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Two new boundary correction techniques are proposed in order to mitigate the order reduction phenomenon associated with the numerical solution of initial boundary value problems for parabolic partial differential equations in arbitrary spatial dimensions with time-dependent Dirichlet boundary conditions. The new techniques are based on the idea of discretizing the PDE problem at the boundary points as similar as possible to that of the interior points of the domain. These new techniques are considered for the time integration with W-methods based on approximate matrix factorization. By suitably modifying the internal stages of the methods on the boundary points, it is illustrated by numerical testing with time-dependent boundary conditions that the new boundary correction techniques are able to keep the same accuracy and order of convergence that the method reaches in the case of homogeneous boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, diagonally implicit two–step peer methods for the numerical solution of initial value problems of order ordinary differential are divided into four types including the combination of explicit and implicit methods in a sequential or parallel environments. In this class of the methods, construction of implicit methods equipped with Runge–Kutta stability property together with A – or L –stability are investigated and examples of such methods are given up to order five. Finally, the efficiency and accuracy of the proposed methods are verified by applying them on some well–known stiff problems. [ABSTRACT FROM AUTHOR]

In this research, we provide a novel approach to the development of effective numerical algorithms for the solution of first-order IVPs. In particular, we detail the fundamental theory behind the development of the aforementioned approaches and show how it can be applied to the Adams–Bashforth approach in three steps. The stability of the new scheme is also analyzed. We compared the performance of our novel algorithm to that of established approaches and found it to be superior. Numerical experiments confirmed that, in comparison to standard approaches to the numerical solution of Initial Value Problems (IVPs), including oscillating solutions, our approach is significantly more effective. [ABSTRACT FROM AUTHOR]

Two-step peer methods for the numerical solution of Initial Value Problems (IVP) combine the advantages of Runge-Kutta (RK) and multistep methods to obtain high stage order and provide in a natural way a dense output. In general, explicit s -stage peer methods require s evaluations of the vector field at each step. Nevertheless, Klinge and coworkers (BIT Numer Math, 2018) have shown that some methods use less function calls s e < s , here called effective stages, by re-using s r = s − s e previously computed stages (shifted stages) from the previous steps in the current one. In this paper we propose a new approach, different from the one used by Klinge and coworkers, to re-use previously computed stages, that we call peer methods with reused stages, showing that methods with reused stages and s e effective stages are equivalent to three-step peer methods with s e stages. Then, we analyze all the families of methods with two effective stages, obtaining methods with s = 3 and orders 4 and 5 in which the free parameters of the families have been used to minimize the coefficient of the leading error term as well as to maximize the absolute stability interval. We have also studied one family of peer methods with s = 4 and three effective stages, obtaining a method with order 6, superconvergent of order 7, and optimized leading error term as well as absolute stability interval. Some numerical experiments show the performance of the obtained methods by comparing them with other previously obtained peer methods as well as other standard Runge-Kutta and multistep methods. [ABSTRACT FROM AUTHOR]

The unsteady Helmholtz type equation of anisotropic functionally graded materials (FGMs) is considered in this study. The study is to find numerical solutions to initial boundary value problems governed by the equation. A combined Laplace and boundary element method is used to solve the problems. The analysis derives a boundary-only integral equation that is used to compute the numerical solutions. The analysis also results in another class of anisotropic FGMs of applications. Some problems are considered. The numerical solutions obtained are accurate and consistent. [ABSTRACT FROM AUTHOR]

The main idea of this paper is to investigate numerical solutions of Noyes Field model for Belousov-Zhabotinsky reaction by implementing the combination of two well-known numerical techniques. The proposed methods are collocation method based on finite elements, which is a useful and very flexible approach for solving partial differential equations (PDE), and operator splitting method which is a widely used procedure in the numerical solution of initial and boundary value problems for PDEs. Especially, for this paper, the application of collocation methods are based on trigonometric cubic B-splines. With the help of two techniques discrete schemes are investigated. Next, we presented stability of discrete schemes with Von- Neumann stability analysis. Also, we present the result of applying methods to Noyes Field model and the error norms L 2 and L ∞ to show how accurate numerical solutions to exact ones and graphical representations associated numerical results are shown. [ABSTRACT FROM AUTHOR]

