM. In the LSBM, the whole computational domain with distributed nodes is divided into a series of overlapping circle-shaped sub-domains. For each sub-domain, the moving least square method and the conventional singular boundary method (SBM) are utilized. Considering that the matrix generated by the CCS-LSBM is sparse, this scheme is granted the tremendous potential to address large-scale problems. Numerical experiments are presented to verify the effectiveness and accuracy of the proposed scheme for problems with mixed boundary conditions, sophisticated analytical solutions, and complex computational domains, respectively. [ABSTRACT FROM AUTHOR]

This paper presents a Cartesian grid based tailored finite point method (TFPM) for singularly perturbed reaction-diffusion equation on complex domains. The method is incorporated with the kernel-free boundary integral algorithm, where the semi-discrete boundary value problems after time integration are reformulated into corresponding Fredholm boundary integral equations (BIEs) of the second kind, however with no algorithmic dependence on the exact analytical expression for the kernels of integrals. The BIEs are iteratively solved by the GMRES method while integral evaluation during each iteration resorts to solving an equivalent interface problem, which in practice is achieved by a series of manipulations in the framework of TFPM including discretization, correction, solution, and interpolation. The proposed method has second-order accuracy for the reaction-diffusion equation as demonstrated by the numerical examples. [ABSTRACT FROM AUTHOR]

This paper is devoted to solving boundary value problems for important fractional differential equations of the Fokker–Planck family, in particular, to studying fractional differential equation for advection–dispersion. The consideration is carried out by the separation of variables (the Fourier method). Most part of this paper is devoted to justification of this method, to proof of a basis of the system of eigenfunctions for the basic equation (Aleroev et al., 2015) for modeling the random walk of a point particle which starts to move at the origin of coordinates in t = 0 on a self-similar fractal set. [ABSTRACT FROM AUTHOR]

This paper focuses on coupling two different computational approaches, namely finite element method (FEM) and meshless finite difference method (MFDM), in one domain. The coupled approach is applied in solving thermomechanical initial–boundary value problem where the heat transport in the domain is non-stationary. In this method, the domain is divided into two subdomains for FEM and MFDM, respectively. Contrary to other coupling techniques, the approach presented in this paper is defined in terms of mathematical problem formulation rather than at the approximation level. In the weak form of thermomechanical initial–boundary value problem (variational principle), the appropriate additional coupling integrals are defined a-priori. Subsequently, the FEM and the MFDM approximations, which may differ from each other, are provided to the formulation. It is assumed that there exists a very thin layer of material between the subdomains, which is not spatially discretized. The width of this layer may be considered the coupling parameter and it is the same for both, thermal and mechanical parts. Similar approach is applied to essential boundary conditions (e.g. prescribed temperature and displacements). Consequently, the consistent formulation of the mixed problem for the coupled FEM–MFDM method is derived. The analysis is illustrated with two- and three-dimensional examples of mechanical and thermomechanical problems. [ABSTRACT FROM AUTHOR]

In this paper, the localized Trefftz method (LTM) is proposed to accurately and efficiently solve two-dimensional boundary value problems, governed by Laplace and biharmonic equations, in complex domains. The LTM is formed by combining the classical indirect Trefftz method and the localization approach, so the LTM, free from mesh and numerical quadrature, has great potential for solving large-scale problems. For problems in multiply-connected domains, the solutions expressions in the proposed LTM is much simpler and more compact than that in the conventional indirect Trefftz method due to the localization concept and the overlapping subdomains. In the proposed LTM, both of the interior nodes and boundary nodes are required and the algebraic equation at each node, represents the satisfaction of governing equation or boundary condition, can be derived by implementing the Trefftz method at every subdomain. By enforcing the satisfaction of governing equations at every interior node and of boundary conditions at every boundary node, a sparse system of linear algebraic equations can be yielded. Then, the numerical solution of the proposed LTM can be efficiently obtained by solving the sparse system. Several numerical examples in simply-connected and multiply-connected domains are provided to demonstrate the accuracy and simplicity of the proposed LTM. Besides, the extremely-accurate solutions of the LTM are simultaneously demonstrated. [ABSTRACT FROM AUTHOR]

