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We consider the repair problem for Reed-Solomon (RS) codes, evaluated on an $\mathbb{F}_{q}$ -linear subspace $U \subseteq \mathbb{F}_{q^{m}}$ of dimension $d$ , where $q$ is a prime power, $m$ is a positive integer, and $\mathbb{F}_{q}$ is the Galois field of size $q$ . For $q > 2$ , we show the existence of a linear repair scheme for the RS code of length $n=q^{d}$ and codimension $q^{s}, s , evaluated on $U$ , in which each of the $n-1$ surviving nodes transmits only $r$ symbols of $\mathbb{F}_{q}$ , provided that $ms\geq d(m-r)$ . For the case $q=2$ , we prove a similar result, with some restrictions on the evaluation linear subspace $U$ . Our proof is based on a probabilistic argument, however the result is not merely an existence result; the success probability is fairly large (at least 1/3) and there is a simple criterion for checking the validity of the randomly chosen linear repair scheme.

The absolute zeta function for a scheme X of finite type over Z satisfying a certain condition is defined as the limit as p → 1 of the zeta function of X ⊗ F p . In 2016, after calculating absolute zeta functions for a few specific schemes, Kurokawa suggested that an absolute zeta function for a general scheme of finite type over Z should have an infinite product structure which he called the absolute Euler product. In this article, formulating his suggestion using a torsion free Noetherian F 1 -scheme defined by Connes and Consani, we give a proof of his suggestion. Moreover, we show that each factor of the absolute Euler product is derived from the counting function of the F 1 -scheme.

In this paper we are interested in investigating the physical shape-changed propagations to the generalized Equal-Width equation through studying the explicit solutions of Wazwaz-Benjamin-Bona-Mahony model. Both models are of considerable importance in many disciplines of research, including ocean engineering and science, and describe the propagation of equally-width waves. We highlight the effect of the coefficients of both nonlinearity and dispersion terms on changing the physical shape of both models by implementing the new exponential-expansion scheme. 2D and 3D graphical plots are provided to validate the findings of the paper.

In this article, developed the compact implicit difference method based Grunwald Letnikov formula (GLF) to compute the solution of the time-fractional diffusion-wave equation (TFDWE) describing wave propagation phenomenan having order α ( 1 α 2 ) . The fractional derivative is in Caputo sense. The theoretical analysis such as stability, consistency, convergence and solvability of the said scheme are discussed and proved that the scheme is conditionally stable and convergent with the order ( τ 2 + ( Δ x ) 4 ) . The numerical results compared with the recent existed method. The results of the numerical examples show that the GLF and the proposed method is very accurate and efficient for time fractional diffusion-wave equation.

In this paper, we present a novel scheme for solving a time-fractional initial–boundary value problem, where the equation contains a sum of Caputo derivatives with orders between 0 and 1. In order to overcome the difficulty of initial layer, we introduce a change of variable in the temporal direction and investigate the regularity of the solutions of the resulting system. A modified L 1 approximation is used to approximate the Caputo derivatives and a standard Galerkin-Spectral method is applied to approximate the spatial derivatives. Unconditional stability and convergence of the fully-discrete scheme are proved by applying a novel discrete fractional Gronwall inequality. Finally, numerical examples are given to confirm our theoretical results.

The fitted mesh numerical method is presented to solve parabolic problems involving an interior layer. A uniform mesh was applied in the temporal direction and fitted mesh type on the space variable yields the discrete problem. The convergence of the method is established and shown to the second-order in the temporal direction with the first-order convergent in the spatial one. Further, the scheme is proven to be parameter independent convergent and also confirmed by numerical experiments. In a nutshell, novelty of the present scheme is uniformly convergent and gives a more accurate solution than some existing methods in the literature.