We present a numerical method based on random projections with Gaussian kernels and physics-informed neural networks for the numerical solution of initial value problems (IVPs) of nonlinear stiff ordinary differential equations (ODEs) and index-1 differential algebraic equations (DAEs), which may also arise from spatial discretization of partial differential equations (PDEs). The internal weights are fixed to ones while the unknown weights between the hidden and output layer are computed with Newton's iterations using the Moore–Penrose pseudo-inverse for low to medium scale and sparse QR decomposition with L 2 regularization for medium- to large-scale systems. Building on previous works on random projections, we also prove its approximation accuracy. To deal with stiffness and sharp gradients, we propose an adaptive step-size scheme and address a continuation method for providing good initial guesses for Newton iterations. The "optimal" bounds of the uniform distribution from which the values of the shape parameters of the Gaussian kernels are sampled and the number of basis functions are "parsimoniously" chosen based on bias-variance trade-off decomposition. To assess the performance of the scheme in terms of both numerical approximation accuracy and computational cost, we used eight benchmark problems (three index-1 DAEs problems, and five stiff ODEs problems including the Hindmarsh–Rose neuronal model of chaotic dynamics and the Allen–Cahn phase-field PDE). The efficiency of the scheme was compared against two stiff ODEs/DAEs solvers, namely, ode15s and ode23t solvers of the MATLAB ODE suite as well as against deep learning as implemented in the DeepXDE library for scientific machine learning and physics-informed learning for the solution of the Lotka–Volterra ODEs included in the demos of the library. A software/toolbox in Matlab (that we call RanDiffNet) with demos is also provided. [ABSTRACT FROM AUTHOR]

Copyright of Transactions of the Japan Society for Industrial & Applied Mathematics is the property of Japan Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This study aims to propose sixth-order two-derivative improved Runge-Kutta type methods adopted with exponentially-fitting and trigonometrically-fitting techniques for integrating a special type of third-order ordinary differential equation in the form u"' (t) = f(t,u(t), u'(t)). The procedure of constructing order conditions comprised of a few previous steps, k-i for third-order two-derivative Runge-Kutta-type methods, has been outlined. These methods are developed through the idea of integrating initial value problems exactly with a numerical solution in the form of linear composition of the set functions еwt and е-wt for exponentially fitted and еiwt and е-iwt for trigonometricallyfitted with w ∊ R Parameters of two-derivative Runge-Kutta type method are adapted into principle frequency of exponential and oscillatory problems to construct the proposed methods. Error analysis of proposed methods is analysed, and the computational efficiency of proposed methods is demonstrated in numerical experiments compared to other existing numerical methods for integrating third-order ordinary differential equations with an exponential and periodic solution. [ABSTRACT FROM AUTHOR]

This work studies Interval Initial Value Problems (IIVPs), where the derivative is given by the generalized Hukuhara derivative (gH-derivative) and the initial condition is given by an interval. The focus of the paper is to provide the numerical approximations for the solutions associated with the gH-derivative of IIVPs. This article considers the Euler numerical method, where the classical arithmetic operation is adapted for intervals. The arithmetic considered here is obtained using sup-J extension principle, where J is a particular family of joint possibility distributions. This family gives raise to different types of interactivity and this work shows what kind of interactivity is necessary in the numerical method, in order to approximate the solution via gH-derivative. To illustrate the results, the paper focuses in the decay Malthusian model. [ABSTRACT FROM AUTHOR]

