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This paper presents an iterative approach for solving the optimal control problem (OCP) with time-delayed terms and bounded control variables. The projection function is used to construct the iterative method to approximate the control law. Employing the approximation of control law the approximation of state and the co-state variables are obtained. To the best of our knowledge, this paper is the first attempt to solve bounded and delayed OCPs with a projection function. The convergence theorem for the proposed method is also proven. Several numerical examples are added, showing the developed method's effectiveness to solve the time-delayed OCPs with bounded control variables. [ABSTRACT FROM AUTHOR]

The paper focuses on investigating the time-fractional (3 + 1)-dimensional cubic and quantic nonlinear Schrödinger equation. We adopt the novel Kudryashov method to generate a distinct class of optical solutions for the current conformable fractional derivative problem. Our method explores various solution forms, including dark, wave, mixed dark-bright, and singular solutions. The soliton solutions we construct are visually represented to illustrate the influence of the fractional order derivative. Further, we elucidate the influence of solution parameters on the wave envelope, providing clear interpretations through 2D graphics presentations. The results underscore the efficacy of our approach in discovering exact solutions for nonlinear partial differential equations, especially in cases where alternative methods prove ineffective. The significance of the present paper lies in its contribution to advancing the understanding of the behavior of optical solutions in nonlinear systems, providing valuable insights for both theoretical and practical applications. In the field of nonlinear optics, this equation can describe the propagation of optical pulses in nonlinear media. It helps in understanding the behavior of intense laser beams as they propagate through materials exhibiting nonlinear optical effects such as self-focusing, self-phase modulation, and optical solitons. [ABSTRACT FROM AUTHOR]

The authors present numerical analysis of singularly perturbed parabolic problems. The previous papers in this direction focussed on the convection-diffusion problems with regular type of boundary layers. It is very difficult and almost impossible as stated in the literature, e.g, in Chapter 14 of 'Fitted numerical methods for singular perturbation problems: error estimates in the maximum norm for linear problems in one and two dimensions' (1996) by Miller, John JH and O'Riordan, Eugene and Shishkin, Grigorii I, to capture singularities due to parabolic layers and develop a robust scheme for reaction-diffusion problems with general values of space shifts. The work is in progress now and this is the first paper in this direction. In this work, we are investigating such problems and develop a robust numerical scheme using Mickens's non-standard finite difference scheme and special mesh. Furthermore, numerical examples have been presented to verify the theoretical findings. [ABSTRACT FROM AUTHOR]

The aim of the paper is to derive minimization algorithms based on the Nesterov accelerated gradient flow [Y. Nesterov, Gradient methods for minimizing composite objective function. Core discussion paper, (2007). Available at ; Y. Nesterov, A method of solving a convex programming problem with convergence rate $ \mathcal {O}(1/k^2) $ O (1 / k 2). In Doklady Akademii Nauk, Vol. 269, Russian Academy of Sciences, 1983, pp. 543–547; Y. Nesterov, Introductory Lectures on Convex Optimization: A basic course, Kluwer Academic Publishers, Massachusetts, 2004.] to compute the ground state of nonlinear Schrödinger equations, which can potentially include a fractional laplacian term. A comparison is developed with standard gradient flow formulations showing that the Nesterov accelerated gradient flow has some interesting properties but at the same time finds also some limitations due to the nature of the problem. A few simulations are finally reported to understand the behaviour of the algorithms and open the path to further complicate questions that require more advanced studies concerning the application of the Nesterov accelerated gradient flow to nonlinear Schrödinger equations. [ABSTRACT FROM AUTHOR]

In this paper, we extend SMSNS [Pourbagher M, Salkuyeh DK. On the solution of a class of complex symmetric linear systems. Appl Math Lett. 2018;76:14–20.] iteration method for solving a class of complex symmetric system of linear equations. We propose a successive-overrelaxation (SOR) acceleration scheme for SMSNS (SMSNSSOR), discuss the convergence conditions of it and give the optimal parameters which make the fast convergence. Numerical results demonstrate that SMSNSSOR iteration method is feasible and effective for solving complex symmetric systems, and performs better than some other usually used iteration methods. [ABSTRACT FROM AUTHOR]

