Nonlinear Initial Value Research Articles

An hp-version spectral collocation method for multi-term nonlinear fractional initial value problems with variable-order fractional derivatives

A general class of multi-term nonlinear fractional initial value problems involving variable-order fractional derivatives are considered. Some sufficient conditions for the existence and uniqueness of exact solution are established. An hp-version spectral collocation method is presented for solving the problem in numerical frames. It employs the shifted Legendre-Gauss interpolations to conquer the influence of the nonlinear term and the variable-order derivatives. The most remarkable feature of the method is its capability to achieve higher accuracy by refining the mesh and/or increasing the degree of the polynomial. The rigorous error estimates are derived for the problem with smooth solutions on arbitrary meshes and weakly singular solutions on quasi-uniform meshes. Numerical results are given to support the theoretical conclusions.

On the approximate solutions of a class of fractional order nonlinear Volterra integro-differential initial value problems and boundary value problems of first kind and their convergence analysis

In this work we consider a class of fractional order Volterra integro-differential equations of first kind where the fractional derivative is considered in the Caputo sense. Here, we consider the initial value problem and the boundary value problem separately. For simplicity of the analysis, we reduce each of these problems to the fractional order Volterra integro-differential equation of second kind by using the Leibniz’s rule. We have obtained sufficient conditions for the existence and uniqueness of the solutions of initial and the boundary value problems. An operator based method has been considered to approximate their solutions. In addition, we provide a convergence analysis of the adopted approach. Several numerical experiments are presented to support the theoretical results.

Modified New Iterative Method for Solving Nonlinear Partial Differential Equations

In this paper, the iterative method, proposed by Gejji and Jafari in 2006, has been modified for solving nonlinear initial value problems. The Laplace transform was used in this modification to eliminate the linear differential operator in the differential equation. The convergence of the solution was discussed according to the modification proposed. To illustrate this modification some examples were presented.

Parametric continuation algorithm for time-delay systems and bifurcation caused by multiple characteristic roots

Stability analysis has been always a key issue in nonlinear dynamics and engineering applications, and it is still a challenging task for time-delay control systems when tuning some parameters like feedback gains and delays. In this paper, firstly we propose a parametric continuation algorithm for calculating the rightmost characteristic root(s), by solving initial value problems of a nonlinear differential equation associated with the characteristic function of the time-delay system. Then we study the static bifurcation caused by multiple characteristic roots that occurs in solving the associated nonlinear initial value problems and present an algorithm for fast calculation of the bifurcation points. We demonstrate the theoretical results with examples arising in vibration control.

A modified perturbation solution to the strongly nonlinear initial value problem of the KdV–KP equation

In this paper, a modified perturbation method to solve strongly nonlinear initial-value-problem of the KdV–KP equation is introduced. The method is a modification of the linearized perturbation expansion method which was developed to solve nonlinear initial value problems with no small parameters. The main idea is to embed an artificial parameter so that the dependent variable can later be expanded in series of this parameter. In addition, another arbitrary constant is included to take care of the secular terms that appear after integration. The current technique assumes the arbitrary constant can also be expanded in terms of the embedded parameter. This step gives an advantage of allowing for more iteration and more correction terms to be added to the perturbation solution. The obtained solution is compared with the exact solution obtained using the Lie group method and the results show good agreement.

An hp-version of the C0-continuous Petrov-Galerkin time stepping method for nonlinear second-order initial value problems

We present an hp-version of the C0-continuous Petrov-Galerkin time stepping method for nonlinear second-order initial value problems. We derive a priori error bound in the H1-norm that is fully explicit with respect to the local time steps and the local approximation orders. Moreover, we prove that the hp-version of the C0-continuous Petrov-Galerkin time stepping method based on geometrically refined time steps and on linearly increasing approximation orders yields exponential rates of convergence for solutions with start-up singularities. Numerical examples confirm the theoretical results.

Analytical solution with existence and uniqueness conditions of non-linear initial value multi-order fractional differential equations using Caputo derivative

The main focus of this study is to apply the two-step Adomian decomposition method (TSADM) for finding a solution of a fractional-order non-linear differential equation by using the Caputo derivative. We are interested in obtaining an analytical solution with two main constraints, that are, without converting the non-linear fractional differential equation to a system of linear algebraic fractional equation, and secondly, with less number of iterations. Moreover, we have investigated conditions for the existence and uniqueness of a solution with the help of some fixed point theorems. Furthermore, the method is demonstrated with the help of some examples. We also compare the results with the Adomian decomposition method (ADM), the modified Adomian decomposition method, and the combination of the ADM and a spectral method. It is concluded that the TSADM provides an analytical solution of fractional-order non-linear differential equation, while the other methods furnish an approximate solution.