This paper presents a sequence of variable time step deferred correction (DC) methods constructed recursively from the second-order backward differentiation formula (BDF2) applied to the numerical solution of initial value problems for first-order ordinary differential equations (ODE). The sequence of corrections starts with the BDF2 then considered as DC2. We prove that this improvement from a p-order solution (DCp) results in a p + 1 -order accurate solution (DC p + 1 ). This one-order increment in accuracy holds for the least stringent BDF2 0-stability conditions. If we introduce additional requirements for the ratio of consecutive variable time step sizes, then the order increment is 2, allowing a direct transition from DCp to DC p + 2 . These requirements include the constant time step DCp methods. We also prove that all these DCp methods are A-stable. We briefly discuss two other DC variants to illustrate how a proper transition from DCp to DC p + 1 is critical to maintaining A-stability at all orders. Numerical experiments based on two manufactured (closed-form) solutions confirmed the accuracy orders of the DCp – for DCp, p = 2 , 3 , 4 , 5 – both with constant or alternating time step sizes. We showed that the theoretical conditions required to obtain an increment of orders 1 and 2 are satisfied in practice. Finally, a test case shows that we can estimate the error on the DCp solution with the DC p + 1 solution, and a last test case that our new methods maintain their order of accuracy for a stiff system. [ABSTRACT FROM AUTHOR]

An invariant solution is derived using the Lie symmetry technique for steady laminar twodimensional and axisymmetric boundary layer jet flow of incompressible power-law fluids with appropriate boundary conditions. Using symmetry, the nonlinear partial differential equation of the jet flow problem is transformed into a nonlinear ordinary differential equation. The resultant nonlinear ordinary differential equation with boundary conditions is converted to an initial value problem using the Lie symmetry technique. A numerical solution for the resulting initial value problem is derived using Fehlberg’s fourth-fifth order Runge-Kutta method through Maple software. The graphical representation of the characteristics of the velocity field for different physical parameters is also discussed. [ABSTRACT FROM AUTHOR]

The paper is devoted to the construction and study of a numerical method for solving two-dimensional Saint–Venant equations. These equations have important applied significance in modern hydraulic engineering and are suitable for describing waves in the atmosphere, rivers and oceans, and for modeling tides. The issues of formulation of the mixed problems for these equations are studied. The system of equations is reduced to a symmetrical form by transforming dependent variables. Then, the matrices of coefficients are represented as the sums of two symmetric semidefinite matrices. This transformation allows constructing an upwind difference scheme in spatial directions to determine the numerical solution of the initial boundary value problem. The stability of the proposed difference scheme in energy norms is rigorously proved. The results of numerical experiments conducted for a model problem are provided to confirm the stability of the proposed method. [ABSTRACT FROM AUTHOR]

This paper presents an alternative method for developing effective embedded optimized Runge-Kutta (RK) algorithms to solve oscillatory problems numerically. The embedded scheme approach has algebraic orders of 5 and 4. By transforming second-order ordinary differential equations (ODEs) into their first-order counterpart, the suggested approach solves first-order ODEs. The amplification error, phase-lag, and first derivative of the phase-lag are all nil in the embedded pair. The alternative method’s absolute stability is demonstrated. The numerical tests are conducted to demonstrate the effectiveness of the developed approach in comparison to other RK approaches. The alternative approach outperforms the current RK methods. [ABSTRACT FROM AUTHOR]

The indirect integration method (IIM) is a numerical method for initial value problem of ordinary differential equation, which is based on the residual iteration of differential equation to optimize the parameters of the highest order derivative function layer, and then integrates to the solution function layer step by step through complex trapezoid quadrature formula. Considering the recurrence format of IIM, a single-layer recurrent neural network is first designed to simulate the first-order IIM; On this basis, a deep recurrent neural network is constructed by stacking single-layer recurrent networks to implement any order IIM algorithm; Taking the structure of deep RNN as the calculation diagram, the error back-propagation paths are graded according to the order of traversal differential state and with the help of BPTT algorithm, the recursive relationship between the path parameter gradients (equivalent to the parameter gradients of different order IIM) is obtained. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Central China Normal University is the property of Huazhong Normal University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The paper presents a new numerical method for the solution of Initial Value Problems (IVPs) of first order Ordinary Differential Equations (ODEs) via a transcendental basis function. FORTRAN 95 programing language was used for the implementation of the derived numerical scheme. The analysis of the properties of the newly derived method was investigated which includes the stability, accuracy and consistency through which the suitability of it was established. This scheme avoids the need for higher derivative of differential equation. This scheme was tested on some illustrative examples and the results compared favourably with exact solutions. We conclude that the derived numerical method is consistent, convergent, quite stable and more accurate for application to numerical problems and can be widely used in solving first order IVPs in ODEs. [ABSTRACT FROM AUTHOR]

