Stable Computer Method for Solving Initial Value Problems with Engineering Applications

Engineering and applied mathematics disciplines that involve differential equations in general,and initial value problems in particular,include classical mechanics,thermodynamics,electromagnetism,and the general theory of relativity.A reliable,stable,efficient,and consistent numerical scheme is frequently required for modelling and simulation of a wide range of real-world problems using differential equations.In this study,the tangent slope is assumed to be the contra-harmonic mean,in which the arithmetic mean is used as a correction instead of Euler’s method to improve the efficiency of the improved Euler’s technique for solving ordinary differential equations with initial conditions.The stability,consistency,and efficiency of the system were evaluated,and the conclusions were supported by the presentation of numerical test applications in engineering.According to the stability analysis,the proposed method has a wider stability region than other well-known methods that are currently used in the literature for solving initial-value problems.To validate the rate convergence of the numerical technique,a few initial value problems of both scalar and vector valued types were examined.The proposed method,modified Euler explicit method,and other methods known in the literature have all been used to calculate the absolute maximum error,absolute error at the last grid point of the integration interval under consideration,and computational time in seconds to test the performance.The Lorentz system was used as an example to illustrate the validity of the solution provided by the newly developed method.The method is determined to be more reliable than the commonly existing methods with the same order of convergence,as mentioned in the literature for numerical calculations and visualization of the results produced by all the methods discussed,Mat Lab-R2011b has been used.

Three Kinds of Discrete Formulae for the Caputo Fractional Derivative

In this paper,three kinds of discrete formulae for the Caputo fractional derivative are studied,including the modified L1 discretisation forα∈(O,1),and L2 discretisation and L2C discretisation forα∈(1,2).The truncation error estimates and the properties of the coeffcients of all these discretisations are analysed in more detail.Finally,the theoretical analyses areverifiedby thenumerical examples.

On the Fractional Derivatives with an Exponential Kernel

The present article mainly focuses on the fractional derivatives with an exponential kernel(“exponential fractional derivatives”for brevity).First,several extended integral transforms of the exponential fractional derivatives are proposed,including the Fourier transform and the Laplace transform.Then,the L2 discretisation for the exponential Caputo derivative with a∈(1,2)is established.The estimation of the truncation error and the properties of the coefficients are discussed.In addition,a numerical example is given to verify the correctness of the derived L2 discrete formula.

Comparing Solutions to the Nonlinear Dissipative Wave Equation

In previous decades, many of the practical problems arising in scientific fields such as physics, engineering, and mathematics have been related to nonlinear fractional partial differential equations. One of these nonlinear partial differential equations, the dissipative wave equation, has been found to have a plethora of useful applications in different fields. A special class of solutions has been studied for the dissipative wave equation including exact solutions and approximate solutions. The aim of this article is to compare the non-polynomial spline method and the cubic B-spline method with the solution of a nonlinear dissipative wave equation. We will conduct a comparison of the stability of the two methods using the Von Neumann stability analysis. In addition, a numerical example will be presented to illustrate the accuracy of these methods.

Multivariate form of Hermite sampling series

In this paper,we establish a new multivariate Hermite sampling series involving samples from the function itself and its mixed and non-mixed partial derivatives of arbitrary order.This multivariate form of Hermite sampling will be valid for some classes of multivariate entire functions,satisfying certain growth conditions.We will show that many known results included in Commun Korean Math Soc,2002,17:731-740,Turk J Math,2017,41:387-403 and Filomat,2020,34:3339-3347 are special cases of our results.Moreover,we estimate the truncation error of this sampling based on localized sampling without decay assumption.Illustrative examples are also presented.

Truncation Errors,Exact and Heuristic Stability Analysis of Two-Relaxation-Times Lattice Boltzmann Schemes for Anisotropic Advection-Diffusion Equation

This paper establishes relations between the stability and the high-order truncated corrections for modeling of the mass conservation equation with the tworelaxation-times(TRT)collision operator.First we propose a simple method to derive the truncation errors from the exact,central-difference type,recurrence equations of the TRT scheme.They also supply its equivalent three-time-level discretization form.Two different relationships of the two relaxation rates nullify the third(advection)and fourth(pure diffusion)truncation errors,for any linear equilibrium and any velocity set.However,the two relaxation times alone cannot remove the leading-order advection-diffusion error,because of the intrinsic fourth-order numerical diffusion.The truncation analysis is carefully verified for the evolution of concentration waves with the anisotropic diffusion tensors.The anisotropic equilibrium functions are presented in a simple but general form,suitable for the minimal velocity sets and the d2Q9,d3Q13,d3Q15 and d3Q19 velocity sets.All anisotropic schemes are complemented by their exact necessary von Neumann stability conditions and equivalent finite-difference stencils.The sufficient stability conditions are proposed for the most stable(OTRT)family,which enables modeling at any Peclet numbers with the same velocity amplitude.The heuristic stability analysis of the fourth-order truncated corrections extends the optimal stability to larger relationships of the two relaxation rates,in agreementwith the exact(one-dimensional)and numerical(multi-dimensional)stability analysis.A special attention is put on the choice of the equilibrium weights.By combining accuracy and stability predictions,several strategies for selecting the relaxation and free-tunable equilibrium parameters are suggested and applied to the evolution of the Gaussian hill.