In this work, we present linear exact repair schemes for one or two erasures in decreasing monomial-Cartesian codes DM-CC, a family of codes which provides a framework for polar codes. In the case of two erasures, the positions of the erasures should satisfy a certain restriction. We present families of augmented Reed-Muller (ARM) and augmented Cartesian codes (ACar) which are families of evaluation codes obtained by strategically adding vectors to Reed-Muller and Cartesian codes, respectively. We develop repair schemes for one or two erasures for these families of augmented codes. Unlike the repair scheme for two erasures of DM-CC, the repair scheme for two erasures for the augmented codes has no restrictions on the positions of the erasures. When the dimension and base field are fixed, we give examples where ARM and ACar codes provide a lower bandwidth (resp., bitwidth) in comparison with Reed-Solomon (resp., Hermitian) codes. When the length and base field are fixed, we give examples where ACar codes provide a lower bandwidth in comparison with ARM. Finally, we analyze the asymptotic behavior when the augmented codes achieve the maximum rate.

In this paper, we propose an efficient numerical method for the solution of one and two dimensional Riesz space fractional advection–dispersion equation. To this end, we use the Crank–Nicolson scheme to discretize this equation in temporal direction and prove that the semi-discrete scheme is unconditionally stable. Then, we apply the spectral element method in spatial directions and obtain the fully discrete scheme. We present an error estimate for the fully discrete scheme. The presented numerical results demonstrate the accuracy and efficiency of the proposed method in comparison with other schemes in literature.

We provide a semi-grammatical description of the set of normal proofs of positive formulae in minimal predicate logic, i.e. a grammar that generates a set of schemes, from each of which we can produce a finite number of normal proofs. This method is complete in the sense that each normal proof-term of the formula is produced by some scheme generated by the grammar. As a corollary, we get a similar description of the set of normal proofs of positive formulae for a large class of theories including simple type theory and System F.

We study the iso-manifold persistence in formulism. We employ a quasilinear iterative scheme to show that the majority of unperturbed tori give rise to invariant tori of the perturbed system starting from the same Riemannian manifold and keep the ratio of certain components of the respective frequencies. We also consider the iso-manifold Melnikov persistence.

In order to find the numerical solutions of the two-dimensional (2D) uniform transmission line equation, we first establish a fully second-order finite difference (FD) scheme in matrix-form and analyze the stability and convergence of the FD solutions by matrix analysis. Then we employ a proper orthogonal decomposition (POD) method to establish a reduced-order extrapolated FD (ROEFD) scheme containing very few unknowns for the 2D uniform transmission line equation and also make use of the matrix analysis to analyze the stability and convergence for the ROEFD solutions. Finally, we employ some numerical experiments to validate the effectiveness of the ROEFD scheme.

In this paper we present a stabilizer-free weak Galerkin (SFWG) finite element scheme for the approximation of the non-self adjoint and indefinite elliptic equations. Supercloseness convergence of the SFWG method is derived. After a suitable postprocessing of the SFWG solution, the resulting post-processed solution converges with two order higher than the optimal order in both H 1 semi-norm and L 2 norm on triangular meshes. Numerical experiments are demonstrated to verify the theoretical findings.

The paper introduces a new weak approximation scheme for Kolmogorov type hypoelliptic diffusions based on an operator splitting method and a Malliavin calculus approach. The approximation works even if the test function is not smooth, in other words, irregular functionals of hypoelliptic diffusions appearing in various fields are effectively approximated. A simple numerical algorithm is provided, which is easily implemented by a simulation method with minimum cost on random number generation. The effectiveness of the scheme is confirmed through numerical examples for irregular functionals of hypoelliptic diffusions.

This paper investigates the stabilization problem for impulsive Boolean networks (BNs) with stochastic disturbances. Firstly, dynamic evolution of the impulsive BN is converted into an algebraic form by using the semi-tensor product method of matrices. By analyzing the reachable set, the notion of distance between each state of the network and the equilibrium is defined, based on which the algorithm is then proposed for designing the appropriate aperiodic impulsive sequences. After that, based on the state-feedback control method, some criteria are presented to stabilize the Boolean control networks with periodic impulses and stochastic disturbances. Numerical examples are also presented to show feasibility of the theoretical results established.

