Nonlinear Initial Value Problems Research Articles

Postprocessing Techniques of High-Order Galerkin Approximations to Nonlinear Second-Order Initial Value Problems with Applications to Wave Equations

Postprocessing Techniques of High-Order Galerkin Approximations to Nonlinear Second-Order Initial Value Problems with Applications to Wave Equations

The Nemytskiĭ Operator and Vector Measure Solutions for Non-Linear Initial Value Problems

Abstract We define and study a Banach-space-valued Nemytskiĭ operator and we find vector measure solutions for associated non-linear initial value problems.

Three-step Collocation Method for Solution of Third Derivative Initial Value Problem

In this research paper, we introduce a novel three-step block method designed to directly tackle third-order initial value problems. This method is crafted through an interpolation and collocation approach, leveraging power series analysis. We conduct a comprehensive examination of the proposed method, ensuring it meets all requisite conditions for rigorous analysis. To assess its efficacy and validity, we employ the method on both highly stiff linear and nonlinear initial value problems, juxtaposing our findings with established approaches in the literature. Our results highlight that the new method exhibits faster convergence compared to existing methods, underscoring its superior performance. Initial value problems, Interpolation and collocation procedure, Three step, Third order

Data Dependence and Existence and Uniqueness for Hilfer Nabla Fractional Difference Equations

In the present article, we establish sufficient conditions for the existence of a unique bounded solution using a prominent fixed-point theorem for the non-linear initial value problem involving the recently introduced Hilfer nabla fractional difference operator. where 0 <ℵ<1, 0 ≤ ß ≤1, ι =ℵ+ ß −ℵß and j : Nx × Rn →Rn. We also analyze the Ulam-Hyers stability of the considered problem and make some interesting observations on the dependence of its solutions on initial conditions and parameters. Finally, we conclude this article by constructing suitable examples to illustrate the applicability of established results.

A Block Hybrid Method with Equally Spaced Grid Points for Third-Order Initial Value Problems

In this paper, we extend the block hybrid method with equally spaced intra-step points to solve linear and nonlinear third-order initial value problems. The proposed block hybrid method uses a simple iteration scheme to linearize the equations. Numerical experimentation demonstrates that equally spaced grid points for the block hybrid method enhance its speed of convergence and accuracy compared to other conventional block hybrid methods in the literature. This improvement is attributed to the linearization process, which avoids the use of derivatives. Further, the block hybrid method is consistent, stable, and gives rapid convergence to the solutions. We show that the simple iteration method, when combined with the block hybrid method, exhibits impressive convergence characteristics while preserving computational efficiency. In this study, we also implement the proposed method to solve the nonlinear Jerk equation, producing comparable results with other methods used in the literature.

Stability of Time-Dependent Motions for Fluid–Rigid Ball Interaction

We aim at the stability of time-dependent motions, such as time-periodic ones, of a rigid body in a viscous fluid filling the exterior to it in 3D. The fluid motion obeys the incompressible Navier–Stokes system, whereas the motion of the body is governed by the balance for linear and angular momentum. Both motions are affected by each other at the boundary. Assuming that the rigid body is a ball, we adopt a monolithic approach to deduce Lq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^q$$\\end{document}–Lr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^r$$\\end{document} decay estimates of solutions to a non-autonomous linearized system. We then apply those estimates to the full nonlinear initial value problem to find temporal decay properties of the disturbance. Although the shape of the body is not allowed to be arbitrary, the present contribution is the first attempt at analysis of the large time behavior of solutions around nontrivial basic states, that can be time-dependent, for the fluid–structure interaction problem and provides us with a stability theorem which is indeed new even for steady motions under the self-propelling condition or with wake structure.

Numerical Analysis of Compound Biochemical Calcium Oscillations Process in Hepatocyte Cells.