In this paper, we study initial–boundary value problem of semi-linear parabolic system (1.1) in Section 1 , and prove that any positive equilibrium of problem (1.1) is an initial datum threshold for the existence and nonexistence of global solution to it. For the precise statement of this result, see Theorem 1.1 in Section 1 of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we investigate magnetohydrodynamic free convective flow of micropolar fluid over a moving flat plate using the Lie group transformations and numerical methods. Instead of using conventional no-slip boundary conditions, we used both the velocity and thermal slip boundary conditions to achieve physically realistic and practically useful results. The governing boundary layer equations are non-dimensionalized and transformed into a set of coupled ordinary differential equations (ODEs) using similarity transformations generated by the Lie group, before being solved numerically using Matlab stiff ODE solver ode15s and Matlab trust-region-reflective algorithm lsqnonlin . The effects of governing parameters on the dimensionless velocity, angular velocity, temperature, skin friction and heat transfer rate are investigated. Our analysis revealed that the dimensionless velocity and angular velocity decrease whilst the dimensionless temperature increases with the velocity slip parameter. Thermal slip reduces the dimensionless velocity and temperature but raises the dimensionless angular velocity. Magnetic field suppresses the velocity but elevates the temperature and angular velocity. Results reported in this paper are in good agreement with the ones reported by the previous authors. [ABSTRACT FROM AUTHOR]

The paper presents a new effective technique for coupling two computational methods with different types of discretization and approximation. It is based on a concept of two adjacent subdomains which are connected with each other by means of a thin layer of material. Each of the subdomain may have a different discretization structure and approximation base. The standard Finite Element Method (FEM) as well as the meshless Finite Difference Method (MFDM) are applied here to be coupled. However, the coupling can be used for any other methods. The same concept is used for applying the essential boundary conditions for both of the methods. The width of the interface layer, depending on discretization density, is evaluated by means of several heuristic assumptions. The paper is illustrated with selected two- and three-dimensional examples. [ABSTRACT FROM AUTHOR]

In this paper we shall consider to improve the pre-asymptotic stability and error estimates of some h p -interior penalty discontinuous Galerkin ( h p -IPDG) methods for the Helmholtz equation with the first order absorbing boundary condition in two and three dimensions given in Feng and Wu (2011). The proposed h p -IPDG methods are defined using a sesquilinear form which is not only mesh-dependent (or h -dependent) but also degree-dependent (or p -dependent). By using a modified duality argument given in Zhu and Wu (2013), pre-asymptotic error estimates are improved for the proposed h p -IPDG methods under the condition of k h p ≤ C 0 ( p k ) 1 p + 1 in this paper, where C 0 is some constant independent of k , h , p , and the penalty parameters. It is shown that the pollution error of the method in the broken H 1 -norm is O ( k 2 p + 1 h 2 p ) if p = O ( 1 ) which coincides with existent dispersion analyses for the DG method on Cartesian grids. Numerical tests are provided to verify the theoretical findings and to illustrate great capability of the IPDG method in reducing the pollution effect. [ABSTRACT FROM AUTHOR]

In this study, we present a new type of domain decomposition algorithm to obtain the ground state solution for Bose–Einstein condensates. Using our proposed algorithm, instead of directly solving the nonlinear eigenvalue problem, one only needs to solve a series of linear boundary value problems on a finite element space sequence using the domain decomposition method, and subsequently solve a small-scale nonlinear eigenvalue problem on an enriched space simultaneously. Because solving large-scale nonlinear eigenvalue problem directly is time intensive, our algorithm can obviously improve the efficiency of producing simulation for Bose–Einstein condensates. In addition, any domain decomposition algorithm for linear boundary value problems can be applied to our algorithm framework, which makes the algorithm considerably flexible. Two numerical experiments are presented in the paper to demonstrate the efficiency and scalability of our proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, a new technique is shown for deriving computable, guaranteed lower bounds of functional type (minorants) for two different cost functionals subject to a parabolic time-periodic boundary value problem. Together with previous results on upper bounds (majorants) for one of the cost functionals, both minorants and majorants lead to two-sided estimates of functional type for the optimal control problem. Both upper and lower bounds are derived for the second new cost functional subject to the same parabolic PDE-constraints, but where the target is a desired gradient. The time-periodic optimal control problems are discretized by the multiharmonic finite element method leading to large systems of linear equations having a saddle point structure. The derivation of preconditioners for the minimal residual method for the new optimization problem is discussed in more detail. Finally, several numerical experiments for both optimal control problems are presented confirming the theoretical results obtained. This work provides the basis for an adaptive scheme for time-periodic optimization problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider the final boundary value problem for non-local Kirchhoff's model of parabolic type with discrete random noise. We first discuss the instability of solutions. Then we present the regularized solution by the trigonometric method in non-parametric regression associated with the truncated expansion method. In addition, under a prior assumption on the exact solution, the convergence rate is obtained. In the last section, the effectiveness and suitability of our results are justified via numerical experiments. [ABSTRACT FROM AUTHOR]

