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• The convergence analysis and the techniques used in Lemma 3.3 by X. Wang et al. in [1] are based essentially on Sobolev's inequality in one dimension and they cannot be extended to two dimensions. • All results based on Lemma 3.3 are false, in particular Theorem 3.3, Theorem 4.1, Theorem 5.1 and Theorem 5.2. • Regarding the conservation of mass and energy, the initial terms Q 0 and E 0 depends on U 0 and on U 1 and the authors consider only Eq. (4) without taking into account Eq. (7). Therfore, there is an error in Theorem 3.1 and Theorem 3.2. • In Theorem 4.1, the authors did not show that u1 is uniquely determined by Eq. (7). In the above article, a second-order convergence in the maximum norm of the two-dimensional regularized long-wave equation is proved. However, due to a wrong inequality in Lemma 3.3 used in X. Wang et al.[1], there are some fundamental errors in this paper, in particular the proof of Theorem 3.3 and the convergence Theorem are no longer correct. The present brief paper contains clarifying comments. [ABSTRACT FROM AUTHOR]

In this paper, a class of non-negativity-preserving and maximum-principle-satisfying finite difference methods have been derived by Vieta theorem for one-dimensional and two-dimensional Fisher's equation. By using the positivity and boundedness of numerical and exact solutions, it is shown that numerical solutions obtained by current methods converge to exact solutions with orders of O (Δ t + (Δ t / h x) 2 + h x 2) for one-dimensional case and O (Δ t + (Δ t / h x) 2 + (Δ t / h y) 2 + h x 2 + h y 2) for two-dimensional case in the maximum norm, respectively. Here, Δ t , h x and h y are meshsizes in t -, x - and y -directions, respectively. Finally, numerical results verify that the proposed method can inherit the monotonicity, boundedness and non-negativity of the continuous problems. • New Du Fort-Frankel methods are devised for Fisher's equation. • They are non-negativity-preserving and maximum-principle-satisfying schemes. • Errors in maximum norm are given for them. • They are very good at long-term simulations. • They are very easy to be implemented. [ABSTRACT FROM AUTHOR]

• A real option is a contract which gives its holder the flexibility to expand the scale of an investment project or production. Real options are often used to hedge risks or capture opportunities in investments. In this paper, we establish a mathematical model for pricing a real option of expansion whose underlying asset price and its volatility/variance satisfy two separate stochastic equations. Based on Ito's lemma and a hedging technique, we show that the option price satisfies a 2 nd -order parabolic partial differential equation (PDE) in two spatial dimensions. We also derive the boundary and terminal conditions for the PDE and some of these conditions are also determined by PDEs. • We propose a novel 9-point finite difference scheme with a upwind technique is designed for solving the PDE system, as well that for determining the terminal (payoff) condition, established. We show that the coefficient matrix of the system from this discretization is an M-matrix and the numerical solution generated by the finite difference scheme converge to the exact one by proving that the scheme is consistent, monotone and stable. • Extensive numerical experiments on the model and numerical methods using a model investment problem in an iron-ore industry have been performed. The numerical results show that our model and numerical methods for solving the model are able to produce numerical results which are financially meaningful. In this paper we develop a PDE-based mathematical model for valuing real options on the expansion of an investment project whose underlying commodity price and its volatility follow their respective geometric Brownian motions. This mathematical model is of the form of a 2-dimensional Black-Scholes equation whose payoff condition is determined also by a PDE system. A novel 9-point finite difference scheme is proposed for the discretization of the spatial derivatives and the fully implicit time-stepping scheme is used for the time discretization of the PDE systems. We show that the coefficient matrix of the fully discretized system is an M -matrix and prove that the solution generated by this finite difference scheme converges to the exact one when the mesh sizes approach zero. To demonstrate the usefulness and effectiveness of the mathematical model and numerical method, we present a case study on a real option pricing problem in the iron-ore mining industry. Numerical experiments show that our model and methods are able to produce numerical results which are financially meaningful. [ABSTRACT FROM AUTHOR]