The Heston model is a well-known two-dimensional financial model. Because the Heston model contains implicit parameters that cannot be determined directly from real market data, calibrating the parameters to real market data is challenging. In addition, some of the parameters in the model are non-linear, which makes it difficult to find the global minimum of the optimization problem within the calibration. In this paper, we present a first step towards a novel space mapping approach for parameter calibration of the Heston model. Since the space mapping approach requires an optimization algorithm, we focus on deriving a gradient descent algorithm. To this end, we determine the formal adjoint of the Heston PDE, which is then used to update the Heston parameters. Since the methods are similar, we consider a variation of constant and time-dependent parameter sets. Numerical results show that our calibration of the Heston PDE works well for the various challenges in the calibration process and meets the requirements for later incorporation into the space mapping approach. Since the model and the algorithm are well known, this work is formulated as a proof of concept. [ABSTRACT FROM AUTHOR]

We investigate in this paper an inverse problem of recovering the space-dependent volatility in option pricing. To enhance precision across the domain, we transform the original problem into an inverse source problem of a bounded degenerate parabolic equation, utilizing linearization and variable substitution. Unlike classical methods, we apply a total variation regularization combined with a novel generalized finite integration technique. This approach accommodates volatility jumps in an overnight rate scenario. Leveraging an optimal control framework, we demonstrate that the inverse problem can be reformulated as an optimal control problem whose existence, necessary conditions, local uniqueness and stability for the minimizer of control functional are obtained. For numerical verification, we derive the Euler equation and design a discretization algorithm with generalized finite integration technique. Numerical examples showcase the robustness of our approach, highlighting advantages in accuracy and effectiveness over other strategies. [ABSTRACT FROM AUTHOR]

This paper demonstrates the mixed formulation of the Brinkman problem using linear equal-order finite element methods in porous media modelling. We introduce Galerkin least-squares (GLS) and least-squares (LS) finite element methods to address the incompatibility of finite element spaces, treating velocity, pressure, and vorticity as independent variables. Theoretical analysis examines coercivity and continuity, providing error estimates. Demonstrating resilience in theoretical findings, these methods achieve optimal convergence rates in the $ L^2 $ L2 norm by incorporating stabilization terms with low-order basis functions. Numerical experiments validate theoretical predictions, showing the effectiveness of the GLS method and addressing finite element space incompatibility. Additionally, the GLS method exhibits promising capabilities in handling the Brinkman equation at low permeability compared to the LS method. The study reveals an increase in the average pressure difference in the Brinkman problem compared to the Stokes equations as the inlet velocity rises, providing insights into the behaviour of Brinkman equations. [ABSTRACT FROM AUTHOR]

First, this paper defines a general nonlinear optimal control problem with state/control constraints and its approximation problem as the Haar wavelet Galerkin optimal control problem (HWGOCP). Then, a Haar wavelet-based Galerkin numerical method has been developed, which converts it to a nonlinear optimization problem. We theoretically prove that a Haar wavelet feasible solution of HWGOCP will exist. We also show that the approximate solutions of HWGOCP are consistent and converge to the optimal solution of the problem. A variety of application problems have been considered, which include optimal control of tumour growth using Chemotherapy drugs, optimal control of infection via the SIS model using treatment, the Brachistochrone problem in mechanics, optimal control of mold using a fungicide, optimal control of pH value of a chemical reaction to determine the quality of a product, etc. [ABSTRACT FROM AUTHOR]

Nonlinear Kawarada equations have been used to model solid fuel combustion processes in the oil industry. An effective way to approximate solutions of such equations is to take advantage of the finite difference configurations. Traditionally, the nonlinear term of the equation is linearized while the numerical stability of a difference scheme is investigated. This leaves certain ambiguity and uncertainty in the analysis. Based on nonuniform grids generated through a quenching-seeking moving mesh method in space and adaptation in time, this paper introduces a completely new stability analysis of the approximation without freezing the nonlinearity involved. Pointwise orders of convergence are investigated numerically. Simulation experiments are carried out to accompany the mathematical analysis to strengthen our conclusions. [ABSTRACT FROM AUTHOR]

In this manuscript, we investigate a stochastic version of the Allen–Cahn equation, which is a nonlinear partial differential equation from mathematical physics. The stochastic model considers temporal multiplicative noise in the form of the derivative of the standard Wiener process. The existence and the uniqueness of solutions for this stochastic model is established rigorously using the theory of distributions. As a corollary from these analytical results, some a priori optimal estimates for the solutions of this model are constructed. In this work, we develop reliable numerical schemes which possess similar features as those of the solutions for the analytical model. Moreover, we establish mathematically the von Neumann stability and the consistency for the schemes proposed in this paper. Both the analytical and numerical results derived in this work are computationally verified through some simulations. [ABSTRACT FROM AUTHOR]