The paper describes a novel method of sampled-data in space (spatial variable) nonlinear control of scalar semilinear parabolic and hyperbolic systems with unknown parameters, distributed disturbances and finite number of measurements along the spatial variable. Differently from recent results based on piecewise constant control laws, the proposed one is used piecewise nonlinear functions choosing by designer for providing some properties in the closed-loop system. In particular, we propose several types of functions providing reduced control. The gain design in the control law is found as a solution of linear matrix inequalities with minimum ultimate bound guarantee. The simulations confirm theoretical results and show the efficiency of the proposed control scheme compared with some existing ones.

In this paper, a new combined second order fully discrete scheme of an H 1 -Galerkin finite element method ( H 1 -GFEM) with two grid method (TGM) is developed for the bacterial equations. Based on Q 11 / Q 1 , 0 × Q 0 , 1 element pair, the existence and uniqueness of the approximation solution are proved, and the superclose and superconvergent estimates of order O ( h 2 + H 4 + τ 2 ) for the original variables in H 1 -norm and intermediate variables in L 2 -norm are derived, respectively. Some numerical results are provided to show that the proposed method has a very good performance and its computing cost is only about half of H 1 -GFEM for the considered problem. Here, the sizes of coarse mesh, fine mesh and time step are denoted by H , h , τ , respectively.

In this paper k always denotes a finite field of characteristic p > 0 . A variety over k means a reduced scheme separated and of finite type over k. An etale cover is always assumed to be finite (as in [SGA1, I, Def. 4.9] ).

In this paper, we focus on the numerical approximation of the Cahn-Hilliard-Hele-Shaw system. Firstly, based on the idea of the stabilized method, an unconditionally stable linear scheme with second-order accuracy in time and space is proposed, which is modified from the Crank-Nicolson scheme. Secondly, we derive that the proposed numerical scheme is unconditionally stable, without any restriction for the time step size. After a careful calculation, we get discrete error estimates of the time step size τ and space step size h. Finally, numerical simulations of energy dissipation and spinodal decomposition are presented to demonstrate the stability, accuracy and efficiency of the proposed scheme.

In this paper, the Finite Volume numerical scheme has been used to solve one-dimensional unsteady state and two-dimensional steady-state heat flow problems with the initial condition and Dirichlet boundary condition. Mainly two schemes are used to compute the numerical solutions of unsteady-state flow and one scheme for steady-state flow. The applicability of these schemes is illustrated using simple test problems. The numerical solutions obtained by these schemes are compared with the analytic solutions to check the accuracy of the developed scheme.

This paper presents a meshless finite point method (FPM) for the numerical analysis of the fractional cable equation. A second-order time discrete scheme is proposed to approximate both integer-order and fractional-order time derivatives. Then, based on the stabilized moving least squares approximation and the meshless smoothed gradients, a new implementation of the FPM is provided to enhance the accuracy and convergence rate in space. Theoretical error of the FPM is analyzed. Numerical results verify the efficiency of the method and show that the method can gain second-order accuracy in time and fourth-order accuracy in space.

The applicability of an efficient sixth convergence order scheme is extended for solving Banach space valued equations. In previous works, the seventh derivative has been used not appearing on the scheme. But we use only the first derivative that appears on the scheme. Moreover, bounds on the error distances and results on the uniqueness of the solution are provided (not given in earlier works) based on ω–continuity conditions. Numerical examples complete this article.

Based on the method of finite differences we propose a numerical blow-up time for a system of nonlinear ordinary differential equations, such system is defined by means of some parameters. Under certain conditions, on the parameters, the system blow-up in finite time. We introduce a numerical time and prove that this converges to the exact blow-up time and an error estimate is also given.