The hepatocyte cells regulate the wide range of liver function by moderating cellular activities such as lipid, protein metabolism, carbohydrate, and interact with other cells for proliferation and maintenance. In hepatocyte cells, the concentration of calcium uptake is quite extensive from various agonists such as active subunit, active phospholipase C, free calcium in the cytosol, and endoplasmic reticulum. The overproduction and degradation of calcium signals can cause homeostasis, liver inflammation, and liver diseases. The spatiotemporal behavior of calcium oscillation reveals the physiological role of these cellular entities in understanding the process of production and degradation. No computational attempt has been registered to date on the compound calcium regulation of these cellular entities including the memory of cells. Hence, the authors proposed a fractional order compartmental model that systematically simulates the exchange of calcium intake in cellular entities. The nonlinear equations of the rate of changes in the active subunit, active phospholipase C, free calcium in the cytosol, and endoplasmic reticulum are coupled to form a nonlinear fractional order initial value problem. The existence and uniqueness, stability analysis of the model is performed that validate the theoretical results and explore the dynamic behaviour of calcium oscillation in each compartment.

Modified Fourth Derivative Block Method and its direct applications to third-order initial value problems

A theoretical order eight Modified Fourth Derivative four-step block method (MFDFBM) has been derived, analysed and numerically applied to solve multiple problems originating from Fluid Dynamics, engineering and other sciences. The MFDFBM was derived by applying collocation and interpolation techniques to a power series approximation. Further introducing fourth derivative terms at each of the collocating points yields a block method with an improved order of accuracy. It was observed that the order of the block method increases with the number of fourth derivative terms introduced into the integration interval. Numerical experiments are presented to test MFDFBM on numerical examples, including non-linear homogeneous thin film flow (NHTFF) problems and two non-linear initial value problems(IVPs). The experiments confirm the good impact of adding the fourth derivative terms, which help improve the order of accuracy of the derived MFDFBM, thereby minimising error and agreeing with analytical solution up to at least seven decimal places.

A Novel Numerical Method for Solving Nonlinear Fractional-Order Differential Equations and Its Applications

In this article, a considerably efficient predictor-corrector method (PCM) for solving Atangana–Baleanu Caputo (ABC) fractional differential equations (FDEs) is introduced. First, we propose a conventional PCM whose computational speed scales with quadratic time complexity O(N2) as the number of time steps N grows. A fast algorithm to reduce the computational complexity of the memory term is investigated utilizing a sum-of-exponentials (SOEs) approximation. The conventional PCM is equipped with a fast algorithm, and it only requires linear time complexity O(N). Truncation and global error analyses are provided, achieving a uniform accuracy order O(h2) regardless of the fractional order for both the conventional and fast PCMs. We demonstrate numerical examples for nonlinear initial value problems and linear and nonlinear reaction-diffusion fractional-order partial differential equations (FPDEs) to numerically verify the efficiency and error estimates. Finally, the fast PCM is applied to the fractional-order Rössler dynamical system, and the numerical results prove that the computational cost consumed to obtain the bifurcation diagram is significantly reduced using the proposed fast algorithm.

Investigation of a one-step hybrid algorithm towards the solution of first order linear and nonlinear initial-value problems of ordinary differential equations (FOLNIVP)

The aim of developing any numerical method is to complement the challenges inherent in obtaining the analytical solution of a differential equation, if at all a closed form solution does exist. In this study we present a one-step implicit code of order eight block algorithms for the purpose of utilizing data at points other than a whole step number. The major advantage of hybrid method is that they possess remarkably small error constants which translate to better approximation. These methods constitute a class of methods whose computational potentialities have probably not yet been fully exploited. Therefore, the performance of the derived block hybrid algorithm is investigated using some numerical examples for the purpose of demonstrating its validity and applicability. The results obtained revealed that the algorithm is suitable for solving first order linear and nonlinear initial value problems (IVPs) of ordinary differential equations.