This paper presents an analytical approach to study the generalized variable coefficient Gardner equation, aiming to investigate the behaviour of plasma phenomena, which presents challenges due to the non-linear nature of the Gardner equation. The point symmetry group associated with the proposed equation is determined by Lie symmetry analysis. The Exp-function method is employed to construct exact travelling wave solutions and to obtain anomalous solutions with non-zero amplitude. Additionally, conservation laws are derived and analysed from the equation in the frame of Noether symmetry. Through simulation, breather, periodic, and localized solitary wave solutions are also presented in graphical form. Our results are expected to elucidate the impact of the equation in the study of the propagation of Alfvén waves with non-uniform density, temperature, and wave-particle interactions in a magnetized plasma. [ABSTRACT FROM AUTHOR]

The noterate is a tool for predicting home mortgage rates, it is often skewed and has missing information. The noterate could be affected by incomplete or inaccurate data, thus leading to inaccurate predictions. Financial organizations including mortgage companies or banks need to consider the risk of uncertainty carefully and make a more accurate prediction based on some suitable models. To deal with this situation, in this paper we compared six computational statistical methods, including the ordinary least square model, maximum likelihood estimation, maximum a posterior, bootstrapping, Metropolis-Hastings, and Gibbs sampling method on a mortgage dataset. Based on the k fold cross-validation technique and four metrics including mean absolute error (MAE), mean squared error (MSE), root mean squared error (RMSE), and mean absolute percentage error (MAPE), the bootstrapping method outperforms other methods. In practice, this method is recommended for predicting noterate. [ABSTRACT FROM AUTHOR]

In this paper, we propose a space-time Galerkin spectral method for the time fractional Fokker–Planck equation. This approach is based on combining temporal Müntz Jacobi polynomials spectral method with spatial Legendre polynomials spectral method. Based on the well-posedness and regularity for the re-scaled problem of a linear model problem which reflects the main difficulty for solving the equivalent equation (i.e. the time fractional convection-diffusion equation): the singularity of the solution in time, we explain in detail why we use the Müntz polynomials to approximate in time. The well-posedness and stability of the discrete scheme as well as its continuous problem are established. Moreover, the error estimation of the space-time approach is derived. We find that the proposed method can attain spectral accuracy regardless of whether the solution of the original equation is smooth or non-smooth. Numerical experiments substantiate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a Euler wavelets method for solving the variational problems is presented. The operational matrices of integration and product of Euler wavelets are calculated. Then, by using Euler wavelets and the operational matrices, the variational problems are reduced into the system of algebraic equations. Furthermore, the convergence analysis and error bound of the Euler wavelets method are given. Some examples are included to demonstrate the applicability and validity of the schemes. [ABSTRACT FROM AUTHOR]

This work presents a new three-term projection algorithm for solving nonlinear monotone equations. The paper is aimed at constructing an efficient and competitive algorithm for finding approximate solutions of nonlinear monotone equations. This is based on a new choice of the conjugate gradient direction which satisfies the sufficient descent condition. The convergence of the algorithm is shown under Lipschitz continuity and monotonicity of the involved operator. Numerical experiments presented in the paper show that the algorithm needs a less number of iterations in comparison with existing algorithms. Furthermore, the proposed algorithm is applied to solve signal recovery problems. [ABSTRACT FROM AUTHOR]

The increase in antibiotic resistance in recent years, mainly due to the non-rational use of antibiotics, is one of the most important global public health threats. In this paper, we propose a mathematical dynamic random model describing the antibiotic resistance evolution of a bacteria and where antibiotic consumption is included is the main driving force in the resistance increase. The random model is solved using the Random Variable Transformation technique and is applied to study the case of Acinetobacter baumannii bacterium resistant to the antibiotic colistin in Valencia, Spain. Using the Multi-Objective Particle Swarm Optimization algorithm, the model has been calibrated with the A. baumannii colistin-resistance and colistin consumption data series. With the optimal model, four possible 7-year future scenarios with different antibiotic consumption trends have been simulated. The model results show how reducing antibiotic consumption does not easily stop the increase in resistance. [ABSTRACT FROM AUTHOR]

