Region Of Absolute Stability Research Articles

A Chebyshev Generated Block Method for Directly Solving Nonlinear and Ill-posed Fourth-Order ODEs

This study presents a block method induced by Chebyshev polynomials of first kind for directly solving initial value problems of fourth-order ordinary differential equations without reducing the problems to a system of first-order differential equations. The method was developed by applying interpolation and collocation procedures to a Chebyshev approximate polynomial. The unknown parameters were obtained using the Gaussian elimination method, then substituted into the approximate solution to get the continuous scheme and evaluated at the selected point to give the discrete scheme. The method was zero stable, consistent, and convergent, p-stable as shown by the region of absolute stability and has an order of seven. The accuracy and usability of the developed method were tested by applying it to solve six numerical examples. The method was found to be efficient as it gives minimal error. The numerical results of the method were compared with other works cited in the literature and found to be better as it gives minor errors.

A class of explicit second derivative general linear methods for non-stiff ODEs

In this paper, we construct explicit second derivative general linear methods (SGLMs) with quadratic stability and a large region of absolute stability for the numerical solution of non-stiff ODEs. The methods are constructed in two different cases: SGLMs with p = q = r = s and SGLMs with p = q and r = s = 2 in which p, q, r and s are respectively the order, stage order, the number of external stages and the number of internal stages. Examples of the methods up to order five are given. The efficiency of the constructed methods is illustrated by applying them to some well-known non-stiff problems and comparing the obtained results with those of general linear methods of the same order and stage order.

Stability analysis and error estimates of implicit-explicit Runge-Kutta least squares RBF-FD method for time-dependent parabolic equation

In this paper, for the time-dependent parabolic equations defined on complex geometries domain, we develop and analyze the least-squares radial basis function finite difference method (RBF-FD) coupled with the implicit-explicit Runge-Kutta (IMEX-RK) time discretization up to third order accuracy, which improves stability and accuracy. We derive the absolute stability region and the optimal time-step constraint for four kinds of IMEX-RK schemes. Compared to the traditional explicit or implicit time discretization, these are not trivial. Under a wide time-step constraint, the stability and the error estimates in l2-norm are established. Finally, several numerical experiments on the regular domain and non-convex domain are performed to validate the theoretical analysis.

An optimized hybrid block method for solving nonlinear singularly perturbed problems using non-uniform Jain-type meshes

This work aims to implement a one-step block method with three optimized hybrid points for solving singularly perturbed nonlinear initial value problems on non-uniform meshes. The basic properties of the method are investigated and it is shown that the proposed method is consistent, zero-stable, convergent and has a large region of absolute stability. The presented numerical results confirm the efficiency and accuracy of the proposed method.

Dynamical Analysis of Peptic Ulcer Disease Model in Nigeria with the Effect of Vaccination and Treatment Program

The study developed a non-linear deterministic mathematical model to investigate the transmission dynamics of peptic ulcer disease in human population by considering both direct and indirect contact transmission with vaccine and treatment as control. In developing the model, the population was compartmentalized into susceptible-vaccinated-exposed-infected-treated-helicobacter pylori concentration in the environment recovered. The model developed is a system of differential equations. The rate of change of the system, existence and uniqueness of solution, region of absolute stability and positivity of the solution was established. Existence of disease-free equilibrium state and basic reproduction number was also established. Mathematical analysis was determined by the basic reproduction number if , the disease-free equilibrium is locally asymptotically stable whereas if then the equilibrium is unstable whereas the numerical analysis for the estimated basic reproduction number of the model is greater than unity in perspective of Nigeria, for which the Herd Immunity Threshold indicated that vaccinating 99.47% of Nigeria population can control spreading of Peptic ulcer disease in the country. The paper recommended amongst other things that if significant changes concerning the issue of threshold target are observed to improve the rate at which peptic ulcer can be minimized from the population where Government and non-governmental organization should encourage the use of peptic ulcer drugs and vaccines among individuals.

An Efficient New Scheme for Direct Simulation of Higher Order Third and Fourth Oscillatory Differential Equations

In this research, we have examined the general block approach for solving higher-order oscillatory differential equations using the linear block approach (LBA). The basic properties of the new method, such as order, error constant, zero-stability, consistency, convergence, linear stability, and region of absolute stability, were also analyzed and satisfied. Some distinct fourth-order oscillatory problems were directly applied to the new method in order to overcome the setbacks of the reduction method. The results obtained were compared with those in the literature, and the new method takes away the burden of solving fourth-order oscillatory differential equations. Therefore, from the results, the new method has shown better accuracy and faster convergence graphically. One of the advantages of the new method is that it does not require much computational burden and is also self-starting. Convergence, Consistency, Direct simulation, Linear stability, Oscillatory differential equations

On the Numerical Approximation of Higher Order Differential Equation

This research examines the general K - step block approach for solving higher order oscillatory differential equations using Linear Block Approach (LBA). The basic properties of the new method such as order, error constant, zero-stability, consistency, convergence, linear stability and region of absolute stability were also analyzed and satisfied. Some distinct fourth order oscillatory differential equation were directly applied on the new method in order to overcome the setbacks in reduction method, where the step size varies. The results obtained were compared with those in literature and the new method takes away the burden of solving fourth order oscillatory differential equations. The accuracy of the new method proved to be better as it outperformed those of existing methods. Therefore, from the results, the new method has shown better accuracy and faster convergence graphically. One of the advantage of the new method is that it does not require much computational burden and it is also self-starting.

