Crank–Nicolson finite difference scheme for the Rosenau–Burgers equation

Numerical solutions of nonhomogeneous Rosenau type equations by quintic B‐spline collocation method

In this study, a numerical scheme based on a collocation finite element method using quintic B‐spline functions for getting approximate solutions of nonhomogeneous Rosenau type equations prescribed by initial and boundary conditions is proposed. The numerical scheme is tested on four model problems with known exact solutions. To show how accurate results the proposed scheme produces, the error norms defined by L2 and L∞ are calculated. Additionally, the stability analysis of the scheme is done by means of the von Neuman method.

Numerical Simulation of Time Fractional BBM-Burger Equation Using Cubic B-Spline Functions

The unidirectional propagation of long waves in certain nonlinear dispersive system is explained by the Benjamin-Bona-Mahony-Burger (BBM-Burger) equation. The purpose of this study is to investigate the BBM-Burger equation numerically using Caputo derivative and B-spline basis functions. The fractional derivative is considered in Caputo form, and $L1$ formula is used for discretization of temporal derivative. The interpolation of space derivative is done with the help of B-spline functions. The effect of $\alpha$ and time on solution profile of travelling wave for different domain of $x$ is discussed in this paper. The numerical results have been presented to show that the cubic B-spline method is effective and efficient in solving the time fractional Benjamin-Bona-Mahony-Burger (BBM-Burger) equation. Moreover, the convergence and stability of the proposed scheme are analyzed. The error norms are also calculated to check the accuracy of the proposed scheme. The numerical results reflect that the proposed scheme can be used for linear and highly nonlinear models.

Mathematical model of back propagation for stock price forecasting

In order to establish a more accurate Stock Price Prediction Model, the Stock Price Prediction mathematical Model SPPM (Stock Price Prediction Model) based on BP neural network with high frequency data is proposed in this paper. The SPPM integrates several neural networks to predict the movement of stock prices over the next few days. The key problems in SPPM—such as data preprocessing, output fusion and the selection of nodes in the hidden layer of neural network—are discussed in detail. The experimental results show that the SPPM predicted the closing price of 2019-03-19 and 2019-03-20 as 207.16 and 207.22, respectively, which is very close to the actual observed value, and the back propagation mathematical model SPPM has a certain practical value. Our conclusion is that the back propagation model can predict the stock price with high accuracy.

On fractional Kakutani–Matsuuchi water wave model: Implementing a reliable implicit finite difference method

In this study, a reliable implicit finite difference method based on the modified trapezoidal quadrature rule, backward Euler differences, nonstandard central approximations, and the Hadamard finite‐part integral is being considered to solve a viscous asymptotical model named as fractional Kakutani–Matsuuchi water wave model. The fractional derivative is used in the Riemann–Liouville sense. Based on the properties of Brouwer's fixed‐point theorem, the existence, uniqueness, convergence, and stability of the proposed method are proved. Furthermore, we show that the global convergence order of the method in maximum norm is O(τ, hmin{β,3 − α}), where 0 < α ≤ 1 and β > 0 are the order of fractional derivative and the Lipschitz constant. Also, τ and h are the time step and space step, respectively. Finally, several examples are used to illustrate the accuracy and performance of the method.

An efficient numerical scheme for the simulation of time-fractional nonhomogeneous Benjamin-Bona-Mahony-Burger model

The Benjamin-Bona-Mahony-Burger (BBM-Burger) equation is important for explaining the unidirectional propagation of long waves in nonlinear dispersion systems. This manuscript proposes an algorithm based on cubic B-spline basis functions to study the nonhomogeneous time fractional model of BBM-Burger via Caputo derivative. The discretization of fractional derivative is achieved by L1 formula, while the temporal and spatial derivatives are interpolated by means of Crank-Nicolson and forward finite difference scheme together with B-spline basis functions. The performance of the Cubic B-spline scheme (CBS) is examined by three test problems with homogeneous initial and boundary conditions. The obtained results are found to be in good agreement with the exact solutions. The behaviour of travelling wave is studied and presented in the form of tables and graphics for various values of α and t. A linear stability analysis, based on the von Neumann scheme, shows that the CBS is unconditionally stable. Moreover, the accuracy of the scheme is quantified by computing error norms.