In this paper, we concern the strong convergence and stability of the balanced Euler method for stochastic pantograph differential equations with Markovian switching (SPDEs-MS). We present the balanced Euler method of SPDEs-MS and consider its moment boundedness under polynomial growth condition plus Khasminskii-type condition. We also study its strong convergence order and its mean-square stability. Two numerical examples are given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Vanilla options become effective immediately after they are entered, while some exotic options will only come to effective some time after they are bought or sold. Forward starting options are one kind of such exotic options started actually at some pre-specified future date. This paper presents an extension of regime-switching jump diffusion model, in which the parameters are driven by a continuous time and stationary Markov chain on a finite state space, by introducing the Wishart process into the instantaneous variance-covariance matrix of the risky asset price and stochastic interest rate. We derive the discounted conditional joint characteristic function and the forward characteristic function of the log-asset price and its the instantaneous variance-covariance process, and thereby the price of forward starting options are well evaluated by the probabilistic approach combined with the Fourier-cosine (COS) method. We also provide efficient Monte Carlo simulation of this proposed model, and simulated solutions to forward starting options pricing within a two-state regime switching framework. Numerical results show that the COS method is accurate and efficient for pricing forward starting options. Finally, we analyse impacts of some main parameters (especially, parameters in the Wishart stochastic volatility) in this proposed model on option prices and Δ values. Also, we consider the forward implied volatility. Furthermore, the forward starting options under the regime-switching jump diffusion model with Wishart stochastic volatility and stochastic interest rate which we derived are more generalized than those recently appeared in the derivatives pricing literature, and thus have wider application. [ABSTRACT FROM AUTHOR]

A reaction–diffusion problem with discontinuous source term and Dirichlet's boundary conditions on the unit square is considered in this paper. The proposed problem has been discretized using a combination of standard Galerkin finite element method (FEM) and non-symmetric discontinuous Galerkin finite element method with an interior penalty (NIPG) with bilinear elements. Layer adapted mesh of Shishkin type has been utilized to discretize the domain. Standard Galerkin FEM is applied on the layer part of the domain where the domain is dense enough and NIPG is applied to the outside layer part. By means of special choice of discontinuity-penalization parameters, the scheme is proved to be uniformly convergent of order $ \mathcal {O}(\varepsilon ^{1/4}N^{-1} + N^{-1} \ln N) $ O (ε 1 / 4 N − 1 + N − 1 ln ⁡ N). Numerical tests are carried out in support of theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, two model order reduction methods based on Laguerre orthogonal polynomials for parabolic equation constrained optimal control problems are studied. The spatial discrete scheme of the cost function subject to a parabolic equation is obtained by Galerkin approximation, and then the coupled ordinary differential equations of the optimal original state and adjoint state with initial value and final value conditions are obtained by Pontryagin's minimum principle. For this original system, we propose two kinds of model order reduction methods based on the differential recurrence formula and integral recurrence formula of Laguerre orthogonal polynomials, respectively, and they are totally different from these existing researches on model order reduction only for initial value problems. Furthermore, we prove the coefficient-matching properties of the outputs between the reduced system and the original system. Finally, two numerical examples are given to verify the feasibility of the proposed methods. [ABSTRACT FROM AUTHOR]

This paper shows the outcomes of computer virus propagation (CVP) model, represented with susceptible, exposed, infected, quarantine and recovered computers (SEIRQ), classes based mathematical model using the stochastic procedures. The systematic study of the CVP based SEIRQ model represents that the equilibrium state of virus-free is stable globally with reproduction not more than one, while the viral symmetry is attractive globally. The numerical performances of the CVP based SEIRQ model are presented by using the stochastic computational framework based on the artificial neural networks (ANNs) together with the Levenberg-Marquardt backpropagation (LBMB) called as ANNs-LBMB. The learning procedures via ANNs-LBMB for solving the CVP based SEIRQ model are implemented to indorse the statics using the testing, authorization, and training. Thirteen numbers of neurons and the data selection for training 72%, testing 12% and validation 16% are selected to solve the model. For the numerical outcomes of the CVP based SEIRQ model using the ANNs-LBMB, a dataset is considered through the Adams approach. The accuracy and reliability performances of the scheme are presented by using the values of the absolute error (AE) along with the observations of state transitions (STs), regression, mean square error (MSE) and error histograms (EHs). [ABSTRACT FROM AUTHOR]