• Added missing literature references and improved literature review. • Fixed formulas going beyond the column. • Added experimental setup diagram. • Added a graph for comparing numerical results with experimental data. • Compacted part of the article with a proof of convergence, reduced the amount of information for readability. This paper presents a detailed and developed method for finding soil nonlinear thermophysical characteristics. Two-layer container complexes are constructed. The side faces of which are thermally insulated, so that one can use the one dimensional heat equation. In order not to solve the boundary value problem with a contact discontinuity and not to lose the accuracy of the solution method, a temperature sensor was placed at the junction of the two media, and a mixed boundary value problem is solved in each region of the container. In order to provide the initial data for the inverse coefficient problem, two temperature sensors are used: one sensor was placed on the open border of the container and recorded the temperature of the soil at this border, and the second sensor was placed at a short distance from the border, which recorded the air temperature. The measurements were carried out in the time interval (0 , 4 t max). At first, the initial-boundary value problem of nonlinear thermal conductivity equation with temperature dependent coefficients of thermal conductivity, heat capacity, heat transfer and material density is investigated numerically. The nonlinear initial-boundary value problem is solved by the finite difference method. Two types of difference schemes are constructed: linearized and nonlinear. The linearized difference scheme is implemented numerically by the scalar Thomas' method, and the nonlinear difference problem is solved by the Newton's method. Assuming the constancy of all thermophysical parameters, except for the thermal conductivity of the material, on the segment (0 , t max) , the quadratic convergence of Newton's method is proved and the thermal conductivity is calculated, while the solution of the linearized difference problem was taken as the initial approximation for the Newton's method. By freezing all other thermophysical parameters, and using the measured temperatures in the segment (t max , 2 t max) , the coefficient of specific heat of the material is found. Proceeding in a similar way, using the initial data on the segment (2 t max , 3 t max) , the density of the soil is determined. Also, based on the measured initial data on the segment (3 t max , 4 t max) , the heat transfer coefficient is determined. Based on the experimentally measured data, at each time interval, the corresponding functional is minimized using the gradient descent method. The differentiation of a nonlinear difference problem with respect to the desired parameter method is used to find the gradient of the functional. This method allows you to find the functional gradient and the damping coefficient of the gradient descent in an explicit form. In this case, the compilation and solution of the conjugate problem, which is a frequently used method for solving the inverse problems, is not needed. The performed numerical calculations show that for small time intervals the solutions of the linearized difference problem differ little from the solution of the nonlinear difference problem (1 - 3%). And for long periods of time, tens of days or months, the solutions of the two methods differ significantly, sometimes exceeding 20%. In addition, all thermophysical characteristics (8 coefficients) were found for a two-layer container with sand and black soil. Also the dependence of the temperature difference at the boundary between the environment and the soil for the heat transfer coefficient was shown. [ABSTRACT FROM AUTHOR]

• Extension of the Matrix Norm (MN) approach and its applications to the nonzero-sum bimatrix games are presented. • Convergence results of the MN approaches are given. • An improved interval for the game value for the zero/nonzero sum matrix games is provided. • The consistency of the approaches is shown by various bimatrix game examples from the literature. • It is shown that a better estimate of the game value can be obtained by repeated applications of the extended matrix norm (EMN) method. In this paper, we extend and apply the Matrix Norm (MN) approach to the nonzero-sum bimatrix games. We present preliminary results regarding the convergence of the MN approaches. We provide a notation for expressing nonzero-sum bimatrix games in terms of two matrix games using the idea of separation of a bimatrix game into two different matrix games. Next, we prove theorems regarding boundaries of the game value depending on only norms of the payoff matrix for each player of the nonzero-sum bimatrix game. In addition to these, we refine the boundaries of the game value for the zero/nonzero sum matrix games. Therefore, we succeed to find an improved interval for the game value, which is a crucial improvement for both nonzero and zero-sum matrix games. As a consequence, we can solve a nonzero-sum bimatrix game for each player approximately without solving any equations. Moreover, we modify the inequalities for the extrema of the strategy set for the nonzero-sum bimatrix games. Furthermore, we adapt the min-max theorem of the MN approach for the nonzero-sum bimatrix games. Finally, we consider various bimatrix game examples from the literature, including the famous battle of sexes, to demonstrate the consistency of our approaches. We also show that the repeated applications of Extended Matrix Norm (EMN) methods work well to obtain a better-estimated game value in view of the obtained convergence results. [ABSTRACT FROM AUTHOR]

In this paper, we present a fourth-order difference scheme for solving the Allen-Cahn equation in both 1D and 2D. The proposed scheme is described by the compact difference operators together with the additional stabilized term. As a matter of fact, the Allen-Cahn equation contains the nonlinear reaction term which is eminently proved that numerical schemes are mostly nonlinear. To solve the complexity of nonlinearity, the Crank-Nicolson/Adams-Bashforth method is applied in order to deal with the nonlinear terms with the linear implicit scheme. The well-known energy-decaying property of the equation is maintained by the proposed scheme in the discrete sense. Additionally, the L ∞ error analysis is carried out in the 1D case in a rigorous way to show that the method is fourth-order and second-order accuracy for the spatial and temporal step sizes, respectively. Concurrently, we examine the L 2 and H 1 error analysis for the scheme in the case of 2D. We consider the impact of the additional stabilized term on numerical solutions. The consequences confirm that an appropriate value of the stabilized term yields a significant improvement. Moreover, relevant results are carried out in the numerical simulations to illustrate the faithfulness of the present method by the confirmation of existing pieces of evidence. [ABSTRACT FROM AUTHOR]

