Special Issue "Numerical Methods for Solving Differential Problems"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: 30 April 2021.

Special Issue Editor

Prof. Dr. Higinio Ramos
E-Mail Website
Guest Editor
Scientific Computing Group, Universidad de Salamanca, Plaza de la Merced Salamanca 37008, Spain
Interests: numerical solution of differential equations; numerical analysis
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Special Issue Information

Dear Colleagues,

It is well known that differential equations are present in numerous fields of science, engineering, economics, and many more. Many physical phenomena are modeled through differential equations, which have the appealing aspect of describing the world around us. There are many different types of such equations beyond the simple classifications between ordinary/partial or linear/nonlinear. Among other types, one can find singular, singularly perturbed, delay, integral or algebraic differential equations. If we add to the differential equation certain conditions that the solution must meet at one or more points, we will have initial value problems or boundary value problems. The combination of the different possibilities above gives rise to a wide catalog of differential problems, for which, in most cases, we cannot find an analytical solution.

The aim of this Special Issue is to update the numerical techniques for solving differential problems in a broad sense, with an emphasis on real-world applications. Contributions that involve a review of the state of the art for the numerical resolution of different types of differential problems will be welcome. As there are other Special Issues in this series devoted to fractional problems and numerical solution of partial differential equations, we have not included this kind of problems here.

Prof. Dr. Higinio Ramos
Guest Editor

Manuscript Submission Information

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Keywords

  • Numerical methods for differential equations
  • Ordinary differential equations
  • Initial value problems
  • Boundary value problems
  • Singular differential equations
  • Singularly perturbed problems
  • Delay differential equations
  • Algebraic differential equations

Published Papers (10 papers)