In this paper, we introduce a general class of the inverse system of nonlinear fractional order partial differential equations (GCISNF-PDEs) with initial-boundary and two overdetermination conditions. An optimization method is considered based on the generalized shifted Legendre polynomials (GSLPs) for solving GCISNV-FPDEs. The concept of the fractional order derivatives (F-Ds) is utilized in the Caputo type. Operational matrices (OMs) of classical derivatives and F-Ds of GSLPs are extracted. Making use of GSLPs, OMs, and Lagrange multipliers method, we reduce the given GCISNF-PDEs into an algebraic system of equations. The proposed approach achieves satisfactory results simply for a small number of the novel GSLPs. In this work, two mathematical examples are illustrated to analyse the introduced method convergence and test its validity as well as applicability. [ABSTRACT FROM AUTHOR]

This paper presents a numerical integration method for the solution of 'Singularly Perturbed Differential-Difference Equations' having dual layers. It is well known that when we use already existing numerical methods to solve such problems, we get oscillatory or unsatisfactory results unless we take a very small step size, which is time-consuming and costly. To get a numerical solution for such a problem, first, the delay and advanced parameters present in the SPDDE are approximated by Taylor's series to get an equivalent 'Singularly Perturbed Differential Equation' of second order. Second, an asymptotically equivalent first-order differential equation is obtained from SPDE using Taylor's transformation. Composite Simpson's 1/3 rule is implemented to get a three-term recurrence relation. The Thomas algorithm is applied to get the solution of the tri-diagonal system of equations. Several model examples are tested and it was found that the numerical solution approximates the available/exact solution very well. [ABSTRACT FROM AUTHOR]

In this paper, the H $ {}_2 $ 2 N $ {}_2 $ 2 method and compact finite difference scheme are proposed for the fourth-order time-fractional diffusion-wave equations. In order to improve the efficiency of calculation, a fast scheme is constructed with utilizing the sum-of-exponentials to approximate the kernel $ t^{1-\gamma } $ t 1 − γ . Based on the discrete energy method, the Cholesky decomposition method and the reduced-order method, we prove the stability and convergence. When $ K_{1} \lt \frac {3}{2} $ K 1 < 3 2 , the convergence order is $ O(\tau ^{3-\gamma }+ h^{4}+ \varepsilon) $ O (τ 3 − γ + h 4 + ϵ) , where $ K_{1} $ K 1 is diffusion coefficient, γ is the order of fractional derivative, τ is the parameters for the time meshes, h is the parameters for the space meshes and ε is tolerance error. Numerical results further verify the theoretical analysis. It is find that the CPU time is extremely little in our scheme. [ABSTRACT FROM AUTHOR]

In this paper, the numerical method for solving a class of generalized fractional advection-diffusion equation (GFADE) is considered. The fractional derivative involving scale and weight factors is imposed for the temporal derivative and is analogous to the Caputo fractional derivative following an integration-after-differentiation composition. It covers many popular fractional derivatives by fixing different weights $ w(t) $ w(t) and scale functions $ z(t) $ z(t) inside. The numerical solution of such GFADE is derived via a collocation method, where conventional Legendre polynomials are implemented. Convergence and error analysis of polynomial expansions are studied theoretically. Numerical examples are considered with different boundary conditions to confirm the theoretical findings. By comparing the above examples with those from existing literature, we find that our proposed numerical method is simple, stable and easy to implement. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the initial value problem of Caputo tempered fractional stochastic differential equations and then study the well-posedness of its solution. Further, a Euler–Maruyama (EM) method is derived for solving the considered problem. The strong convergence order of the derived EM method is proved to be $ \alpha -\frac {1}{2} $ α−12 with $ \frac {1}{2} \lt \alpha \lt 1 $ 12<α<1. Additionally, a fast EM method is also developed which is based on the sum-of-exponentials approximation. Finally, numerical experiments are given to support the theoretical findings of the above two methods and verify computational efficiency of the fast EM method. The fast EM method can greatly improve the computational performance of the original EM method. [ABSTRACT FROM AUTHOR]

In this paper, a modified two-grid algorithm based on block-centred finite difference method is developed for the fourth-order nonlinear extended Fisher–Kolmogorov equation. To further improve the computational efficiency, an effective second-order accurate backward difference formula is considered. The modified two-grid method based on Newton iteration is constructed to linearize the nonlinear system. The method solves a miniature nonlinear system on a coarse grid accompanying a larger time step to get the numerical solution, then computes a linear system constructed by the previous result with the Taylor expansion on a fine grid accompanying a smaller time step to get the correct numerical solution. Theoretical analysis shows that the modified two-grid algorithm can achieve second-order convergence accuracy both in time and space domain. Several numerical experiments are provided to verify the theoretical result and the high efficiency of this approach. The practical problem illustrates the actual applicable value of the algorithm. [ABSTRACT FROM AUTHOR]