First-arrival traveltime tomography has been widely used for upper crustal velocity modeling, but it usually suffers from the problem of complex surface topography. To overcome this problem, we have developed a new topography-dependent eikonal tomography scheme that combines a developed accurate and efficient traveltime modeling method and introduces a flexible and robust adjoint inversion scheme in the presence of irregular topography. A surface-flattening scheme is used to handle the irregular surface, where the real model is discretized by curvilinear grids and the irregular free surface is mathematically flattened through the transformation from Cartesian to curvilinear coordinates. Based on this parameterization, the forward traveltime modeling is conducted by a monotone fast-sweeping method that discretizes the factored topography-dependent eikonal equation with a point-source condition. This algorithm can circumvent the source-singularity problem and decrease the numerical error in the vicinity of a point source in the curvilinear system. Then, the gradient-based inversion is used to minimize the misfit function, which is achieved by a matrix-free adjoint-state method without cumbersome ray tracing and explicit estimation of the Fréchet derivative matrix in the curvilinear coordinate system. The new tomographic scheme is evaluated through numerical examples with different seismic structures with complex topography, and then applied to a wide-angle profile acquired in the northeastern Tibetan Plateau. The results validate the effectiveness and efficiency of our tomography scheme in constructing shallow crustal velocity models with irregular topography.

In this paper we show that evolution algebras over any given field $\Bbbk$ are universally finite. In other words, given any finite group $G$, there exist infinitely many regular evolution algebras $X$ such that $Aut(X)\cong G$. The proof is built upon the construction of a covariant faithful functor from the category of finite simple (non oriented) graphs to the category of (finite dimensional) regular evolution algebras. Finally, we show that any constant finite algebraic affine group scheme $\mathbf{G}$ over $\Bbbk$ is isomorphic to the algebraic affine group scheme of automorphisms of a regular evolution algebra., Comment: Minor corrections. Bibliography updated. To appear in Proc. Amer. Math. Soc

We carry out error estimates for a linear, second-order, unconditionally energy stable, semi-discrete time stepping scheme, which is based on the Lagrange Multiplier approach and could also be viewed as a special case of the invariant energy quadratization approach, for the modified phase field crystal equation. At each time step, the proposed scheme only requires solving several linear equations. Rigorous proofs are presented to demonstrate the unique solvability, mass conservation, and unconditional energy stability of the scheme. Various numerical experiments in 2D and 3D are performed to validate the accuracy, unconditional energy stability and mass conservation of the proposed numerical strategy.

This paper is concerned with numerical solutions of one-dimensional (1D) and two-dimensional (2D) nonlinear coupled Schrodinger-Boussinesq equations (CSBEs) by a type of linearly energy- and mass- preserving finite difference methods (EMP-FDMs) because the existing EMP-FDMs for CSBEs are nonlinear and time-consuming, and corresponding theoretical analyses are not easy to generalize high-dimensional problems. Firstly, a linearized EMP-FDM is created for solving 1D CSBEs. By using the discrete energy analysis method, it is shown that this scheme is uniquely solvable and convergent with an order of O ( Δ t 2 + h x 2 ) in L ∞ -, H 1 - and L 2 -norms, and corresponding numerical energy and mass are conservative. Then, by generalizing this type of EMP-FDM, a linearized EMP-FDM is developed for solving 2D CSBEs. Theoretical findings including the convergence, the discrete conservative laws, and the solvability of this numerical algorithm are strictly derived in detail by using the discrete energy analysis method as well. Finally, numerical results confirm the efficiency of our algorithms and the exactness of the theoretical results.

Here, we use the modified Kudryashov’s algorithm and addendum to Kudryashov’s technique to obtain new optical solitons of the Ginsburg–Landau fractional complex of non-linearity (CGLE) laws in Kerr, which demonstrate various phenomena in physics such as non-linear waves, the transformation of the secondary phase, superconductivity, surfaces, liquid crystals, and field-theory strings. Dark, bright, singular solitons and combo bright-singular solitons are retrieved. The modified Kudryashov’s algorithm provides dark, singular solitons, and combo bright-singular solitons, although the addendum to Kudryashov’s technique ensures bright and singular solitons. These solutions are practiced for three optional definitions of derivative viz. conformable derivative, beta derivative, and M-truncated. Also shown are graphic representations of the solutions obtained.