In the present study, firstly, based on the continuous time random walk (CTRW) theory, general diffusion equations are derived. The time derivative is taken as the general Caputo-type derivative introduced by Kochubei and the spatial derivative is the general Laplacian defined by removing the conditions (1.5) and (1.6) from the definition of the general fractional Laplacian proposed in the paper (Servadei and Valdinoci, 2012). Secondly, the existence of solutions of the Cauchy problem for the general diffusion equation is proved by extending the domain of the general Laplacian to a general Sobolev space. The results for positivity and boundedness of the solutions are also obtained. In the last, the existence result for solutions of the initial boundary value problem (IBVP) for the general diffusion equation on a bounded domain is established by using the Friedrichs extension of the general fractional Laplacian introduced in the book (Bisci et al., 2016). [ABSTRACT FROM AUTHOR]

In this paper, the fractional calculus approach is employed in the constitutive relationship of the fluid model and the Darcy's law. The flow of the fractional Jeffreys' fluid induced by the impulsive motion of a flat plate in a porous half space is studied in form of an initial and boundary value problem with fractional derivatives. Using the Laplace transform method, we obtain an exact solution of the model in term of Fox's H -function. As a byproduct, the solutions of the analogous flow for the generalized second grade fluid, the fractional Maxwell fluid and the Newtonian fluid in the porous half space are also deduced. In addition, the influence of the material parameters and the fractional parameters on the fluid motion is investigated, as well as a comparison among the fractional Jeffreys' fluid, the fractional Maxwell fluid and the Newtonian fluid in porous medium is also analyzed by graphical illustrations. [ABSTRACT FROM AUTHOR]

Abstract: Understanding the structure of diffusion flames is often complicated by the dependence of flame structure on the boundary conditions, e.g. composition, temperature and flow field (e.g., strain rate in a counterflow flame.) In this paper, the local carbon-to-oxygen atom ratio (C/O ratio) is applied as a variable to interpret the flame and soot zone structures of counterflow diffusion flames from numerical results with detailed chemical kinetics and transport. Radical pool and soot precursor zones are shown to be clearly delineated in C/O ratio space. The boundary of these two zones, as well as the flame location, are shown to be independent of both stoichiometric mixture fraction ( ) and strain rate when interpreted in C/O space. The kinetic ratio is used to study the characteristics of key chemical reactions and to identify regions of equilibrium for these reactions. The results of this paper indicate that the C/O ratio is a valuable variable for interpreting flame structure and soot precursor chemistry for diffusion flames. [Copyright &y& Elsevier]

Abstract: This paper is motivated from some recent papers treating the boundary value problems for impulsive fractional differential equations. We first make a counterexample to show that the formula of solutions in cited papers are incorrect. Second, we establish a general framework to find the solutions for impulsive fractional boundary value problems, which will provide an effective way to deal with such problems. Third, some sufficient conditions for the existence of the solutions are established by applying fixed point methods. Meanwhile, data dependence is obtained by using a new generalized singular Gronwall inequality. Finally, three examples are given to illustrate the results. [Copyright &y& Elsevier]

Abstract: There has been a recent increase in interest in the use of distributed order differential equations (particularly in the case where the derivatives are given in the Caputo sense) to model various phenomena. Recent papers have provided insights into the numerical approximation of the solution, and some results on existence and uniqueness have been proved. In each case, the representation of the solution depends, among other parameters, on Caputo-type initial conditions. In this paper we discuss the existence and uniqueness of solutions and we propose a numerical method for their approximation in the case where the initial conditions are not known and, instead, some Caputo-type conditions are given away from the origin. [Copyright &y& Elsevier]

Abstract: The objective of this paper is to present a numerical study of a class of boundary value problems of singularly perturbed differential difference equations (SPDDE) which arise in computational neuroscience in particular in the modeling of neuronal variability. The mathematical modeling of the determination of the expected time for the generation of action potential in the nerve cells by random synaptic inputs in dendrites includes a general boundary-value problem for singularly perturbed differential difference equation with shifts. The problem considered in this paper exhibit turning point behavior which add to the complexity in the construction of numerical approximation to the solution of the problem as well as in obtaining theoretical estimates on the solution. Exponentially fitted finite difference scheme based on Il’in-Allen-Southwell fitting is used on a specially designed mesh. Some numerical examples are given to validate convergence and computational efficiency of the proposed numerical scheme. Effect of the shifts on the layer structure is illuminated for the considered examples. [Copyright &y& Elsevier]