Lévy flights are generalised random walk processes where the independent stationary increments are drawn from a long-tailed α-stable jump length distribution. We consider the formulation of Lévy flights, for 0 < α < 1, in terms of a space-fractional diffusion equation which fundamental solutions are the probability density functions. First, we present how to obtain the governing equation of Lévy motion from the Fourier transform of the jump distribution. Then, we derive a family of implicit numerical methods to determine the numerical solutions and we study their consistency and stability. Although numerical algorithms for the case 1 < α < 2 have been widely discussed, very few works paid attention to the case we discuss here. We present numerical experiments to show the performance of the numerical methods and to highlight the advantages and disadvantages of the different approaches. In the end we determine the numerical solutions of an initial value problem, that considers an approximation of the Dirac delta function as the initial condition, in order to obtain approximations of the probability density functions. [ABSTRACT FROM AUTHOR]

This document contains a detailed comparison between the initial numerical solution methods of ordinary differential equations, which start from the Euler method approach, which is based on the solution of differential equations by the Taylor method, the others two methods to be compared are improvements of this method, that of Euler, and they are the method of Heun, and the method of the midpoint. It will be observed from the solution of test differential equations its respective error with respect to the analytical solution, obtaining an error index dictated by the mean square error EMC. Through this document we will know the best numerical approximation to the analytical solution of the different PVI (initial value problems) raised, also fixing a solution pattern for certain problems, that is, the appropriate method for each type of problem will be stipulated. [ABSTRACT FROM AUTHOR]

This paper addresses spectral collocation techniques to treat with the fractional initial value problem of distributed-order. We introduce three algorithms based on shifted fractional order Jacobi orthogonal functions outputted by Jacobi polynomials. The shifted fractional order Jacobi–Gauss–Radau collocation method is developed for approximating the fractional initial value problem of distributed-order. The principal target in our techniques is to transform the fractional initial value problem of distributed-order to a system of algebraic equations. Some numerical examples are given to test the accuracy and applicability of our technique. It is known that the accuracy of numerical approaches for nonsmooth solution is deteriorated. Employing fractional order Jacobi functions instead of the classical Jacobi stopped this deterioration. [ABSTRACT FROM AUTHOR]

We consider a scalar function depending on a numerical solution of an initial value problem, and its second-derivative (Hessian) matrix for the initial value. The need to extract the information of the Hessian or to solve a linear system having the Hessian as a coefficient matrix arises in many research fields such as optimization, Bayesian estimation, and uncertainty quantification. From the perspective of memory efficiency, these tasks often employ a Krylov subspace method that does not need to hold the Hessian matrix explicitly and only requires computing the multiplication of the Hessian and a given vector. One of the ways to obtain an approximation of such Hessian-vector multiplication is to integrate the so-called second-order adjoint system numerically. However, the error in the approximation could be significant even if the numerical integration to the second-order adjoint system is sufficiently accurate. This paper presents a novel algorithm that computes the intended Hessian-vector multiplication exactly and efficiently. For this aim, we give a new concise derivation of the second-order adjoint system and show that the intended multiplication can be computed exactly by applying a particular numerical method to the second-order adjoint system. In the discussion, symplectic partitioned Runge–Kutta methods play an essential role. [ABSTRACT FROM AUTHOR]