We present a new generalization of the extensible bin packing with unequal bin sizes problem. In our generalization the cost of exceeding the bin size depends on the index of the bin and not only on the amount in which the size of the bin is exceeded. This generalization does not satisfy the assumptions on the cost function that were used to present the existing polynomial time approximation scheme (PTAS) for the extensible bin packing with unequal bin sizes problem. In this work, we show the existence of an efficient PTAS (EPTAS) for this new generalization and thus in particular we improve the earlier PTAS for the extensible bin packing with unequal bin sizes problem into an EPTAS. Our new scheme is based on using the shifting technique followed by a solution of polynomial number of $n$-fold programming instances. In addition, we present an asymptotic fully polynomial time approximation scheme (AFPTAS) for the related bin packing type variant of the problem.

In this paper, we present a high-order approach for solving one- and two-dimensional time-space fractional diffusion equations (FDEs) with Caputo-Riesz derivatives. To design the scheme, the Caputo temporal derivative is approximated using a high-order method, and the spatial Riesz derivative is discretized by the second-order weighted and shifted Grunwald difference (WSGD) method. It is proved that the scheme is unconditionally stable and convergent with the order of O ( τ α h 2 + τ 4 ) , where τ and h are time and space step sizes, respectively. We illustrate the accuracy and effectiveness of the method by providing several numerical examples.

This paper proposes a higher-order numerical approximation scheme to solve singularly perturbed reaction–diffusion boundary value problems. The proposed scheme is a combination of a fourth-order numerical difference method and classical central difference method applied on a non-equidistant grid. The non-equidistant grid is generated by equi-distributing a non-negative monitor function. The theoretical and empirical error analysis demonstrate that the proposed scheme has fourth-order uniform convergence with respect to the perturbation parameter.

This paper develops a high order semi-implicit weighted compact nonlinear scheme (WCNS) for one- and two-dimensional viscous Burgers’ equations. This semi-implicit WCNS combines a fifth-order WCNS with a third-order implicit-explicit Runge–Kutta (IMEX-RK) time-stepping method. The advection terms of viscous Burgers’ equations are treated explicitly, while the viscosity terms are treated implicitly. Stability analysis shows that the CFL condition of the semi-implicit WCNS is controlled only by the advection terms. Compared to the explicit time-stepping method, the semi-implicit method has the advantage in terms of computational efficiency. Numerical results validate the accuracy and efficiency of the semi-implicit WCNS. This method can also solve viscous Burgers’ equations with large Reynolds numbers and has high-resolution shock-capturing ability.

Let p be a prime number and k a perfect field of characteristic p. In the present paper, we study deformations of finite flat commutative group schemes over k to the ring W of Witt vectors with coefficients in k. We prove that, for a given principally quasi-polarizable p-torsion finite flat commutative group scheme over k, it holds that the group scheme is pseudo-rigid — i.e., roughly speaking, has a unique, up to isomorphism over W, deformation to W — if and only if the group scheme is superspecial.

In this paper, we provide a framework of designing the local discontinuous Galerkin scheme for integral fractional Laplacian $(-\Delta)^{s}$ with $s\in(0,1)$ in two dimensions. We theoretically prove and numerically verify the numerical stability and convergence of the scheme with the convergence rate no worse than $\mathcal{O}(h^{k+\frac{1}{2}})$., Comment: 11 pages

A new normal plane constraint for assessing the geometric non-linear behaviour of structures is proposed. In this procedure, the vector passing through a point placed on the equilibrium path and the iteration point is perpendicular to the locus of the iterative points. The capability of the suggested constraint equation was investigated numerically. To compare its abilities with other well-known techniques (the normal plane, cylindrical arc-length and updated normal plane methods), several planar and space trusses, plate and shells were analysed. The results proved the high accuracy and efficiency of the developed scheme. The powers of the methods in passing the limit points were also investigated. In addition, to assess the capabilities of the methods more accurately, they were all ranked based on the total number of increments and iterations required.