TWO-STEP SCHEME FOR BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS

In this paper,a stochastic linear two-step scheme has been presented to approximate backward stochastic differential equations(BSDEs).A necessary and sufficient condition is given to judge the L 2-stability of our numerical schemes.This stochastic linear two-step method possesses a family of 3-order convergence schemes in the sense of strong stability.The coefficients in the numerical methods are inferred based on the constraints of strong stability and n-order accuracy(n∈N^(+)).Numerical experiments illustrate that the scheme is an efficient probabilistic numerical method.

Nonuniform sampling and approximation in Sobolev space from perturbation of the framelet system

The Sobolev space H^(■)(R^(d)),where■>d/2,is an important function space that has many applications in various areas of research.Attributed to the inertia of a measurement instrument,it is desirable in sampling theory to recover a function by its nonuniform sampling.In the present paper,based on dual framelet systems for the Sobolev space pair(H^(s)(R^(d)),H^(-s)(R^(d))),where d/2<s<■,we investigate the problem of constructing the approximations to all the functions in H^(■)(R^(d))by nonuniform sampling.We first establish the convergence rate of the framelet series in(H^(s)(R^(d)),H^(-s)(R^(d))),and then construct the framelet approximation operator that acts on the entire space H^(■)(R^(d)).We examine the stability property for the framelet approximation operator with respect to the perturbations of shift parameters,and obtain an estimate bound for the perturbation error.Our result shows that under the condition d/2<s<■,the approximation operator is robust to shift perturbations.Motivated by Hamm(2015)’s work on nonuniform sampling and approximation in the Sobolev space,we do not require the perturbation sequence to be in■^(α)(Z^(d)).Our results allow us to establish the approximation for every function in H^(■)(R^(d))by nonuniform sampling.In particular,the approximation error is robust to the jittering of the samples.

Numerical Solution of Partial Differential Equations in Arbitrary Shaped Domains Using Cartesian Cut-Stencil Finite Difference Method.Part II:Higher-Order Schemes

Compact higher-order(HO)schemes for a new finite difference method,referred to as the Cartesian cut-stencil FD method,for the numerical solution of the convection-diffusion equation in complex shaped domains have been addressed in this paper.The Cartesian cut-stencil FD method,which employs 1-D quadratic transformation functions to map a non-uniform(uncut or cut)physical stencil to a uniform computational stencil,can be combined with compact HO Pad´e-Hermitian formulations to produce HO cut-stencil schemes.The modified partial differential equation technique is used to develop formulas for the local truncation error for the cut-stencil HO formulations.The effect of various HO approximations for Neumann boundary conditions on the solution accuracy and global order of convergence are discussed.The numerical results for second-order and compact HO formulations of the Cartesian cut-stencil FD method have been compared for test problems using the method of manufactured solutions.

Sampling formulas for 2D quaternionic signals associated with various quaternion Fourier and linear canonical transforms

The main purpose of this paper is to study different types of sampling formulas of quaternionic functions,which are bandlimited under various quaternion Fourier and linear canonical transforms.We show that the quaternionic bandlimited functions can be reconstructed from their samples as well as the samples of their derivatives and Hilbert transforms.In addition,the relationships among different types of sampling formulas under various transforms are discussed.First,if the quaternionic function is bandlimited to a rectangle that is symmetric about the origin,then the sampling formulas under various quaternion Fourier transforms are identical.If this rectangle is not symmetric about the origin,then the sampling formulas under various quaternion Fourier transforms are different from each other.Second,using the relationship between the two-sided quaternion Fourier transform and the linear canonical transform,we derive sampling formulas under various quaternion linear canonical transforms.Third,truncation errors of these sampling formulas are estimated.Finally,some simulations are provided to show how the sampling formulas can be used in applications.

Numerical Method for the Time Fractional Fokker-Planck Equation

In this paper,a new numerical algorithm for solving the time fractional Fokker-Planck equation is proposed.The analysis of local truncation error and the stability of this method are investigated.Theoretical analysis and numerical experiments show that the proposed method has higher order of accuracy for solving the time fractional Fokker-Planck equation.