The unsteady anisotropic Helmholtz type equation of spatially varying coefficients is considered in this study. The study is to find numerical solutions to initial boundary value problems governed by the equation. Such problems are relevant for anisotropic functionally graded materials. A mathematical analysis is used to transform the variable coefficient equation to a constant coefficient equation which is then Laplace-transformed (LT) to get a time-independent equation. The latest equation is then written as a boundary-only integral equation of a time-free fundamental solution. A boundary element method (BEM) which is derived from the integral equation and combined with the Stehfest formula for numerical Laplace transform inversion is then employed to find the numerical solutions. Some problems are considered. The combined LT-BEM is easy to implement. The results show that the numerical solutions obtained are accurate. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the time fractional system of fluid-conveying single-walled carbon nanotube (SWCNT), the generalization of SWCNT system which plays important roles in many applied fields. The corresponding Lie symmetries admitted by this fractional system in Riemann–Liouville sense are obtained and symmetry reductions are performed. In addition, based on the above symmetries, the conservation laws are derived using new Noether theorem. Furthermore, analytical solution and numerical series solution to the initial value problem of time fractional SWCNT system in Caputo sense are constructed by applying invariant subspace method and q-homotopy analysis method, respectively. [ABSTRACT FROM AUTHOR]

In the search of light yet strong and tough materials the nature produced soft composites with the staggered architecture. In such design, a soft protein matrix connects rigid mineral platelets similar to the cement connecting bricks in construction. In the present work, we examine strength and toughness (the total energy dissipated in fracture) of soft composites from the "first principles". The latter means that we do not postulate material strength and toughness in advance — they become an outcome of numerical solutions of the initial boundary value problem. We formulate the boundary value problem on the basis of the material-sink approach enforcing the damage description in constitutive equations. In addition to the classical nature-made staggered architecture, we examine some alternative possible designs of platelets inside the soft matrix. Comparisons show that the classical staggered design is the best one (among considered). Nature wins at the moment. [Display omitted] • Fracture in 3D inclusions-reinforced matrix composites is modeled. • The modeling is carried out using the material-sink method. • A nonlinear viscoelastic theory is adopted. • Offset pattern is the toughest arrangement exhibiting multiple stages of fracture. • In offset pattern, stiffness increases with the presence of the fractured areas. [ABSTRACT FROM AUTHOR]

In this paper, the Heisenberg ferromagnetic spin chain equation, which is a model with different magnetic interactions in the classical and semiclassical limits, is investigated using the generalized exponential rational function method. The reduction of the governing equation to a simpler ordinary differential equation by wave transformation is the first step of the procedure. A plurality of the exact solution is obtained by using the relevant method. Physical interpretations of some obtained solutions are also included. Particularly, upon choosing appropriate parameters, various plots are depicted. We achieve also a numerical solution corresponding to the initial value problem by the Elzaki–Adomian decomposition method and give comparative results in a table. Moreover, by using Maple, some graphical simulations are done to see the behavior of these solutions with choosing the suitable parameters. [ABSTRACT FROM AUTHOR]

1D water oil displacement in porous media is usually described by the Buckley-Leverett equation or the Rapoport-Leas equation when capillary diffusion is included. The rectilinear geometry is not representative for near well oil displacement problems. It is therefore of interest to describe the radially symmetric Buckley-Leverett or Rapoport-Leas equation in cylindrical geometry (radial Buckley-Leverett problem). We can show that under appropriate conditions, one can apply a similarity transformation (r , t) → η = r 2 / (2 t) that reduces the PDE in radial geometry to an ODE, even when capillary diffusion is included (as opposed to the situation in the rectilinear geometry (Yortsos, Y.C. (Phys. Fluids 30(10),2928–2935 1987)). We consider two cases (1) where the capillary diffusion is independent of the saturation and (2) where the capillary diffusion is dependent on the saturation. It turns out that the solution with a constant capillary diffusion coefficient is fundamentally different from the solution with saturation-dependent capillary diffusion. Our analytical approach allows us to observe the following conspicuous difference in the behavior of the dispersed front, where we obtain a smoothly dispersed front in the constant diffusion case and a power-law behavior around the front for a saturation-dependent capillary diffusion. We compare the numerical solution of the initial value problem for the case of saturation-dependent capillary diffusion obtained with a finite element software package to a partially analytical solution of the problem in terms of the similarity variable η. [ABSTRACT FROM AUTHOR]