Direct Z-scheme g-C3N4/V2O5 photocatalysts were prepared through a sonication-assisted calcination method. The obtained samples were characterised by X-ray diffraction (XRD),Ultraviolet-visible diffuse reflectance spectroscopy (UV-Vis-DRS), Scanning electron microscope (SEM), andPhotoluminescence spectroscopy (PL). Oxidations of tetracycline hydrochloride (TC) were employed to evaluate the photocatalytic activities of the obtained g-C3N4/V2O5materials. Different weight ratios (5, 10, 15, and 20%) of g-C3N4/V2O5 loaded composites were prepared, in which a 15% (CV-15) loaded composite was found to show optimal catalytic performance for the reaction. The degradation conversation of TC has achieved approximately 79.67% in CV-15 after a 2-hour reaction. g-C3N4/V2O5 photocatalystwas more active than the individual g-C3N4 and V2O5 materials, which could be attributed to the efficient separation of photogenerated electron-hole pairs shown in the photocatalytic mechanism of TC degradation.

In this paper, a new kind of hybrid method based on the weighted essentially non-oscillatory (WENO) type reconstruction is proposed to solve hyperbolic conservation laws. Comparing the WENO schemes with/without hybridization, the hybrid one can resolve more details in the region containing multi-scale structures and achieve higher resolution in the smooth region; meanwhile, the essentially oscillation-free solution could also be obtained. By adapting the original smoothness indicator in the WENO reconstruction, the stencil is distinguished into three types: smooth, non-smooth, and high-frequency region. In the smooth region, the linear reconstruction is used and the non-smooth region with the WENO reconstruction. In the high-frequency region, the mixed scheme of the linear and WENO schemes is adopted with the smoothness amplification factor, which could capture high-frequency wave efficiently. Spectral analysis and numerous examples are presented to demonstrate the robustness and performance of the hybrid scheme for hyperbolic conservation laws.

In this paper, we consider the operator splitting scheme based on barycentric Lagrange interpolation collocation method for the two-dimensional Allen-Cahn equation. The original problem is split into linear and nonlinear subproblems: the linear part is solved by barycentric Lagrange interpolation collocation method in space and Crank-Nicolson scheme in time; the nonlinear part is solved analytically due to the availability of a closed-form solution. The error estimates of the proposed scheme are studied. Numerical experiments are carried out to demonstrate the accuracy and efficiency of the two operator splitting schemes.

In this paper, we present an optimal fourth-order parameter-uniform non-monotone scheme on equidistributed meshes for singularly perturbed reaction–diffusion boundary value problems exhibiting boun...

Rate distortion theory is concerned with optimally encoding signals from a given signal class $$\mathcal {S}$$ S using a budget of R bits, as $$R \rightarrow \infty $$ R → ∞ . We say that $$\mathcal {S}$$ S can be compressed at rates if we can achieve an error of at most $$\mathcal {O}(R^{-s})$$ O ( R - s ) for encoding the given signal class; the supremal compression rate is denoted by $$s^*(\mathcal {S})$$ s ∗ ( S ) . Given a fixed coding scheme, there usually are some elements of $$\mathcal {S}$$ S that are compressed at a higher rate than $$s^*(\mathcal {S})$$ s ∗ ( S ) by the given coding scheme; in this paper, we study the size of this set of signals. We show that for certain “nice” signal classes $$\mathcal {S}$$ S , a phase transition occurs: We construct a probability measure $$\mathbb {P}$$ P on $$\mathcal {S}$$ S such that for every coding scheme $$\mathcal {C}$$ C and any $$s > s^*(\mathcal {S})$$ s > s ∗ ( S ) , the set of signals encoded with error $$\mathcal {O}(R^{-s})$$ O ( R - s ) by $$\mathcal {C}$$ C forms a $$\mathbb {P}$$ P -null-set. In particular, our results apply to all unit balls in Besov and Sobolev spaces that embed compactly into $$L^2 (\varOmega )$$ L 2 ( Ω ) for a bounded Lipschitz domain $$\varOmega $$ Ω . As an application, we show that several existing sharpness results concerning function approximation using deep neural networks are in fact generically sharp. In addition, we provide quantitative and non-asymptotic bounds on the probability that a random $$f\in \mathcal {S}$$ f ∈ S can be encoded to within accuracy $$\varepsilon $$ ε using R bits. This result is subsequently applied to the problem of approximately representing $$f\in \mathcal {S}$$ f ∈ S to within accuracy $$\varepsilon $$ ε by a (quantized) neural network with at most W nonzero weights. We show that for any $$s > s^*(\mathcal {S})$$ s > s ∗ ( S ) there are constants c, C such that, no matter what kind of “learning” procedure is used to produce such a network, the probability of success is bounded from above by $$\min \big \{1, 2^{C\cdot W \lceil \log _2 (1+W) \rceil ^2 - c\cdot \varepsilon ^{-1/s}} \big \}$$ min { 1 , 2 C · W ⌈ log 2 ( 1 + W ) ⌉ 2 - c · ε - 1 / s } .