The numerical solution of the Helmholtz equations is challenging to compute when the wave numbers contained in the governing equation are large. In this paper, we present a parallel algorithm for this problem. A class of sixth-order hybrid compact finite-difference schemes for the Helmholtz equations is presented based on the Taylor expansion. To improve the efficiency of solving the large-wave-number problem, we implemented a parallel algorithm based on the Message Passing Interface environment to solve the discrete system. The validity and accuracy of the proposed method are verified by numerical examples. The method is also applicable to solving problems with oscillatory solutions, which are characterized by numerical instability as the wave number increases. [ABSTRACT FROM AUTHOR]

In this paper, we investigate two-step Runge–Kutta methods to solve Volterra integro-differential equations. Two-step Runge–Kutta methods increase the order of convergence in comparing the classical Runge–Kutta method without extra computational cost. First, the local order conditions and convergence theorem are derived. Then, stability properties of two-step Runge–Kutta methods corresponding to the basic and convolution test equations are analysed. Furthermore, one-stage method with order four and two-stage method with order six are constructed and we plot the stability regions. Numerical examples are presented to confirm the theoretical analyses. [ABSTRACT FROM AUTHOR]

Recent research on continuous attractors is focused on single neural networks. Our brain has a lot of functional regions and these regions mutually cooperate to each other and are coupled to the whole brain. Each region is a subnetwork of the brain. The question is if each region has continuous attractor, in which condition, the whole network can possess a continuous attractor? Based on this question, the emphasis of this paper is to investigate the continuous attractors of new networks formed by coupling different neural networks under the T–S fuzzy rule. By exploring the properties of the transfer function, conditions for the new coupled network to possess continuous attractor are successfully obtained. To verify the correctness and validity of the proposed conclusions, the results of this paper are further verified by simulation. [ABSTRACT FROM AUTHOR]

This paper constructs a modified partially truncated Euler-Maruyama (EM) method for stochastic differential equations with piecewise continuous arguments (SDEPCAs), where the drift and diffusion coefficients grow superlinearly. We divide the coefficients of SDEPCAs into global Lipschitz continuous and superlinearly growing parts. Our method only truncates the superlinear terms of the coefficients to overcome the potential explosions caused by the nonlinearities of the coefficients. The strong convergence theory of this method is established and the 1/2 convergence rate is presented. Furthermore, an explicit scheme is developed to preserve the mean square exponential stability of the underlying SDEPCAs. Several numerical experiments are offered to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study oscillation and non-oscillation of Runge–Kutta methods for linear mixed type impulsive differential equations with piecewise constant arguments. The conditions for oscillation and non-oscillation of numerical solutions are obtained. Also conditions under which Runge–Kutta methods can preserve the oscillation and non-oscillation of linear mixed type impulsive differential equations with piecewise constant arguments are obtained. Moreover, the interpolation function of numerical solutions is introduced and the properties of the interpolation function are discussed. It turns out that the zeros of the interpolation function converge to ones of the analytic solution with the same order of accuracy as that of the corresponding Runge–Kutta method. To confirm the theoretical results, the numerical examples are given. [ABSTRACT FROM AUTHOR]

This article focuses on the investigation of two-dimensional elliptic singularly perturbed problems that incorporate positive and negative shifts, the solution of this class of problems may demonstrate regular/parabolic/degenerate or interior boundary layers. The goal of this article is to establish the development of numerical techniques for two-dimensional elliptic singularly perturbed problems with positive and negative shifts having regular boundary layers. The three numerical schemes are proposed to estimate the solution of this class of problems based on the fitted operator and fitted mesh finite-difference methods. The fitted operator finite difference method is analysed for convergence. The effect of shift terms on the solution behaviour is demonstrated through numerical experiments. The paper concludes by providing several numerical results that demonstrate the performance of proposed numerical schemes. [ABSTRACT FROM AUTHOR]

