Initial Value Problem Research Articles

A polynomial-based operational matrix approach to solve fractional-order Bratu-type problems

Abstract This article presents a computationally efficient numerical algorithm that relies on fractional-order Lagrange polynomials (FOLPs) for solving initial and boundary value problems involving nonlinear fractional-order Bratu equation. This Bratu equation is a core component of the fabrication process framework for electroscope nanofibers. We create an operational matrix of fractional integration using the FOLPs to tackle the relevant initial and boundary value problems. Additionally, when using the collocation approach, it is reduced to a set of algebraic equations that can be solved using Newton’s iterative method. The Caputo notion of the fractional derivative is utilized. The effectiveness and accuracy of the present strategy are then illustrated using a few numerical examples. It has been found that for accurate numerical computation, just a few terms of Lagrange’s polynomials are needed.

Instability in nonlinear Schrödinger breathers

We consider the focusing Nonlinear Schrödinger equation posed on the one dimensional line, with nonzero background condition at spatial infinity, given by a homogeneous plane wave. For this problem of physical interest, we study the initial value problem for perturbations of the background wave in Sobolev spaces. It is well-known that the associated linear dynamics for this problem describes a phenomenon known in the literature as modulational instability, also recently related to the emergence of rogue waves in ocean dynamics. In qualitative terms, small perturbations of the background state increase its size exponentially in time. In this paper we show that, even if there is no time decay for the linear dynamics due to the modulationally unstable regime, the equation is still locally well-posed in Hs, s > 1/2. We apply this result to give a rigorous proof of the unstable character of two well-known NLS solutions: the Peregrine and Kuznetsov-Mabreathers.

ON SOME COMPARISON THE METHODS OF RUNGE-KUTTA AND MULTI-STEP TYPES.

There are exactly two popular classes of methods to solve the initial-value problem for Ordinary Differential Equations, which are usually called the Runge-Kutta and Multistep methods. Each method of these classes has its advantages and disadvantages. Note that at the intersection of these methods, there is one method the explicit Euler method. The main difference between this class of methods is that in the class of Multistep methods, there are implicit methods. However, this cannot be said about the classic Runge-Kutta method. Here have investigated these class methods, considering to construction of stable methods with a high degree. And also recommended to construct a method that preserves some properties of the Runge-Kutta methods and also some properties of the Multistep Methods with constant coefficients. By using the Runge-Kutta methods are one-step, here by changing the values of step size, recommended to construct methods at the intersection of these methods. It is also shown that depending on the nature of solving problems, these methods can coincide.

Damage Behaviour of Quasi-Brittle Composites: Mathematical and Computational Aspects

In the present paper, an evaluation of the damage behaviour of quasi-brittle composites exposed to mechanical, thermal, and other loads is studied by means of viscoelastic and/or viscoplastic material models, applying some non-local regularisation techniques to the initiation and development of damages. The methods above are presented as a strong tool for a deeper understanding of material structures in miscellaneous engineering disciplines like civil, mechanical, and many others. Nevertheless, all of the software packages reflect certain compromises between the need for effective computational tools, with parameters obtained from inexpensive experiments, within the possibilities and the complexity of both physical and geometrical descriptions of structure deformation within processes. The article is devoted to the mathematical aspects regarding a considerably wide class of computational modelling problems, emphasising the following ones: (i) the existence and the uniqueness of solutions of engineering problems formulated in terms of the deterministic initial and boundary value problems of partial differential equations theory; (ii) the problems of convergence of computational algorithms applied to (i). Both aspects have numerous references to possible generalisations and investigations connected with open problems.

Rapidly Decreasing Solution of the Initial Boundary value Problem for an Infinite System of Nonlinear Evolution Equations

Rapidly Decreasing Solution of the Initial Boundary value Problem for an Infinite System of Nonlinear Evolution Equations

A Block Hybrid Method for the Direct Solution of y^(”)=f(x,y,y^(’))

This paper introduces a Block Hybrid Method for solving general second-order ordinary differential equations (ODEs) with initial value problems. The method is based on a continuous formulation of the second-order hybrid generalized Adams method, incorporating one off-grid point per step. Discrete schemes are derived from the continuous form and its first derivative, forming the block methods. Analysis shows the method is consistent, zero-stable, and convergent. Numerical results highlight its superiority over existing methods.