• Numerical algorithms: Design efficient numerical methods for a novel linearly elastic shallow shell and clamped plate models, which could describe 3D problem by a 2D approximation. • Numerical analyses: Existence, uniqueness and convergence of numerical solutions. • Numerical experiments: Paraboloid shell, cylindrical shell and circular plate, rectangular plat. For the two-dimensional (2D) linearly elastic shallow shell and clamped plate models, we have provided the nonconforming finite element methods (FEMs) in a previous study. However, the conforming FEMs for above models have not been given. This is what we aim to do in this paper, namely proposing a conforming FEM for shallow shell and clamped plate models. Specifically, we approximate the first two displacement components by conforming linear element, and the third displacement component by conforming Hsieh-Clough-Tocher element. Furthermore, we rigorously deduce the convergent theorem independent of mesh steps. Finally, we confirm the theoretical results with some numerical experiments. [ABSTRACT FROM AUTHOR]

• A new unilateral constraint frictional contact model for a version of normal velocity is studied. • A coupled system that consists of a hemivariational inequality and a variational fihemivariational inequality is derived. • The unique weak solvability of the contact problem is obtained. A new model that describes a dynamic frictional contact between a viscoelastic body and an obstacle is investigated in this paper. We consider a nonlinear viscoelastic constitutive law which involves a convex subdifferential inclusion term and thermal effects. The contact condition is modeled with unilateral constraint condition for a version of normal velocity. The boundary conditions that describe the contact, friction and heat flux are govern by the generalized Clarke multivalued subdifferential. We derive a coupled system of two nonlinear first order evolution inclusions problems, which consists of a parabolic variational-hemivariational inequality for the displacement and a hemivariational inequality of parabolic type for the temperature. Then, the unique weak solvability of the contact problem is obtained by virtue of a fixed point theorem and the surjectivity result of multivalued maps. Finally, we deliver a continuous dependence result on a coupled system when the data are subjected to perturbations. [ABSTRACT FROM AUTHOR]

• The fourth-order compact difference scheme for symmetric regularized long wave (SRLW) equation for a single nonlinear velocity form are developed. • The scheme is four-time level linear scheme. • The discrete conservation, priori estimate, solvability, convergence with fourth-order in space and second-order in time and stability of the present scheme are proved in detail. • Numerical examples are given to support the theoretical analysis. In this paper, we develop a fourth-order accurate compact difference scheme for the symmetric regularized long wave (SRLW) equation for a single nonlinear velocity form. The discrete conservation, priori estimate, solvability, convergence and stability of the present scheme are proved by the discrete energy method. Numerical examples are given to support the theoretical analysis. [ABSTRACT FROM AUTHOR]

The present paper is devoted to constructing L2 type difference analog of the Caputo fractional derivative. The fundamental features of this difference operator are studied and it is used to construct difference schemes generating approximations of the second and fourth order in space and the (3 − α) th-order in time for the time fractional diffusion equation with variable coefficients. Difference schemes were also constructed for the variable-order diffusion equation and the generalized fractional-order diffusion equation of the Sobolev type. Stability of the schemes under consideration as well as their convergence with the rate equal to the order of the approximation error are proven. The received results are supported by the numerical computations performed for some test problems. [ABSTRACT FROM AUTHOR]

In this paper, we present a respectively scaled splitting-like (RSS-like) iteration method for block two-by-two linear systems from time-periodic parabolic optimal control problems. The detailed spectral properties of the RSS-like preconditioned matrix are analyzed and the unconditionally convergent properties of the RSS-like iteration method are described. Furthermore, we propose the optimal parameters of the RSS-like preconditioner. Numerical experiments are used to compare with some classical and recent efficient methods to show the efficiency of the new methods. [ABSTRACT FROM AUTHOR]

• A novel numerical approach for solving SIVIE has been proposed first time ever. • Operational matrices are used to transform the SIVIE into a system of algebraic equations. • Resultant system of algebraic equations has been solved by the collocation method. • This technique has the advantage of being simple to implement. • Error estimate, convergence, and stability analysis are also established. In this paper, a numerical method is implemented to solve the stochastic Itô-Volterra integral equations. In this approach, operational matrices have been applied to reduce the stochastic Itô-Volterra integral equations to linear algebraic equations. Then collocation method is applied to solve the algebraic equations. The error, convergence, and stability analysis of the proposed method are discussed. Also, the steps of the proposed method have been presented in the form of an algorithm. Numerical examples are introduced to confirm the efficiency and reliability of the proposed scheme. [ABSTRACT FROM AUTHOR]