Various order of implicit method has been formulated for solving initial value problems having an initial singular point. The method provides better result than those obtained by used implicit formulae developed based on Euler and Runge-Kutta methods. Romberg scheme has been used for obtaining more accurate result. [ABSTRACT FROM AUTHOR]

In this paper, we present an explicit six-step singularly P-stable Obrechkoff method of tenth algebraic order for solving second-order linear periodic and oscillatory initial value problems of ordinary differential equations. The advantage of this new singularly P-stable Obrechkoff method is that it is a high-order explicit method, and thus does not require additional predictor stages. The numerical stability and phase properties of the new method is analyzed. Four numerical examples show that the new explicit method is more accurate than Obrechkoff schemes of the same order when applied to the numerical solution of second-order initial value problems with highly oscillatory solutions. [ABSTRACT FROM AUTHOR]

The article explores how combination of hierarchical structure comprising mechanical and optical function inspire technological classes of devices and interfaces based on robust, light, and spectrally responsive natural substrates. It mentions the strategies for coloration devoid of inorganic pigments.

The authors propose a numerical method to approximate the solution of specific bivariate Volterra integral equations which arise in the numerical solution of the initial value problem for the Korteweg-de Vries equation. A preliminary study of the domain of the unknown function is carried out and a general algorithm based on the Gauss-Legendre and Newton-Cotes quadrature formulae is developed. Numerical tests are also given in order to show the efficiency of the method. [ABSTRACT FROM AUTHOR]

An algorithm using the differential transformation which is convenient for finding numerical solutions to initial value problems for functional differential equations is proposed in this paper. We focus on retarded equations with delays which in general are functions of the independent variable. The delayed differential equation is turned into an ordinary differential equation using the method of steps. The ordinary differential equation is transformed into a recurrence relation in one variable using the differential transformation. Approximate solution has the form of a Taylor polynomial whose coefficients are determined by solving the recurrence relation. Practical implementation of the presented algorithm is demonstrated in an example of the initial value problem for a differential equation with nonlinear nonconstant delay. A two-dimensional neutral system of higher complexity with constant, nonconstant, and proportional delays has been chosen to show numerical performance of the algorithm. Results are compared against Matlab function DDENSD. [ABSTRACT FROM AUTHOR]

In this paper, an effective numerical iterative method for solving nonlinear initial value problems (IVPs) is presented. The proposed iterative scheme, called the Jacobi‐Picard iteration (JPI) method, is based on the Picard iteration technique, orthogonal shifted Jacobi polynomials, and shifted Jacobi‐Gauss quadrature formula. In comparison with traditional methods, the JPI method uses an iterative formula for updating next step approximations and calculating integrals of the shifted Jacobi polynomials are performed via an exact relation. Also, a vector‐matrix form of the JPI method is provided in details which reduce the CPU time. The performance of the presented method has been investigated by solving several nonlinear IVPs. Numerical results show the efficiency and the accuracy of the proposed iterative method. [ABSTRACT FROM AUTHOR]

We consider the initial value problem for the Klein‐Gordon equation in de Sitter spacetime. We use the central difference scheme on the temporal discretization. We also discretize the spatial variable using the finite element method with implicit and the Crank‐Nicolson schemes for the numerical solution of the initial value problem. In order to show the accuracy for the results of the solutions, we also examine the finite difference methods. We observe that the numerical results obtained by using these methods are compatible. [ABSTRACT FROM AUTHOR]

In this study, a trial solution satisfying the initial conditions of a differential equation is defined. Next, a cost function is specified regarding with the trial solution depending on the parameters of a feed-forward neural network. In the training stage of the neural network, Gravitational Search Algorithm (GSA) is used as a global optimization method for minimizing the network cost. The main contributions of our paper are getting the numerical solution of Initial Value Problems (IVPs) without derivative and obtaining the solution at any point in the interval by taking advantages of using GSA and Neural Networks(NNs) together. GSA trains the NNs without derivation knowledge using the inputs obtained by the discretization of the domain. However, NNs produce the numerical solutions at any point of problem domain without depending on the step size, which is used for discretization stage. In numerical implementations of IVPs, the obtained results are discussed comparing with the analytical solution, forward Euler and Runge-Kutta 4 methods. [ABSTRACT FROM AUTHOR]