Система моделирования пространственной динамики бактериальной популяции при вариации режимов антимикробной обработки

<p>This work is devoted to the development and implementation of a model for the evolution of a bacterial population grown on a nutrient medium under conditions of controlled biomass inhibition by an antimicrobial agent. To formalize the model a continuous deterministic approach is used. The mathematical model is described by an initial boundary value problem for a system of reaction-diffusion equations defining the spatio-temporal distributions of nutrient substrate and biomass, taking into account integration with a pharmacokinetic model for a single antimicrobial treatment. The model was implemented by the finite element method using the finite element analysis system – COMSOL Multiphysics platform. A series of computational experiments were performed to establish numerical patterns of changes in bacterial mass concentration with variation in antimicrobial dose. A discussion of the potential application of this approach to investigate the issue of bacterial resistance to antibiotics is presented.</p>

Long time behavior of solutions to the generalized Boussinesq equation

In this paper, we investigate the generalized Boussinesq equation (gBq) as a model for the water wave problem with surface tension. Our study begins with the analysis of the initial value problem within Sobolev spaces, where we derive improved conditions for global existence and finite-time blow-up of solutions, extending previous results to lower Sobolev indices. Furthermore, we explore the time-decay behavior of solutions in Bessel potential and modulation spaces, establishing global well-posedness and time-decay estimates in these function spaces. Using Pohozaev-type identities, we demonstrate the non-existence of solitary waves for specific parameter regimes. A significant contribution of this work is the numerical generation of solitary wave solutions for the gBq equation using the Petviashvili iteration method. Additionally, we propose a Fourier pseudo-spectral numerical method to study the time evolution of solutions, particularly addressing the gap interval where theoretical results on global existence or blow-up are unavailable in the Sobolev spaces. Our numerical results provide new insights by confirming theoretical predictions in covered cases and filling gaps in unexplored scenarios. This comprehensive analysis not only clarifies the theoretical and numerical landscape of the gBq equation but also offers valuable tools for further investigations.

A Comprehensive Study of Nonlinear Mixed Integro-Differential Equations of the Third Kind for Initial Value Problems: Existence, Uniqueness and Numerical Solutions

Nonlinear mixed integro-differential equations (NM-IDEs) of the third kind present a complex challenge during solving initial value problems (IVPs), particularly after converting them from standard forms. In this work, we address the existence and uniqueness of a type of NM-IDEs employing Picard’s method. Additionally, we estimate the solution using the homotopy analysis method (HAM) and analyze the convergence of the approach. To demonstrate the credibility of the theoretical results, various applications are given, and the numerical results are displayed in a group of figures and tables to highlight that solving IVPs by first converting them to NM-IDEs and using the HAM is a computationally efficient approach.

New Solution for Charge Transport in DNA Model with Generalized Morse Potential

This paper is concerned with the global weak solution for charge transport DNA model with vibrational and rotational coupling motion. For that we use the theory of semigroup according to rewrite in vector form the system obtained from the dynamics of the Peyrard-Bishop model for vibrational motion of DNA dynamics. We consider an abstract initial value problem, which we show that, under suitable assumptions the system supports a single global weak solution satisfying the given initial condition, for that one, we consider the Sobolev spaces which will solve the Cauchy problem. We present our result developed in Matlab for the calculation of the average stretching amplitude for a Morse potential dependent on a parameter q (generalized Morse potential). The emphasis is on the inverse relationship between of the average distance of a DNA breathing and the “q” values of the generalized Morse potential.