In this paper, we propose a single timescale stochastic quasi-Newton method for solving the stochastic optimization problems. The objective function of the problem is a composition of two smooth functions and their derivatives are not available. The algorithm sets to approximate sequences to estimate the gradient of the composite objective function and the inner function. The matrix correction parameters are given in BFGS update form for avoiding the assumption that Hessian matrix of objective is positive definite. We show the global convergence of the algorithm. The algorithm achieves the complexity O (ϵ − 1) to find an ϵ − approximate stationary point and ensure that the expectation of the squared norm of the gradient is smaller than the given accuracy tolerance ϵ. The numerical results of nonconvex binary classification problem using the support vector machine and a multicall classification problem using neural networks are reported to show the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we propose a truncated Euler–Maruyama scheme for stochastic differential equations driven by fractional Brownian motion with super-linear drift coefficient. Meanwhile, the convergence rate of the numerical method is established. Numerical example is demonstrated to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose and present a non-overlapping substructuring-type iterative algorithm for the Cahn–Hilliard (CH) equation, which is a prototype for phase-field models. It is of great importance to develop efficient numerical methods for the CH equation, given the range of applicability of CH equation has. Here we present a formulation for the linear and non-linear Dirichlet–Neumann (DN) methods applied to the CH equation and study the convergence behaviour in one and two spatial dimensions in multiple subdomains. We show numerical experiments to illustrate our theoretical findings and effectiveness of the method. [ABSTRACT FROM AUTHOR]

In this paper, a block-by-block scheme is proposed for a class of nonlinear fractional integro-differential equations. This method is based on the Gauss–Lobatto numerical integration method, which shows the high accuracy at all time intervals. Also, the method convergence for this type of equations is proved and it is shown that the order of convergence is at least eight. Finally, the high accuracy, fast calculations and good performance of the method are investigated by solving some numerical examples. [ABSTRACT FROM AUTHOR]

The effectiveness of this paper lies in presenting a new solution for the singular fractional two-dimensional linear continuous-time systems using the conformable derivative and Sumudu transform. The proposed technique combines the new advantageous features of conformal derivative and double-delta-Kronecker, which efficiently handles singularities and Sumudu transform, and provides an efficient solution for 2D singular Fornasini–Marchesini fractional models. Applying these approaches, we then derive new explicit expressions for the fundamental matrices of the considered model. The applicability and usefulness of our proposed methods are validated and evaluated by numerical simulations in order to show the accuracy of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we develop a two-grid virtual element method for nonlinear variable-order time-fractional diffusion equation on polygonal meshes. The L1 graded mesh scheme is considered in the time direction, and the VEM is used to approximate spatial direction. The two-grid virtual element algorithm reduces the solution of the nonlinear time fractional problem on a fine grid to one linear equation on the same fine grid and an original nonlinear problem on a much coarser grid. As a result, our algorithm not only saves total computational cost, but also maintains the optimal accuracy. Optimal L 2 error estimates are analysed in detail for both the VEM scheme and the corresponding two-grid VEM scheme. Finally, numerical experiments presented confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the application of an iterative collocation method based on the use of Lagrange polynomials for the numerical solution of a class of nonlinear third-kind Volterra integral equations. The approximate solution is given by explicit formulas. The error analysis of the proposed numerical method is studied theoretically. Some numerical examples are given to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we extend the Mickens' methodology to construct a second-order nonstandard finite difference (NSFD) method, which preserves dynamical properties including positivity, local asymptotic stability and especially, global asymptotic stability of a general single-species model. This NSFD method is based on a novel weighted non-local approximation of the right-hand side function in combination with the renormalization of the denominator function. The weight guarantees the dynamic consistency and the nonstandard denominator function ensures the convergence of order 2 of the NSFD method. The result is that we obtain a second-order and dynamically consistent NSFD method. It is proved that the NSFD method is simple and efficient and can be extended for solving a broad range of mathematical models arising in real-world applications. Also, we combine the constructed second-order NSFD method with Richardson's extrapolation technique to generate high-order numerical approximations. Finally, the theoretical findings are illustrated and supported by numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, the conforming virtual element method (VEM) is considered to solve the two-dimensional fractional cable equation involving two Riemann–Liouville fractional derivatives. We adopt the Backward Euler Method and the classical L 1 scheme for the numerical discrete scheme of the time derivative. Meanwhile, the conforming VEM, which is generated for arbitrary order of accuracy and the arbitrary polygonal meshes, is analysed for the discretization of the spatial direction. Based on the energy projection operator, the fully discrete formula is proved to be unconditionally stable, and the optimal convergence results are derived with regard to the L 2 -norm in detail. Finally, some numerical experiments are implemented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