A Study on Convergence and Region of Absolute Stability of Implicit Two – Step Multiderivative Method for Solving Stiff Initial Value Problems of First Order Ordinary Differential Equations.

This study investigated and related the convergence and region of absolute stability of an implicit two - step multiderivative method, used in solving sampled stiff initial value problem of first order ordinary differential equation. It adopted the general implicit multiderivative linear multistep method at step - number (k) = 2 with derivative order (l) varied from 1 - 6 to develop six different variants of the method. Boundary locus method was adopted to determine the intervals of absolute stability which were plotted on the complex plane to show the regions of absolute stability. The variant methods were used to solve sampled stiff initial value problem of first order ordinary differential equation. The resulting numerical solutions were compared with the exact solution to determine accuracy and convergence of the methods. The study showed that two - step first derivative, two - step second derivative, two - step third derivative, two - step fifth derivative method and two - step sixth derivative methods yielded more accurate and convergent results with wider regions of absolute stability than two - step fourth derivative.

A New Hybrid Block Method via Combined Hermite Polynomials and Exponential Functions as Basis Function

This paper presents the derivation and implementation of a hybrid block method for solving stiff and oscillatory first-order initial value problems of ordinary differential equations (ODEs). The hybrid block method was derived by continuous collocation and interpolation using combined Hermite polynomials and exponential functions as the basis function to produce a continuous implicit Linear Multistep Method (LMM) of order nine and implement in block form. The basic properties of the derived method were studied, and the hybrid block integrator was demonstrated to be zero-stable, convergent, consistent, and to have an A-stable region of absolute stability, which made it suitable for stiff and oscillatory ordinary differential equations. The use of a combined basis in the generation of LMMs is worthy of universal acceptance. The technique indicates that, utilizing an interpolation and collocation approach, continuous LMMs can be derived from combinations of any polynomials and exponential functions. On two sampled stiff and oscillatory problems, the new integrator was tested. The numerical results indicate that our new hybrid block integrator is computationally efficient and outperforms existing methods in terms of accuracy

APPLICATION OF MAPLE PROGRAM AND MATLAB CODE ON THE STABILITY ANALYSIS OF AN EXPLICIT FOURTH-STAGE SECOND-ORDER RUNGE-KUTTA METHOD

The essence of this paper is to analyze the stability of a derived explicit fourth-stage second-order Runge-Kutta methodusing MAPLE PROGRAM and MATLAB CODE. The property of stability is essential and necessary requirement for anynumerical method to be termed reliable. The analysis revealed that the method is absolutely stable. The test equation was applied on the method and a polynomial equation was generated. The MAPLE program was used to resolve thepolynomial equation resulting to sets of real and complex roots. The MATLAB code was used to plot the stability curvewhich shows the region of absolute stability. Hence, the whole analysis revealed that the method is not just stable, but itis absolutely stable.

Numerical Approximation for Direct Solution of Third-Order Ordinary Differential Equations Using New Class of Falkner-Type Block Method

In this article, a new class of Falkner-type methods for direct solution of third order ordinary differential equation is developed. The approach of collocation and interpolation technique is adopted to derive the new Falkner-type methods, which is implemented in block mode to get approximation at grids points simultaneously. The resulting scheme is zero-stable, consistent and convergent with good region of absolute stability. The tabular and graphical presentations of the numerical results to the problems considered, demonstrate the effectiveness and good accuracy of the scheme in comparison with other methods

A Comparative Study on the Solutions of 4th Order Differential Equations With Boundary Conditions

In the present study, we have investigated the differential equations of order four to evolve the methods to achieve the solution for differential equations. Absolute Stability Region (ASR) of the differential equations has been examined. Numerical Differentiation (ND) and Differential Transform Method (DTM) which are suggested and derived in this article are much suitable to understand the solutions of differential equations of fourth order. Both the methods are applied to some differential equations, numerical examples and results are presented to outline the capability and robustness of our strategies and compared them with that of exact solution.

Generalization of a class of uniformly optimized [formula omitted]-step hybrid block method for solving two-point boundary value problems

This study aims to develop, analyze and implement an efficient method for approximating two-point boundary value problems of ordinary differential equations. The method contains six and twelve implicit formulas, respectively, for the one-step and two-step schemes. The continuous approximations, using the shifted Chebyshev polynomial as the basis function, were obtained via evaluations at three different points on the selected one-step method, including two optimized hybrid points. Evaluations were carried out on six different points on the selected two-step method, including four generalized optimized hybrid points. Qualitative analysis of the method proves the proposed methods are consistent, zero-stable, convergent and have a larger region of absolute stability. The quantitative analysis shows that the methods compared favorably well and established some superiority strength with the existing methods.

