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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 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 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.

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, 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.

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.

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.

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

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

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.

In this paper we investigate the finite element approximation of time fractional optimal control problem with integral state constraint. A space-time finite element scheme for the control problem is developed with piecewise constant time discretization and piecewise linear spatial discretization for the state equation. A priori error estimate for the space-time discrete scheme is derived. Projected gradient algorithm is used to solve the discrete optimal control problem. Numerical experiments are carried out to illustrate the theoretical findings.

In this work, an efficient alternating direction implicit (ADI) finite difference scheme is proposed to solve the three-dimensional time-fractional telegraph equation. The fully-discrete scheme is established via the L1 discrete formula in temporal direction and finite difference method in spatial direction. In order to reduce the computational cost of solving three-dimensional problems, an ADI algorithm is designed and applied. Then, via the energy method, we prove the stability and convergence of the scheme in L 2 and H 1 norms, respectively. Eventually, several numerical examples are given to validate the theoretical results.

We analyze a fully discrete pseudo-spectral scheme for the numerical solution of the initial-periodic boundary value problem for a nonlinear family of Boussinesq systems. The equations are discretized in space by the Fourier collocation method and in time by an explicit fourth order Runge–Kutta scheme. For the resulting fully discrete scheme, we prove error bounds of spectral accuracy in space and of fourth-order accuracy in time.

This paper considers the numerical treatment of the time-dependent Gross–Pitaevskii equation. In order to conserve the time invariants of the equation as accurately as possible, we propose a Crank–Nicolson-type time discretization that is combined with a suitable generalized finite element discretization in space. The space discretization is based on the technique of Localized Orthogonal Decompositions and allows to capture the time invariants with an accuracy of order O ( H 6 ) \mathcal {O}(H^6) with respect to the chosen mesh size H H . This accuracy is preserved due to the conservation properties of the time stepping method. Furthermore, we prove that the resulting scheme approximates the exact solution in the L ∞ ( L 2 ) L^{\infty }(L^2) -norm with order O ( τ 2 + H 4 ) \mathcal {O}(\tau ^2 + H^4) , where τ \tau denotes the step size. The computational efficiency of the method is demonstrated in numerical experiments for a benchmark problem with known exact solution.

In this paper, we develop a higher order numerical method for the fractional Bagley-Torvik equation. The main tools used include a new fourth-order approximation for the fractional derivative based on the weighted shifted Grünwald-Letnikov difference operator and a discrete cubic spline approach. We show that the theoretical convergence order improves those of previous work. Five examples are further presented to illustrate the efficiency of our method and to compare with other methods in the literature.

We obtain asymptotic accuracy of the poissonisation in the infinite occupancy scheme. All results are obtained for integer-valued random variables having a regularly varying distribution.

We establish necessary and sufficient conditions for the solvability of the nonlinear boundary-value problem in the critical case and develop a scheme for the construction of solutions of this problem. By using the Newton–Kantorovich method, we propose a new iterative scheme for the determination of solutions to the weakly nonlinear boundary-value problem for a system of ordinary differential equations in the critical case. As examples of application of the constructed iterative scheme, we obtain approximations to the solutions of a periodic boundary-value problem for the Duffing and Lienard equations. To check the accuracy of the established approximations to the solutions of the periodic boundary-value problem for the Duffing and Lienard equations, we use discrepancies in the original equations.

In this study, a new modification of the homotopy perturbation method (HPM) is introduced for various order boundary value problems (BVPs). In this modification, HPM is hybrid with least square optimizer and named as the least square homotopy perturbation method (LSHPM). The proposed scheme is tested against various linear and nonlinear BVPs (second to seventh order DEs). Validity of the obtained solutions is confirmed by finding absolute errors. To analyze the efficiency of the proposed scheme, tested problems have also been solved through HPM and results are compared with LSHPM. Furthermore, obtained results are also compared with other numerical schemes available in literature. Analysis reveals that LSHPM is a consistent and effective scheme which can be used for more complex BVPs in science and engineering.

Concentration properties of functionals of general Poisson processes are studied. Using a modified Phi-Sobolev inequality a recursion scheme for moments is established, which is of independent interest. This is applied to derive moment and concentration inequalities for functionals on abstract Poisson spaces. Applications of the general results in stochastic geometry, namely Poisson cylinder models and Poisson random polytopes, are presented as well. (C) 2021 Elsevier B.V. All rights reserved.

A fully discrete ADI scheme is proposed for solving the two-dimensional time-fractional diffusion equation with weakly singular solutions, where L1 scheme on graded mesh is adopted to tackle the initial singularity. An improved discrete fractional Gronwall inequality is employed to give an α-robust H 1 -norm convergence analysis of the fully discrete ADI scheme, where the error bound does not blow up when the order of fractional derivative α → 1 − . Numerical results show that the theoretical analysis is sharp.

In this paper, we provide a semi-implicit scheme based on the Euler–Maruyama method to study second-order stochastic systems driven by multiplicative independent additive noises. Using the Milstein’s fundamental convergence theorem, mean-square convergence analysis of the proposed approach is investigated. We examine the convergence of the stochastic system by considering the Lipschitz and linear growth bound conditions, Holder’s and Gronwall’s inequalities and also the Ito isometry. Finally, the numerical simulations demonstrate the potential of the introduced scheme to investigate the behavior of the second-order stochastic systems.