The goal of this work is to look at how to solve Barbashin equations using a mix of Volterra–Fredholm integro-differential equations and Volterra equations. For the first time, the 1D and 2D block-pulse functions are employed in a hybrid technique to solve this two-dimensional integral problem concurrently. We construct a new matrix called the converter operating matrix for this purpose. The Barbashin equation is reduced to a linear system that can be solved using well-known methods employing direct computing. The proposed method's convergence theorem is demonstrated under appropriate conditions. In addition, some numerical results are presented in the paper, which supports the strong theoretical properties of our approach. [ABSTRACT FROM AUTHOR]

Several properties of the products of finite maximal prefix, maximal biprefix, semaphore, synchronous, maximal infix and maximal outfix codes are discussed respectively. We show that, for two nonempty subsets X and Y of A * such that the product XY being thin, if XY is a maximal biprefix code, then X and Y are maximal biprefix codes. Also, it is shown that, for two finite nonempty subsets X and Y of A * such that the product XY being unambiguous, if XY is a semaphore code then X and Y are semaphore codes. Finally, two open problems to the product of finite semaphore and maximal infix codes are presented. [ABSTRACT FROM AUTHOR]

This paper studies equity basket options – i.e. multi-dimensional derivatives whose payoffs depend on the value of a weighted sum of the underlying stocks – and develops a new and innovative approach to ensure consistency between options on individual stocks and the index comprising them. Specifically, we show how to resolve a well-known problem that when individual constituent distributions of an equity index are inferred from the single-stock option markets and combined in a multi-dimensional local/stochastic volatility model, the resulting basket option prices will not generate a skew matching that of the options on the equity index corresponding to the basket. [ABSTRACT FROM AUTHOR]

In this paper, we propose two momentum-preserving finite difference schemes for solving one-dimensional Camassa–Holm equation with periodic boundary conditions. A two-level nonlinear difference scheme and a three-level linearized difference scheme are constructed by using the method of order reduction. For nonlinear scheme, we combine mid-point rule and a specific difference operator, which ensures that our obtained scheme is of second-order convergence in both temporal and spatial directions. For linearized scheme, we apply a linear implicit Crank–Nicolson scheme in the temporal direction, then unique solvability and momentum conservation are analysed in detail. Numerical experiments are provided for Camassa–Holm equation admitting different types of solutions, which demonstrate the convergence order and accuracy of the proposed methods coincide with theoretical analysis. Moreover, numerical results show that the nonlinear scheme exhibits better accuracy for mass conservation, while the linearized scheme is more time-saving in computation. [ABSTRACT FROM AUTHOR]

The fractional Schrödinger equation is an important fractional nonlinear evolution equation, and the study of its numerical solution has profound scientific meaning and wide application prospects. This paper proposes a new high-accuracy difference method for nonhomogeneous time-fractional Schrödinger equation (TFSE). The Caputo time-fractional derivative is discretized by high-order L 2 − 1 σ formula and the fourth-order compact difference approximation is applied for spatial discretization. A new nonlinear compact difference scheme with temporal second-order and spatial fourth-order accuracy is constructed, which is solved by the efficient linearized iterative algorithm. The unconditional stability and convergence are analysed by the energy method. The unique existence and maximum-norm estimate of new compact difference scheme solution are obtained. Theoretical analysis shows that the convergence accuracy of new compact difference scheme is O (τ 2 + h 4) with the strong regularity assumption. Numerical experiments verify theoretical results and indicate that the proposed method is an efficient numerical method for solving TFSE. [ABSTRACT FROM AUTHOR]

To address the split best proximity point and monotone variational inclusion problems in real Hilbert spaces, we present and investigate projection and viscosity approximation methods. Under a few reasonable assumptions, we prove some weak and strong convergence theorems for the aforementioned methods. The efficiency of the proposed method is demonstrated by some numerical examples. Some well-known recent results in this area have been improved, generalized, and extended as an outcome of this paper. [ABSTRACT FROM AUTHOR]

This paper deals with the numerical solution of the general Emden–Fowler equation using the Haar wavelet collocation method. This method transforms the differential equation into a system of nonlinear equations. These equations are further solved by Newton's method to obtain the Haar coefficients, and finally the solution to the problem is acquired using these coefficients. We have taken many examples of fifth- and sixth-order equations and implemented our method on those examples. The graphs show the efficiency of the solution for resolution L = 3 and the maximum absolute error of our approach. The error tables give a good picture of the accuracy of this approach. [ABSTRACT FROM AUTHOR]