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Research

Open AccessArticle
A Singularly P-Stable Multi-Derivative Predictor Method for the Numerical Solution of Second-Order Ordinary Differential Equations
Mathematics 2021, 9(8), 806; https://doi.org/10.3390/math9080806 - 08 Apr 2021
Viewed by 294
Abstract
In this paper, a symmetric eight-step predictor method (explicit) of 10th order is presented for the numerical integration of IVPs of second-order ordinary differential equations. This scheme has variable coefficients and can be used as a predictor stage for other implicit schemes. First, [...] Read more.
In this paper, a symmetric eight-step predictor method (explicit) of 10th order is presented for the numerical integration of IVPs of second-order ordinary differential equations. This scheme has variable coefficients and can be used as a predictor stage for other implicit schemes. First, we showed the singular P-stability property of the new method, both algebraically and by plotting the stability region. Then, having applied it to well-known problems like Mathieu equation, we showed the advantage of the proposed method in terms of efficiency and consistency over other methods with the same order. Full article
(This article belongs to the Special Issue Numerical Methods for Solving Differential Problems)
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Open AccessArticle
A Family of Functionally-Fitted Third Derivative Block Falkner Methods for Solving Second-Order Initial-Value Problems with Oscillating Solutions
Mathematics 2021, 9(7), 713; https://doi.org/10.3390/math9070713 - 25 Mar 2021
Viewed by 247
Abstract
One of the well-known schemes for the direct numerical integration of second-order initial-value problems is due to Falkner. This paper focuses on the construction of a family of adapted block Falkner methods which are frequency dependent for the direct numerical solution of second-order [...] Read more.
One of the well-known schemes for the direct numerical integration of second-order initial-value problems is due to Falkner. This paper focuses on the construction of a family of adapted block Falkner methods which are frequency dependent for the direct numerical solution of second-order initial value problems with oscillatory solutions. The techniques of collocation and interpolation are adopted here to derive the new methods. The study of the properties of the proposed adapted block Falkner methods reveals that they are consistent and zero-stable, and thus, convergent. Furthermore, the stability analysis and the algebraic order conditions of the proposed methods are established. As may be seen from the numerical results, the resulting family is efficient and competitive compared to some recent methods in the literature. Full article
(This article belongs to the Special Issue Numerical Methods for Solving Differential Problems)
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Open AccessArticle
High Order Two-Derivative Runge-Kutta Methods with Optimized Dispersion and Dissipation Error
Mathematics 2021, 9(3), 232; https://doi.org/10.3390/math9030232 - 25 Jan 2021
Viewed by 278
Abstract
In this work we consider explicit Two-derivative Runge-Kutta methods of a specific type where the function f is evaluated only once at each step. New 7th order methods are presented with minimized dispersion and dissipation error. These are two methods with constant coefficients [...] Read more.
In this work we consider explicit Two-derivative Runge-Kutta methods of a specific type where the function f is evaluated only once at each step. New 7th order methods are presented with minimized dispersion and dissipation error. These are two methods with constant coefficients with 5 and 6 stages. Also, a modified phase-fitted, amplification-fitted method with frequency dependent coefficients and 5 stages is constructed based on the 7th order method of Chan and Tsai. The new methods are applied to 4 well known oscillatory problems and their performance is compared with the methods in that of Chan and Tsai.The numerical experiments show the efficiency of the derived methods. Full article
(This article belongs to the Special Issue Numerical Methods for Solving Differential Problems)
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Open AccessArticle
Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs
Mathematics 2021, 9(2), 174; https://doi.org/10.3390/math9020174 - 16 Jan 2021
Viewed by 377
Abstract
In this paper, a new class of Runge–Kutta-type collocation methods for the numerical integration of ordinary differential equations (ODEs) is presented. Its derivation is based on the integral form of the differential equation. The approach enables enhancing the accuracy of the established collocation [...] Read more.
In this paper, a new class of Runge–Kutta-type collocation methods for the numerical integration of ordinary differential equations (ODEs) is presented. Its derivation is based on the integral form of the differential equation. The approach enables enhancing the accuracy of the established collocation Runge–Kutta methods while retaining the same number of stages. We demonstrate that, with the proposed approach, the Gauss–Legendre and Lobatto IIIA methods can be derived and that their accuracy can be improved for the same number of method coefficients. We expressed the methods in the form of tables similar to Butcher tableaus. The performance of the new methods is investigated on some well-known stiff, oscillatory, and nonlinear ODEs from the literature. Full article
(This article belongs to the Special Issue Numerical Methods for Solving Differential Problems)
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Open AccessArticle
Asymptotic Behavior of Solution to Nonlinear Eigenvalue Problem
Mathematics 2020, 8(11), 2064; https://doi.org/10.3390/math8112064 - 19 Nov 2020
Viewed by 301
Abstract
We study the following nonlinear eigenvalue problem: u(t)=λf(u(t)),u(t)>0,tI:=(1,1),u [...] Read more.
We study the following nonlinear eigenvalue problem: u(t)=λf(u(t)),u(t)>0,tI:=(1,1),u(±1)=0, where f(u)=log(1+u) and λ>0 is a parameter. Then λ is a continuous function of α>0, where α is the maximum norm α=uλ of the solution uλ associated with λ. We establish the precise asymptotic formula for L1-norm of the solution uα1 as α up to the second term and propose a numerical approach to obtain the asymptotic expansion formula for uα1. Full article
(This article belongs to the Special Issue Numerical Methods for Solving Differential Problems)
Open AccessArticle
Implicit Three-Point Block Numerical Algorithm for Solving Third Order Initial Value Problem Directly with Applications
Mathematics 2020, 8(10), 1771; https://doi.org/10.3390/math8101771 - 14 Oct 2020
Cited by 2 | Viewed by 354
Abstract
Recently, direct methods that involve higher derivatives to numerically approximate higher order initial value problems (IVPs) have been explored, which aim to construct numerical methods with higher order and very high precision of the solutions. This article aims to construct a fourth and [...] Read more.