This paper is concerned with the application of the fast multipole method (FMM) to the pseudospectral methods. The spectral differentiation in pseudospectral methods involves (dense) matrix–vector products, which, if evaluated directly, entail O (N 2) operations. The core of this work lies in the fact that the FMM leads to a significant speed-up in CPU time with a major reduction in the amount of computer memory needed when performing matrix–vector products. The FMM is exploited to estimate the barycentric weights and the pseudospectral differentiation, which reduces the complexity to O (N). Furthermore, we show the convergence analysis and present the applications to the second-order boundary value problems. Numerical examples confirm the advantages of the method. [ABSTRACT FROM AUTHOR]

We study in this paper a multilayer discretization of second order elliptic problems, aimed at providing reliable multilayer discretizations of shallow fluid flow problems with diffusive effects. This discretization is based upon the formulation by transposition of the equations. It is a Petrov–Galerkin discretization in which the trial functions are piecewise constant per horizontal layers, while the test functions are continuous piecewise linear, on a vertically shifted grid. We prove the well posedness and optimal error order estimates for this discretization in natural norms, based upon specific inf–sup conditions. We present some numerical tests with parallel computing of the solution based upon the multilayer structure of the discretization, for academic problems with smooth solutions, with results in full agreement with the theory developed. [ABSTRACT FROM AUTHOR]

In this paper the quasi-static initial and boundary value problem for an elasto-plastic mixed hardening material is reformulated within the constitutive framework of small strains. The plastic factor plays the basic role in describing the rate independent evolution equations for the plastic strain and hardening variables. The plastic factor is equivalently represented as the solution of an appropriate local inequality involving the yield function. The main idea was to introduce the variational inequality at any time t to be solved for the velocity field and the complementary plastic factor. There is the plastic factor in a strain-driven process. The solution procedure proposed here to solve the initial and boundary value problem is based on the solutions of the variational inequality at time t , coupled with an update algorithm in order to evaluate the current state of the material for an incremental deformation process. This time the return mapping algorithm is avoided as the values of the plastic factor and the velocity are known at time t. As we developed a procedure to simultaneously solve the equilibrium equation coupled with the rate-independent evolution equations, no necessity to compute the algorithmic elasto-plastic tangent moduli occurs. The numerical simulations are done for the mixed hardening elasto-plastic model involving Armstrong–Frederick kinematic hardening. To validate the proposed numerical algorithms, we compare the solutions based on the variational inequality and those based on return mapping algorithm, computed for the same Prager kinematic hardening law. [ABSTRACT FROM AUTHOR]

Abstract B-spline Gaussian collocation software has been widely used in the numerical solution of boundary value ordinary differential equations (BVODEs) and 1D partial differential equations (PDEs) for several decades. Such packages represent the numerical solution in terms of a piecewise polynomial (B-spline) basis with basis coefficients determined through the use of Gaussian collocation. The software package, BACOL, developed over a decade ago, was the first 1D PDE package of this type to provide both temporal and spatial error control. A recently developed package, BACOLI, improves upon the efficiency of BACOL through the use of new types of spatial error estimation and control. The complexity of the interactions among the component numerical algorithms used by these packages (particularly the spatial and temporal error estimation and control algorithms) implies that extensive testing and analysis of the test results is an essential factor in the ongoing development of these packages In this paper, we investigate the performance of BACOL and BACOLI with respect to several important machine independent algorithmic measures, examine the effectiveness of the new spatial error estimation and control strategies, and investigate the influence of the choice of the degree of the B-spline basis on the performance of the solvers. These results will provide new insights into how to improve BACOLI, potentially lead to improvements in Gaussian collocation BVODE solvers, and guide further development of B-spline Gaussian collocation software with error control for 2D PDEs. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the unique continuation property for the Zakharov–Kuznetsov equation posed in a bounded domain. For this purpose, we establish a boundary Carleman estimate for the linear Zakharov–Kuznetsov operator. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with the diffusive minimal sediment model with no-flux boundary conditions. The steady-state bifurcation with two-dimensional kernel is firstly obtained with respect to this model by the space decomposition and the implicit function theorem. The existence of the Hopf bifurcation is next attained, and the direction and the stability of the Hopf bifurcation are analyzed in detail. Numerical simulations are done to support and complement the theoretical results. [ABSTRACT FROM AUTHOR]