Theoretical and experimental studies on a novel potential barrier control for a vehicle model and piezoelectric smart structure

Abstract A novel control methodology, inspired by the quantum mechanical phenomenon of particle-wave duality, wherein particles and waves utilize their energy to traverse potential barriers, has been proposed. This control strategy is investigated both numerically and experimentally to mitigate undesirable vibrations in a three-degree-of-freedom vehicle system and a smart beam structure. The mathematical framework for the vehicle system is derived using Lagrange's equations, while the experimental setup is designed for the smart beam. Disturbance inputs in the form of step and sinusoidal road profiles are introduced into the vehicle model to simulate real-world conditions. For the smart beam, endpoint vibrations are induced as an initial value problem by releasing the beam's endpoint from a predefined initial displacement. In order to achieve vibration suppression, control signals, parameterized based on system-specific frequency and period characteristics, are applied to both systems. The optimal control parameters are determined by evaluating the vibration damping performance. The findings of this study demonstrate that the proposed control strategy is not only effectively applicable to the experimental validation of the piezoelectric smart structure but also holds significant potential for numerical integration into mechanical systems, such as vehicle dynamics models.

An asymptotic characterisation of the Kerr spacetime

Abstract We provide a characterisation of the Kerr spacetime close to future null infinity using the asymptotic characteristic initial value problem in a conformally compactified spacetime. Stewart's gauge is used to set up the past-oriented characteristic initial value problem. By a theorem of M. Mars characterising the Kerr spacetime, we provide conditions for the existence of an asymptotically timelike Killing vector on the development of the initial data by demanding that the spacetime is endowed with a Killing spinor. The conditions on the characteristic initial data ensuring the existence of a Killing spinor are, in turn, analysed. Finally, we write the conditions on the initial data in terms of the free data in the characteristic initial value problem. As a result, we characterise the Kerr spacetime using only a section of future null infinity and its intersection with an outgoing null hypersurface.

Modeling of hysteresis effects in magneto-active polymers: constitutive theory and variational principles

Abstract This work covers the variational-based modeling of magneto-mechanical hysteresis effects in hard magnetic magneto-active polymers (MAPs). We discuss basic ingredients of the constitutive theory within the concept of generalized standard materials that necessitates suitable definitions of (i) the total energy density function and (ii) the dissipation potential. A key feature of the developed energy-based constitutive model is its modular structure that allows to associate individual components of the total energy density with the magnetic field strength in vacuum, the magnetization of the material, and specific components of the total stress tensor. The utilization of spectral invariants allows the formulation of a compact constitutive model that is able to accurately capture the highly nonlinear material behavior of hard magnetic MAPs with stochastic microstructures. The developed constitutive functions are subsequently embedded in incremental variational principles. This is supplemented by the discussion of corresponding conforming finite element methods. The performance of the developed variational-based models is demonstrated by solving some application-oriented initial boundary value problems.

Accurate Chebfun solutions to third-order nonlinear BVPs on the half-line. Applications in boundary layer theory

Abstract The paper is devoted to obtaining accurate numerical solutions to some third-order nonlinear boundary problems on the real semi-axis of Blasius type. It aims to address these issues more accurately as boundary value problems rather than converting them into initial value problems. The Blasius boundary layer problem with slip and nonslip boundary conditions, the Sakiadis and Falkner-Skan problems, a pseudoplastic boundary layer problem, etc, are solved using the Chebyshev technology, in the form of Chebfun, along with the domain truncation. The choice of this method is justified vis-à-vis the difficulties of solving some singular and nonlinear third-order boundary value problems. The convergence rate and the convergence order of Newton’s method in solving the nonlinear algebraic system obtained by discretisation are established. The shear stress is accurately estimated along with the first and second derivatives of the solutions. Some comments on the stability of numerical outcomes concerning the length of the integration domain are provided. The results are presented graphically and tabulated for the most challenging case (non-linearity involving fractional power). The method correctly captures the large gradients of the solution.