Recently, direct methods that involve higher derivatives to numerically approximate higher order initial value problems (IVPs) have been explored, which aim to construct numerical methods with higher order and very high precision of the solutions. This article aims to construct a fourth and fifth derivative, three-point implicit block method to tackle general third-order ordinary differential equations directly. As a consequence of the increase in order acquired via the implicit block method of higher derivatives, a significant improvement in efficiency has been observed. The new method is derived in a block mode to simultaneously evaluate the approximations at three points. The derivation of the new method can be easily implemented. We established the proposed method’s characteristics, including order, zero-stability, and convergence. Numerical experiments are used to confirm the superiority of the method. Applications to problems in physics and engineering are given to assess the significance of the method. Full article
(This article belongs to the Special Issue Numerical Methods for Solving Differential Problems)
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Open AccessFeature PaperArticle
Efficient k-Step Linear Block Methods to Solve Second Order Initial Value Problems Directly
Mathematics 2020, 8(10), 1752; https://doi.org/10.3390/math8101752 - 12 Oct 2020
Cited by 1 | Viewed by 399
Abstract
There are dozens of block methods in literature intended for solving second order initial-value problems. This article aimed at the analysis of the efficiency of k-step block methods for directly solving general second-order initial-value problems. Each of these methods consists of a [...] Read more.
There are dozens of block methods in literature intended for solving second order initial-value problems. This article aimed at the analysis of the efficiency of k-step block methods for directly solving general second-order initial-value problems. Each of these methods consists of a set of 2k multi-step formulas (although we will see that this number can be reduced to k+1 in case of a special equation) that provides approximate solutions at k grid points at once. The usual way to obtain these formulas is by using collocation and interpolation at different points, which are not all necessarily in the mesh (it may also be considered intra-step or off-step points). An important issue is that for each k, all of them are essentially the same method, although they can adopt different formulations. Nevertheless, the performance of those formulations is not the same. The analysis of the methods presented give some clues as how to select the most appropriate ones in terms of computational efficiency. The numerical experiments show that using the proposed formulations, the computing time can be reduced to less than half. Full article
(This article belongs to the Special Issue Numerical Methods for Solving Differential Problems)
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Open AccessArticle
Analysis of Perturbed Volterra Integral Equations on Time Scales
Mathematics 2020, 8(7), 1133; https://doi.org/10.3390/math8071133 - 10 Jul 2020
Cited by 1 | Viewed by 392
Abstract
This paper describes the effect of perturbation of the kernel on the solutions of linear Volterra integral equations on time scales and proposes a new perspective for the stability analysis of numerical methods. Full article
(This article belongs to the Special Issue Numerical Methods for Solving Differential Problems)
Open AccessArticle
Numerical Approach for Solving Delay Differential Equations with Boundary Conditions
Mathematics 2020, 8(7), 1073; https://doi.org/10.3390/math8071073 - 02 Jul 2020
Viewed by 587
Abstract
In the present paper, a fifth-order direct multistep block method is proposed for solving the second-order Delay Differential Equations (DDEs) directly with boundary conditions using constant step size. In many life sciences applications, a delay plays an essential role in modelling natural phenomena [...] Read more.
In the present paper, a fifth-order direct multistep block method is proposed for solving the second-order Delay Differential Equations (DDEs) directly with boundary conditions using constant step size. In many life sciences applications, a delay plays an essential role in modelling natural phenomena with data simulation. Thus, an efficient numerical method is needed for the numerical treatment of time delay in the applications. The proposed direct block method computes the numerical solutions at two points concurrently at each computed step along the interval. The types of delays involved in this research are constant delay, pantograph delay, and time-dependent delay. The shooting technique is utilized to deal with the boundary conditions by applying a Newton-like method to guess the next initial values. The analysis of the proposed method based on the order, consistency, convergence, and stability of the method are discussed in detail. Four tested problems are presented to measure the efficiency of the developed direct multistep block method. The numerical simulation indicates that the proposed direct multistep block method performs better than existing methods in terms of accuracy, total function calls, and execution times. Full article
(This article belongs to the Special Issue Numerical Methods for Solving Differential Problems)
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Open AccessArticle
Semi-Implicit Multistep Extrapolation ODE Solvers
Mathematics 2020, 8(6), 943; https://doi.org/10.3390/math8060943 - 08 Jun 2020
Viewed by 435
Abstract
Multistep methods for the numerical solution of ordinary differential equations are an important class of applied mathematical techniques. This paper is motivated by recently reported advances in semi-implicit numerical integration methods, multistep and extrapolation solvers. Here we propose a novel type of multistep [...] Read more.
Multistep methods for the numerical solution of ordinary differential equations are an important class of applied mathematical techniques. This paper is motivated by recently reported advances in semi-implicit numerical integration methods, multistep and extrapolation solvers. Here we propose a novel type of multistep extrapolation method for solving ODEs based on the semi-implicit basic method of order 2. Considering several chaotic systems and van der Pol nonlinear oscillator as examples, we implemented a performance analysis of the proposed technique in comparison with well-known multistep methods: Adams–Bashforth, Adams–Moulton and the backward differentiation formula. We explicitly show that the multistep semi-implicit methods can outperform the classical linear multistep methods, providing more precision in the solutions for nonlinear differential equations. The analysis of stability regions reveals that the proposed methods are more stable than explicit linear multistep methods. The possible applications of the developed ODE solver are the long-term simulations of chaotic systems and processes, solving moderately stiff differential equations and advanced modeling systems. Full article
(This article belongs to the Special Issue Numerical Methods for Solving Differential Problems)
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