A new convergent two-level finite difference scheme is proposed for the numerical solution of initial value problem of the generalized Rosenau-KdV-RLW equation. The new scheme is second-order, linear, conservative, and unconditionally stable. It contains one free parameter. The impact of the parameter to error of the numerical solution is studied. The prior estimate of the finite difference solution is obtained. The existence, uniqueness, and convergence of the scheme are proved by the discrete energy method. Accuracy and reliability of the scheme are tested by simulating the solitary wave graph of the equation. Wave generation subject to initial Gaussian condition has been studied numerically. Different wave generations are observed depending on the dispersion coefficients and the nonlinear advection term. Numerical experiments indicate that the present scheme is conservative, efficient, and of high accuracy, and well simulates the solitary waves for a long time. [ABSTRACT FROM AUTHOR]

Entropy stable schemes for the numerical solution of initial value problems of nonlinear, possibly strongly degenerate systems of convection–diffusion equations were recently proposed in Jerez and Parés's study. These schemes extend the theoretical framework of Tadmor's study to convection–diffusion systems. They arise from entropy conservative schemes by adding a small amount of viscosity to avoid spurious oscillations. The main condition for feasibility of entropy conservative or stable schemes for a given model is that the corresponding first‐order system of conservation laws possesses a convex entropy function and corresponding entropy flux, and that the diffusion matrix multiplied by the inverse of the Hessian of the entropy is positive semidefinite. As a new contribution, it is demonstrated in the present work, first, that these schemes can naturally be extended to initial‐boundary value problems with zero‐flux boundary conditions in one space dimension, including an explicit bound on the growth of the total entropy. Second, it is shown that these assumptions are satisfied by certain diffusively corrected multiclass kinematic flow models of arbitrary size that describe traffic flow or the settling of dispersions and emulsions, where the latter application gives rise to zero‐flux boundary conditions. Numerical examples illustrate the behavior and accuracy of entropy stable schemes for these applications. [ABSTRACT FROM AUTHOR]

Abstract A new class of two-step linear symmetric methods is introduced for the numerical solution of second-order initial value problems that have highly oscillatory solutions. In this class, for the first time in the literature, we calculate the coefficients of the method by a combination of a trigonometrically fitted (TF) method and a vanishing of the phase lag and some of its derivatives (VSDPL) method, and we construct a new class of methods that has the properties of TF methods and VSDPL methods, which we call the TFPL method. This method is of algebraic order 8, and has an important P-stability property. The main structure of the method is multiderivative, and the combined phases were applied to expand the stability interval and to achieve P-stability. The advantage of the method in comparison with similar methods in terms of efficiency, accuracy, and stability is shown by its implementation in some important problems, the undamped Duffing equation, etc. [ABSTRACT FROM AUTHOR]

An optimized two-step hybrid block method for the numerical solution of third-order initial value problems is presented. The method takes into regard three hybrid points which are selected suitably to optimize the local truncation errors of the main formulas for the block. The method is zero-stable and consistent with sixth algebraic order. Some numerical examples are debated to demonstrate the efficiency and the accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

Copyright of Karaelmas Science & Engineering Journal / Karaelmas Fen ve Mühendislik Dergisi is the property of Karaelmas Science & Engineering Journal and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

There are several mathematical models that describe the propagation of cracks. For many engineering applications, up to a certain point, it is not necessary to have great accuracy in predictions about the behavior of the evolution of a crack, but a reliable prediction, within certain limits, of such behavior. This work presents theoretical results consisting in obtaining lower and upper bounds that "envelop" the first and second order statistical moment estimators of the crack size function based on the fast crack bounds method. These bounds are polynomials defined in the variable "number of cycles" that consider the uncertainties of the parameters that describe the crack propagation models. The performance of the bounds for the statistical moments of the crack size is evaluated through the relative deviation between the bounds and the approximate numerical solutions of the initial value problems (IVP) that describe the crack evolution laws. For this work, the Collipriest model is used. The Monte Carlo simulation method is used to create samples of the selected parameters to obtain the crack size for both the bounds and the Runge-Kutta method. [ABSTRACT FROM AUTHOR]