The improved convex combination method applied to crack propagation problems

In this work, the Improved Convex Combination method is presented to obtain approximate, closed-form solutions, for crack propagation problems, applying the Paris-Erdogan model. In most of the crack propagation problems, it is not possible to achieve explicit solutions due to the complexity of the terms. In this regard, it is required that numerical methods be used to secure approximate numerical solutions. This article presents a method that provides a proposal that can be employed and extended to other problems which, in turn, are defined by differential equations. The approximate solutions are obtained as an exact solution of a modified initial value problem established by the equation of Ricatti. The performance is evaluated in relation to problems with known exact solutions and approximate numerical solutions.

Application of the One-Step Second-Derivative Method for Solving the Transient Distribution in Markov Chain

Markov chains are an application of stochastic models in operation research, helping the analysis and optimization of processes with random events and transitions. The method that will be deployed to obtain the transient solution to a Markov chain problem is an important part of this process. The present paper introduces a novel Ordinary Differential Equation (ODE) approach to solve the Markov chain problem. The probability distribution of a continuous-time Markov chain with an infinitesimal generator at a given time is considered, which is a resulting solution of the Chapman-Kolmogorov differential equation. This study presents a one-step second-derivative method with better accuracy in solving the first-order Initial Value Problems (IVPs) compared to other approaches found in the literature, which is verified by the obtained solutions. The determination of the transient solutions for Markov chains is presented using the proposed method. The results show better accuracy in solving the transient distribution in Markov chains, which implies that there is an improved assurance in adopting this approach in future studies of the Markov chain modeling process for predicting future events based on the current state of a process. Future studies on Markov chain modeling could adopt the introduced method to predict future events based on the current state of a process.

Development of Hybrid Block Methods for Oscillatory Solutions of Second-Order Differential Equations

This study explores the application of hybrid block method in solving oscillatory second-order initial value problems (IVPs), a category of differential equations relevant in modeling various real-life phenomena, such as mechanical and electrical oscillations. Traditional numerical methods often face challenges in accuracy and stability when applied to oscillatory problems, prompting a need for advanced methods like hybrid block method. This research developed a hybrid block method that offers improved error control, stability and convergence for oscillatory differential equations. Error analysis is conducted to assess the method's effectiveness, comparing it with existing approaches to highlight its robustness in handling oscillatory behavior. The proposed method demonstrates an expanded stability region, making it suitable for complex, real-world applications that require high precision. The study's findings emphasize the potential of block hybrid methods in advancing numerical solutions for differential equations, providing valuable insights for further research and application in science and engineering

Two Higher-Order Single-Step Embedded Block Hybrid Integrators for First-Order Ordinary Differential Equations

In this research work, the extended Backward Differentiation Formula (EBDF) is used to develop an embedded block hybrid method (EBHM) for step numbers and for the solution of first-order initial-value problems (IVPs) in ordinary differential equations (ODEs). The methods are obtained by continuous approximation using multi-step collocation techniques. The new embedded block hybrid method for and consist of five and six discrete formulae, respectively, which are simultaneously used as integrators. Analyses of the EBHM5 and EBHM6 indicate that the methods are respectively of orders six and seven, hence consistent. In addition, both methods are zero-stable and therefore convergent. Linear stability analyses indicate that the methods are A-stable, hence suitable for stiff problems. The new methods are used to solve some numerical examples, and the absolute errors show a better rate of convergence when compared with existing methods in the literature.

Global well-posedness of strong solutions to the 3D full compressible magnetohydrodynamics equations with zero heat-conduction

In this paper, we study an initial value problem of three-dimensional full compressible magnetohydrodynamics (MHD) fluids with zero heat-conduction. The global existence of strong solutions with large oscillations to the Cauchy problem was obtained under the conditions that the initial energy is suitably small.

On Fractal Derivatives and Applications

ABSTRACTIn this paper, we develop some fundamental tools for fractal derivatives, such as a Leibniz‐type product rule and a chain rule. Also, a local Picard theorem for an initial value problem via fractal derivatives is obtained. Finally, the Gompertz and logistic models associated to those differential operators are considered. We also show how effective these models are compared to other well‐known (classical and conformable) models when applied to an example of real data.