ON THE NUMERICAL SIMULATION OF TIME-SPACE FRACTIONAL COUPLED NONLINEAR SCHRÖDINGER EQUATIONS UTILIZING WENDLAND’S COMPACTLY SUPPORTED FUNCTION COLLOCATION METHOD

This research describes an efficient numerical method based on Wendland’s compactly supported functions to simulate the time-space fractional coupled nonlinear Schrödinger (TSFCNLS) equations. Here, the time and space fractional derivatives are considered in terms of Caputo and Conformable derivatives, respectively. The present numerical discussion is based on the following ways: we first approximate the Caputo fractional derivative of the proposed equation by a scheme order O(∆t2−α), 0 < α < 1 and then the Crank-Nicolson scheme is employed in the mentioned equation to discretize the equations. Second, applying a linear difference scheme to avoid solving nonlinear systems. In this way, we have a linear, suitable calculation scheme. Then, the conformable fractional derivatives of the Wendland’s compactly supported functions are established for the scheme. The stability analysis of the suggested scheme is also examined in a similar way to the classic Von-Neumann technique for the governing equations. The efficiency and accuracy of the present method are verified by solving two examples.

A new implicit nonlinear discrete scheme for Rosenau–Burgers equation based on multiple integral finite volume method

In this paper, we study the initial-boundary value problem of the Rosenau–Burgers equation by the multiple integral finite volume method (MIFVM). The MIFVM can keep the original equation property very well. We propose a two-level implicit nonlinear discrete scheme, which preserves the energy decline property of the original equation. Existence and uniqueness of the numerical solution are derived. The convergence with the order of O(τ2 + h3) and unconditional stability of the numerical scheme are verified. Numerical examples demonstrate that the scheme is reliable and effective.

On the convergence of operator splitting for the Rosenau–Burgers equation

We present convergence analysis of operator splitting methods applied to the nonlinear Rosenau–Burgers equation. The equation is first splitted into an unbounded linear part and a bounded nonlinear part and then operator splitting methods of Lie‐Trotter and Strang type are applied to the equation. The local error bounds are obtained by using an approach based on the differential theory of operators in Banach space and error terms of one and two‐dimensional numerical quadratures via Lie commutator bounds. The global error estimates are obtained via a Lady Windermere's fan argument. Lastly, a numerical example is studied to confirm the expected convergence order.

A New Linear Difference Scheme for Generalized Rosenau-Kawahara Equation

We introduce in this paper a new technique, a semiexplicit linearized Crank-Nicolson finite difference method, for solving the generalized Rosenau-Kawahara equation. We first prove the second-order convergence in L∞-norm of the difference scheme by an induction argument and the discrete energy method, and then we obtain the prior estimate in L∞-norm of the numerical solutions. Moreover, the existence, uniqueness, and satiability of the numerical solution are also shown. Finally, numerical examples show that the new scheme is more efficient in terms of not only accuracy but also CPU time in implementation.

Solitary Wave Formation from a Generalized Rosenau Equation

A generalized viscous Rosenau equation containing linear and nonlinear advective terms and mixed third- and fifth-order derivatives is studied numerically by means of an implicit second-order accurate method in time that treats the first-, second-, and fourth-order spatial derivatives as unknown and discretizes them by means of three-point, fourth-order accurate, compact finite differences. It is shown that the effect of the viscosity is to decrease the amplitude, curve the wave trajectory, and increase the number and width of the waves that emerge from an initial Gaussian condition, whereas the linear convective term pushes the wave front towards the downstream boundary. It is also shown that the effect of the nonlinear convective term is to increase the steepness of the leading wave front and the number of sawtooth waves that are generated behind it, while that of the first dispersive term is to increase the number of waves that break up from the initial condition as the coefficient that characterizes this term is decreased. It is also shown that, for reasons of stability, the second dispersion coefficient must be much smaller than the first one and its effects on wave propagation are relatively small.

A Modified Three-Level Average Linear-Implicit Finite Difference Method for the Rosenau-Burgers Equation

We introduce a new technique, a three-level average linear-implicit finite difference method, for solving the Rosenau-Burgers equation. A second-order accuracy on both space and time numerical solution of the Rosenau-Burgers equation is obtained using a five-point stencil. We prove the existence and uniqueness of the numerical solution. Moreover, the convergence and stability of the numerical solution are also shown. The numerical results show that our method improves the accuracy of the solution significantly.

Numerical Simulation for General Rosenau‐RLW Equation: AnAverage Linearized Conservative Scheme

Numerical solutions for the general Rosenau‐RLW equation are considered and an energy conservative linearized finite difference scheme is proposed. Existence of the solutions for the difference scheme has been shown. Stability, convergence, and a priori error estimate of the scheme are proved using energy method. Numerical results demonstrate that the scheme is efficient and reliable.

C‐N Difference Schemes for Dissipative Symmetric Regularized Long Wave Equations with Damping Term

We study the initial‐boundary problem of dissipative symmetric regularized long wave equations with damping term. Crank‐Nicolson nonlinear‐implicit finite difference scheme is designed. Existence and uniqueness of numerical solutions are derived. It is proved that the finite difference scheme is of second‐order convergence and unconditionally stable by the discrete energy method. Numerical simulations verify the theoretical analysis.