Abstract We introduce the fractional delayed matrix cosine and sine functions and use the variation of constants method to obtain an explicit formula for a solution to the initial value problem for a Riemann–Liouville sequential fractional linear time-delay system of order 1 < 2 α ≤ 2. Our results extend those for fractional linear time-delay systems and novel for the classical case α = 1. [ABSTRACT FROM AUTHOR]

In this paper, a new class of two-derivative two-step Runge-Kutta (TDTSRK) methods for the numerical solution of non-stiff initial value problems (IVPs) in ordinary differential equation (ODEs) is considered. The TDTSRK methods are a special case of multi-derivative Runge-Kutta methods proposed by Kastlunger and Wanner (1972). The methods considered herein incorporate only the first and second derivatives terms of ODEs. These methods possess large interval of stability when compared with other existing methods in the literature. The experiments have been performed on standard problems, and comparisons were made with some standard explicit Runge-Kutta methods in the literature. [ABSTRACT FROM AUTHOR]

Abstract A Linear Multistep Hybrid Block method with four intra-step grid points is presented for approximating directly the solution of fourth order Initial Value Problems (IVPs). Multiple Finite Difference formulas are derived and combined in a block formulation to form a numerical integrator that provides direct solution to fourth order IVPs over sub-intervals. The properties and convergence of the proposed method are discussed. The superiority of this method over existing methods is established numerically on different test problems. [ABSTRACT FROM AUTHOR]

In this research, a modified rational interpolation method for the numerical solution of initial value problem is presented. The proposed method is obtained by fitting the classical rational interpolation formula in Chebyshev polynomials leading to a new stability function and new scheme. Three numerical test problems are presented in other to test the efficiency of the proposed method. The numerical result for each test problem is compared with the exact solution. The approximate solutions are show competitiveness with the exact solutions of the ODEs throughout the solution interval. [ABSTRACT FROM AUTHOR]

We consider initial-boundary-value problems for three classes of inhomogeneous degenerate equations of mixed parabolic-hyperbolic type: mixed-type equations with degenerate hyperbolic part, mixed-type equations with degenerate parabolic part, and mixed-type equations with power degeneration. In each case, we state a criterion of uniqueness of a solution to the problem. We construct solutions as series with respect to the system of eigenfunctions of the corresponding one-dimensional spectral problem. We prove that the uniqueness of the solution and the convergence of the series depend on the ratio of sides of the rectangular from the hyperbolic part of the mixed domain. In the proof of the existence of solutions to the problem, small denominators appear that impair the convergence of series constructed. In this connection, we obtain estimates of small denominators separated from zero and the corresponding asymptotics, which allows us, under certain conditions, to prove that the solution constructed belongs to the class of regular solutions. [ABSTRACT FROM AUTHOR]

We demonstrate how to use the Fokas method to analyze initial-boundary value problems for integrable evolution equations posed on a time-dependent interval. In particular, we implement this method to a general dispersive linear evolution equation with spatial derivatives of arbitrary order and to the nonlinear Schrödinger equation, formulated in the time-dependent domain l1(t) < x < l2(t), 0 < t < T, where l1(t) and l2(t) are given, real, twice differentiable functions whose first derivatives are monotonic, and T is a positive fixed constant. [ABSTRACT FROM AUTHOR]

We find the stationary solutions of the Chern-Simons gauged O(3) sigma model in one space dimension and study an initial value problem under the gauge condition A1 = 0. We also study low regularity local well-posedness using bilinear estimates for wave-Sobolev space H 1 , θ . As a by-product, we construct a finite energy global solution by considering the conservation of energy. [ABSTRACT FROM AUTHOR]