Romanovski-Jacobi spectral schemes for high-order differential equations

We develop direct solution techniques for solving high-order differential equations with constant coefficients using the spectral tau method. The spatial approximation is based on Romanovski-Jacobi polynomials {Rnα,β(x)}n=0N with α>−1, β<−2N−α−1, x∈(0,∞) and n is the polynomial degree. Then, a hybrid approach combining the Romanovski-Jacobi tau method with the Romanovski-Jacobi collocation technique is introduced for the numerical solution of high-order differential equations with variable coefficients. The main characteristic behind this approach is that it reduces the problems to those of solving systems of algebraic equations, which greatly simplifies the problem. Finally, the Romanovski-Jacobi collocation method is presented for solving nonlinear high-order initial value problems. Numerical examples are included to demonstrate the validity and applicability of the techniques, and comparisons are made with the existing results. The method is easy to implement and yields very accurate results.

Encapsulated generalized summation-by-parts formulations for curvilinear and non-conforming meshes

We extend the construction of so-called encapsulated global summation-by-parts operators to the general case of a mesh which is not boundary conforming. Owing to this development, energy stable discretizations of nonlinear and variable coefficient initial boundary value problems can be formulated in simple and straightforward ways using high-order accurate operators of generalized summation-by-parts type. Encapsulated features on a single computational block or element may include polynomial bases, tensor products as well as curvilinear coordinate transformations. Moreover, through the use of inner product preserving interpolation or projection, the global summation-by-parts property is extended to arbitrary multi-block or multi-element meshes with non-conforming nodal interfaces.

Nonlinear boundary conditions for initial boundary value problems with applications in computational fluid dynamics

We derive new boundary conditions and implementation procedures for nonlinear initial boundary value problems (IBVPs) with non-zero boundary data that lead to bounded solutions. The new boundary procedure is applied to nonlinear IBVPs in skew-symmetric form, including dissipative terms. The complete procedure has two main ingredients. Firstly, the energy rate in terms of a surface integral with boundary terms is derived. Secondly, we bound the surface integral by deriving new nonlinear boundary procedures for boundary conditions with non-zero data. The new nonlinear boundary procedure generalises the well known characteristic boundary procedure for linear problems to the nonlinear setting.To introduce the procedure, a skew-symmetric scalar IBVP encompassing the linear advection equation and Burgers equation is analysed. Once the continuous analysis is done, we show that energy stable nonlinear discrete approximations follow by using summation-by-parts operators combined with weak boundary conditions. The scalar analysis is subsequently repeated for general nonlinear systems of equations. Finally, the new boundary procedure is applied to four important IBVPs in computational fluid dynamics: the incompressible Euler and Navier-Stokes, the shallow water and the compressible Euler equations.

Novel Approximate Solutions for Nonlinear Initial and Boundary Value Problems

This paper investigates an effective computational method (ECM) based on the standard polynomials used to solve some nonlinear initial and boundary value problems appeared in engineering and applied sciences. Moreover, the effective computational methods in this paper were improved by suitable orthogonal base functions, especially the Chebyshev, Bernoulli, and Laguerre polynomials, to obtain novel approximate solutions for some nonlinear problems. These base functions enable the nonlinear problem to be effectively converted into a nonlinear algebraic system of equations, which are then solved using Mathematica®12. The improved effective computational methods (I-ECMs) have been implemented to solve three applications involving nonlinear initial and boundary value problems: the Darcy-Brinkman-Forchheimer equation, the Blasius equation, and the Falkner-Skan equation, and a comparison between the proposed methods has been presented. Furthermore, the maximum error remainder () has been computed to prove the proposed methods' accuracy. The results convincingly prove that ECM and I-ECMs are effective and accurate in obtaining novel approximate solutions to the problems.

Third (fourth) order accurate two-step super-stable cubic spline polynomial approximation for the second order non-linear initial-value problems

Third (fourth) order accurate two-step super-stable cubic spline polynomial approximation for the second order non-linear initial-value problems

Solving First-Order Nonlinear Fuzzy Initial Value Problems Using Two-Step Block Method with Presence of Higher Derivatives

Fuzzy differential equation models are suitable where uncertainty exists for real-world phenomena. Numerical techniques are used to provide an approximate solution to these models in the absence of an exact solution. However, existing studies that have developed numerical techniques for solving FIVPs possess an absolute error accuracy that could be improved. This is as a result of the low order and non-self-starting properties of the developed numerical techniques by previous studies. For this reason, this study, develops an Obrechkoff-type two-step implicit block method with the presence of second and third derivative for the numerical solution of first-order nonlinear fuzzy initial value problems. The convergence properties for the proposed block method are described in detail. Then the proposed method is adopted to solve first-order nonlinear fuzzy initial value problems with triangular and trapezoidal fuzzy numbers. The obtained results indicates that the proposed method effectively solves first-order nonlinear fuzzy initial value problems with better accuracy.