Numerical Verification of Solutions for Nonlinear Parabolic Problems

In this paper, we present a numerical verification method of solutions for nonlinear parabolic initial boundary value problems. Decomposing the problem into a nonlinear part and an initial value part, we apply Nakao’s projection method, which is based on the full-discrete finite element method with constructive error estimates, to the nonlinear part and use the theoretical analysis for the heat equation to the initial value part, respectively. We show some verified examples for solutions of nonlinear problems from initial value to the neighborhood of the stationary solutions, which confirm us the actual effectiveness of our method.

Analytic approximate solutions of diffusion equations arising in oil pollution

In this article, modified versions of variational iteration algorithms are presented for the numerical simulation of the diffusion of oil pollutions. Three numerical examples are given to demonstrate the applicability and validity of the proposed algorithms. The obtained results are compared with the existing solutions, which reveal that the proposed methods are very effective and can be used for other nonlinear initial value problems arising in science and engineering.

Initial Value Problem For Nonlinear Fractional Differential Equations With ψ-Caputo Derivative Via Monotone Iterative Technique

In this article, we discuss the existence and uniqueness of extremal solutions for nonlinear initial value problems of fractional differential equations involving the ψ -Caputo derivative. Moreover, some uniqueness results are obtained. Our results rely on the standard tools of functional analysis. More precisely we apply the monotone iterative technique combined with the method of upper and lower solutions to establish sufficient conditions for existence as well as the uniqueness of extremal solutions to the initial value problem. An illustrative example is presented to point out the applicability of our main results.

Applications Modified Adomian Decomposition Method for Solving the Second-Order Ordinary Differential Equations

In this paper, we will display an efficient application of the Modified Adomian Decomposition method (MADM) for solving linear and non-linear Singular initial value problem in the Second order ordinary differential physical equations such as the Schrödinger’s Equation, Hermit differential Equation and capacitance regulation equation. It is shown that the MADM efficency, simple, easy to use and accuracy because the series solutions converge to the specific answer for every problem if it exist or numerical solution if not exist.

Non-Newtonian coefficient condition for a stable long-time behavior of a single bubble: existence and characteristics of stable solutions

The present paper takes up the underlying nonlinear initial value problem from a preceding author’s work about the dynamics of a single bubble in a highly viscous liquid medium under different pressure impacts. The arising ordinary differential equation is mainly based on the constitutive relation of a second-order liquid that in particular includes two non-Newtonian material constants. In this article, the significance of these coefficients is mathematically analyzed in detail by proving the existence of stable solutions of the named initial value problem. This is achieved by special transformations of the differential equation at hand and the introduction of appropriate Lyapunov functions. It particularly turns out that a combined condition of the non-Newtonian coefficients and diverse restrictions to the external pressure impact are decisive for the validity of the existence results. Furthermore, the convergence speed of solutions is investigated by considering the linearized equation associated with the present initial value problem and by applying a special variant of Gronwall’s lemma. The main theoretical result, being the prementioned strong condition for the non-Newtonian coefficients, is finally compared to real data sets.

A new improved generalized decomposition method (improved-GDM) for hydromagnetic boundary layer flow

Purpose The purpose of this study is to investigate the magneto-hydrodynamics boundary layer Falkner–Skan flow over a flat plate numerically by using the Runge–Kutta method featuring shooting technique and analytically via a new modified analytical technique called improved generalized Adomian decomposition method (improved-GDM). Design/methodology/approach It is well established that the generalized decomposition method (GDM) (Yong-Chang et al., 2008), which uses a new kind of decomposition strategy for the nonlinear function, has proved its efficiency and superiority when compared to the standard ADM method. In this investigation, based on the idea of improved-ADM method developed by Lina and Song (Song and Wang, 2013), the authors proposed a new analytical algorithm of computation named improved-GDM. Thereafter, the proposed algorithm is tested by solving the nonlinear problem of the hydro-magnetic boundary layer flow over a flat plate. Findings The proposed improved generalized decomposition method (I-GDM) introduces a convergence-control parameter “ω’’ into the GDM, which accelerates the convergence of solution and reduces considerably the computation time. In fact, the key of this method is mainly based on the best selection of the convergence-control parameter ω. Originality/value The paper presents a new efficient algorithm of computation that can be considered as an alternative for solving the nonlinear initial boundary layer value problems. Obtained results show clearly the accuracy of the proposed method.