Numerous methods have been introduced in the literature for numerical solution of two-dimensional hyperbolic differential equations (telegraph equations). Improved techniques using explicit group methods derived from the standard and skewed (rotated) finite difference operators have been developed over the last few years in solving the linear systems that arise from the discretization of the several types of partial differential equations. In this paper, we present a preliminary study of the formulation of new preconditioned scheme based on explicit group relaxationmethods for the difference solution of the telegraph equations. The efficient and robustness of these new formulations over the existing explicit group schemes demonstrated through numerical experiments.

Let R be a regular local ring. Let G be a reductive R-group scheme. A conjecture of Grothendieck and Serre predicts that a principal G-bundle over R is trivial if it is trivial over the quotient field of R. The conjecture is known when R contains a field. We prove the conjecture for a large class of regular local rings not containing fields in the case when G is split., Comment: The final version to be published in Transactions of the AMS. Results about quadratic forms are strengthened. In the section on Bertini type theorems a correction in the case of a non-perfect residue field is made. Other minor corrections and improvements

Most of numerical methods for diffusion equations, refer to vertex unknowns directly or indirectly, and their accuracy is ultimately determined by the approximation to vertex unknowns. Based on the “twin-fitting” method, a simple and high accurate treatment for the vertex unknowns is developed and is applied to the nine-point scheme for diffusion problem. Numerical experiments show that the resulting nine-point scheme is high accurate for diffusion problems with discontinuous diffusion coefficients on distorted meshes.

The purpose of this paper is to recommend an iterative scheme to approximate a common element of the solution sets of the split problem of variational inclusions, split generalized equilibrium problem and fixed point problem for non-expansive mappings. We prove that the sequences generated by the recommended iterative scheme strongly converge to a common element of solution sets of stated split problems. In the end, we provide a numerical example to support and justify our main result. The result studied in this paper generalizes and extends some widely recognized results in this direction.

In this paper, the difference method with intrinsic parallelism is studied in order to meet the needs of quickly solving the variable-coefficient compound KdV-Burgers equation. The classical Crank-Nicolson (C-N) scheme is segmented on the basis of the alternating segment difference technique. And four different types of Saul'yev asymmetric schemes are alternately used at the segment points. Then we obtain the alternating segment Crank-Nicolson (ASC-N) difference scheme for the variable-coefficient compound KdV-Burgers equation. Theoretical analyses prove the existence and uniqueness of solution to ASC-N scheme, the linear absolute stability and the second-order convergence in time and space. Numerical experiments show that when the computational accuracy of the ASC-N scheme is similar to that of the C-N scheme, the computational efficiency of the ASC-N scheme is significantly higher than that of the C-N scheme. It shows that the difference method with intrinsic parallelism in this paper can effectively solve the variable-coefficient compound KdV-Burgers equation.

In this article, a hybrid Haar wavelet collocation method (HWCM) is proposed for the ill-posed inverse problem with unknown source control parameters. Applying numerical techniques to such problems is a challenging task due to the presence of nonlinear terms, unknown control parameter sources along the solution inside the given region. To find the numerical solution, derivatives are discretized adopting implicit finite-difference scheme and Haar wavelets. The computational stability and theoretical rate of convergence of the proposed HWCM are discussed in detail. Two numerical experiments are incorporated to show the well-condition behavior of the matrix obtained from HWCM and hence not required to supplement some regularization procedures. Moreover, the numerical solutions of the considered experiments illustrate the reliability, suitability, and correctness of HWCM.