A generalized CAS wavelet method for solving ψ-Caputo fractional differential equations

PurposeThe purpose of the present work is to introduce a wavelet method for the solution of linear and nonlinear psi-Caputo fractional initial and boundary value problem.Design/methodology/approachThe authors have introduced the new generalized operational matrices for the psi-CAS (Cosine and Sine) wavelets, and these matrices are successfully utilized for the solution of linear and nonlinear psi-Caputo fractional initial and boundary value problem. For the nonlinear problems, the authors merge the present method with the quasilinearization technique.FindingsThe authors have drived the orthogonality condition for the psi-CAS wavelets. The authors have derived and constructed the psi-CAS wavelets matrix, psi-CAS wavelets operational matrix of psi-fractional order integral and psi-CAS wavelets operational matrix of psi-fractional order integration for psi-fractional boundary value problem. These matrices are successfully utilized for the solutions of psi-Caputo fractional differential equations. The purpose of these operational matrices is to make the calculations faster. Furthermore, the authors have derived the convergence analysis of the method. The procedure of implementation for the proposed method is also given. For the accuracy and applicability of the method, the authors implemented the method on some linear and nonlinear psi-Caputo fractional initial and boundary value problems and compare the obtained results with exact solutions.Originality/valueSince psi-Caputo fractional differential equation is a new and emerging field, many engineers can utilize the present technique for the numerical simulations of their linear/non-linear psi-Caputo fractional differential models. To the best of the authors’ knowledge, the present work has never been introduced and implemented for psi-Caputo fractional differential equations.

A new acceleration of variational iteration method for initial value problems

This paper is devoted to accelerating the variational iteration method (VIM) to solve a nonlinear initial value problem. For this purpose, the redundant calculations of the conventional VIM are removed, and the complex integration is evaluated recursively and quickly with a suitable numerical approximation. The convergence of the proposed method is proven, and an efficient implementation algorithm is presented. This method is successfully applied to solve three applied equations, including the Riccati equation, the Lane–Emden equation, and the SIR epidemic model with a constant vaccination rate. Numerical simulations show that this improvement has significantly increased the method’s speed and enlarged its convergence region.

A new approach for solving boundary and initial value problems by coupling the he method and Sawi transform

This paper discusses and implements a newly developed technique using the He method with Sawi Transform. The main aim is to solve some initial and boundary problems. This combination exhibits an accurate strategy to obtain a precise solution for linear and nonlinear problems. To validate the proposed Hybrid method, a 4- examples are discussed, these including: Burger’s equation, telegraph equation, Kelin-Gordan equation, Duffing oscillator with cubic nonlinear term. The obtained results improve the exactness and the accuracy of the proposed combinations, and the proposed method is capable to solve a large number of linear and nonlinear initial and boundary value problems.

A new adaptive nonlinear numerical method for singular and stiff differential problems

In this work, a new adaptive numerical method is proposed for solving nonlinear, singular, and stiff initial value problems often encountered in real life. Starting with a fixed step size, the new method’s performance can be significantly enhanced by introducing an adaptive step-size approach. The qualitative properties of the proposed method have been investigated to determine the efficiency and reliability of the method. The proposed method is of fifth-order accuracy, zero stable, L-stable, and consistent. In addition, the proposed method is convergent, and its stability properties are also shown through its Order Stars. Finally, numerical experiments are conducted to illustrate the performance of the method. The results obtained show that the proposed method compares favourably with existing methods.