Stability of Minkowski space and polyhomogeneity of the metric

We study the nonlinear stability of the $$(3+1)$$ -dimensional Minkowski spacetime as a solution of the Einstein vacuum equation. Similarly to our previous work on the stability of cosmological black holes, we construct the solution of the nonlinear initial value problem using an iteration scheme in which we solve a linearized equation globally at each step; we use a generalized harmonic gauge and implement constraint damping to fix the geometry of null infinity. The linear analysis is largely based on energy and vector field methods originating in work by Klainerman. The weak null condition of Lindblad and Rodnianski arises naturally as a nilpotent coupling of certain metric components in a linear model operator at null infinity. Upon compactifying $$\mathbb {R}^4$$ to a manifold with corners, with boundary hypersurfaces corresponding to spacelike, null, and timelike infinity, we show, using the framework of Melrose’s b-analysis, that polyhomogeneous initial data produce a polyhomogeneous spacetime metric. Finally, we relate the Bondi mass to a logarithmic term in the expansion of the metric at null infinity and prove the Bondi mass loss formula.

Hermite wavelet method for the numerical solution of nonlinear singular initial value problems

Hermite wavelet method for the numerical solution of nonlinear singular initial value problems

A Framework for Nonlocal, Nonlinear Initial Value Problems

We study the Cauchy problem for non-linear non-local operators that may be degenerate. Our general framework includes cases where the jump intensity is allowed to depend on the values of the solution itself, e.g. the porous medium equation with the fractional Laplacian and the parabolic fractional $p$-Laplacian. We show the existence, uniqueness of bounded solutions and study their further properties. Several new examples of non-local, non-linear operators are provided.

Layer-Parallel Training of Deep Residual Neural Networks

Residual neural networks (ResNets) are a promising class of deep neural networks that have shown excellent performance for a number of learning tasks, e.g., image classification and recognition. Mathematically, ResNet architectures can be interpreted as forward Euler discretizations of a nonlinear initial value problem whose time-dependent control variables represent the weights of the neural network. Hence, training a ResNet can be cast as an optimal control problem of the associated dynamical system. For similar time-dependent optimal control problems arising in engineering applications, parallel-in-time methods have shown notable improvements in scalability. This paper demonstrates the use of those techniques for efficient and effective training of ResNets. The proposed algorithms replace the classical (sequential) forward and backward propagation through the network layers by a parallel nonlinear multigrid iteration applied to the layer domain. This adds a new dimension of parallelism across layers that is attractive when training very deep networks. From this basic idea, we derive multiple layer-parallel methods. The most efficient version employs a simultaneous optimization approach where updates to the network parameters are based on inexact gradient information in order to speed up the training process. Using numerical examples from supervised classification, we demonstrate that the new approach achieves similar training performance to traditional methods, but enables layer-parallelism and thus provides speedup over layer-serial methods through greater concurrency.

Jacobi‐Picard iteration method for the numerical solution of nonlinear initial value problems

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.

Approximation of Functions Belonging to Generalized Hölder’s Class $$H_{\alpha }^{(\omega )}[0,1)$$ by First Kind Chebyshev Wavelets and Its Applications in the Solution of Linear and Nonlinear Differential Equations

In this paper, two new estimators $$E^{(1)}_{2^{k-1},0}$$ and $$ E^{(2)}_{2^{k-1},M}$$ of a function f belonging to generalized Holder’s class $$H_{\alpha }^{(\omega )}[0,1)$$ by first kind Chebyshev wavelets have been determined. These estimators are new and best possible approximations in wavelet analysis. Applying this algorithm, a solution of linear and non-linear second order differential equations have been obtained. These solutions are approximately the same as the exact solution of the differential equation. Two non-linear initial value problems are solved by Chebyshev wavelet method when Collocation method, Galerkin’s method and other known methods do not work. This is a significant achievement of this paper in wavelet analysis.

Superconvergent discontinuous Galerkin methods for nonlinear parabolic initial and boundary value problems

Abstract In this article, we discuss error estimates for nonlinear parabolic problems using discontinuous Galerkin methods which include HDG method in the spatial direction while keeping time variable continuous. When piecewise polynomials of degree k ⩾ 1 are used to approximate both the potential as well as the flux, it is shown that the error estimate for the semi-discrete flux in L∞(0, T; L2)-norm is of order k + 1. With the help of a suitable post-processing of the semi-discrete potential, it is proved that the resulting post-processed potential converges with order of convergence $\begin{array}{} \displaystyle O\big(\!\sqrt{{}\log(T/h^2)}\,h^{k+2}\big) \end{array}$ in L∞(0, T; L2)-norm. These results extend the HDG analysis of Chabaud and Cockburn [Math. Comp. 81 (2012), 107–129] for the heat equation to non-linear parabolic problems.