Numerical Integration of Stiff Differential Systems Using Non-Fixed Step-Size Strategy

Over the years, researches have shown that fixed (constant) step-size methods have been efficient in integrating a stiff differential system. It has however been observed that for some stiff differential systems, non-fixed (variable) step-size methods are required for efficiency and for accuracy to be attained. This is because such systems have solution components that decay rapidly and/or slowly than others over a given integration interval. In order to curb this challenge, there is a need to propose a method that can vary the step size within a defined integration interval. This challenge motivated the development of Non-Fixed Step-Size Algorithm (NFSSA) using the Lagrange interpolation polynomial as a basis function via integration at selected limits. The NFSSA is capable of integrating highly stiff differential systems in both small and large intervals and is also efficient in terms of economy of computer time. The validation of properties of the proposed algorithm which include order, consistence, zero-stability, convergence, and region of absolute stability were further carried out. The algorithm was then applied to solve some samples mildly and highly stiff differential systems and the results generated were compared with those of some existing methods in terms of the total number of steps taken, number of function evaluation, number of failure/rejected steps, maximum errors, absolute errors, approximate solutions and execution time. The results obtained clearly showed that the NFSSA performed better than the existing ones with which we compared our results including the inbuilt MATLAB stiff solver, ode 15s. The results were also computationally reliable over long intervals and accurate on the abscissae points which they step on.

Three-Step Numerical Approximant Based on Block Hybrid Backward Differentiation Formula for Stiff System of Ordinary Differential Equations

As long as the field of Engineering, Science and Technology exists, the place of Mathematical modelling that involves stiff systems cannot be overemphasized. Models involving stiff system may result in ordinary differential equations (ODEs) or sometimes as system of ordinary differential equations which must be solved by experts working in that field.However, solving these models using analytical approach may sometimes be challenging or even near impossible. Therefore, it puts a great measure of importance on research into numerical algorithmsfor solution of this class of ordinary differential equations.Premised on the above mentioned, we have formulated, in this paper, a class of backward differentiation formula (BDF) which is a three-step numerical approximant for stiff systems of ODEs. The method was obtained through continuous collocation approach with Legendre polynomial as basis function. We incorporated three off-grid points at interpolation in order that we may retain theBDF’s single function evaluation characteristic. Analyzing basic properties of numerical methods led us to see that the method was consistent, having a uniform order six, zero-stable and in turn, convergent. The method's region of absolute stability was determined using the general linear method, which was plotted and shown to be stable over a vast area. The approach enumerates the solution of stiff of systems ODEs block by block using some discrete schemes that are secured from the corresponding continuous scheme. The method was tested using numerical experiments, and the results, when compared to exact or analytical answers as well as some methods published in the literature, proved that the method is efficient and accurate.

Explicit Nordsieck second derivative general linear methods for ODEs

The paper deals with the construction of explicit Nordsieck second derivative general linear methods with \(s\) stages of order \(p\) with \(p=s\) and high stage order \(q=p\) with inherent Runge–Kutta or quadratic stability properties. Satisfying the order and stage order conditions together with inherent stability conditions leads to methods with some free parameters, which will be used to obtain methods with a large region of absolute stability. Examples of methods with \(r\) external stages and \(p=q=s=r-1\) up to order five are given, and numerical experiments in a fixed stepsize environment are presented. doi: https://doi.org/10.1017/S1446181122000049

Implicit-explicit second derivative general linear methods with strong stability preserving explicit part

In this paper, we discuss the class of implicit–explicit (IMEX) methods for systems of ordinary differential equations where the explicit part has strong stability preserving (SSP) property, while the implicit part of the method has Runge–Kutta stability property together with A-, or L-stability. The explicit part of these methods is treated by the explicit second derivative general linear methods and the implicit part is treated by the implicit general linear methods. We construct methods of order p=s and stage order q=p and q=p−1 with r=2, of order p=q=s−1=r−1 and of order p=q+1=s−1=r−1 up to the order p=4 with large SSP coefficients with respect to the large region of absolute stability when the implicit and explicit parts interact with each other. In the case of methods with p=q=r−1=s−1 and p=q+1=s−1=r−1, the implicit part has inherent Runge–Kutta stability property, and is L-stable. Finally, we test the proposed SSP IMEX schemes on some one dimensional linear and nonlinear problems, and the results for the expected order of convergence are presented.

A Class of Seventh Order Hybrid Extended Block Adams Moulton Methods for Numerical Solutions of First Order Delay Differential Equations

This paper considers the computational solution of some first order delay differential equations (DDEs) using a class of seventh order Hybrid Extended Block Adams Moulton Methods (HEBAMM) without the application of interpolation techniques in evaluating the delay term. The delay term was evaluated by a valid expression of sequence. By matrix inversion techniques, the discrete schemes of the proposed method were obtained through its continuous derivations with the help of linear multistep collocation procedure. The convergence and stability analysis of the method were investigated. After the demonstration of the proposed method in solving some first order DDEs, it was observed that the higher step number k = 4 integrated with a hybrid extended future point performed better and faster in terms of efficiency, accuracy, consistency, convergence, region of absolute stability and Central Processing Unit Time (CPUT) than the lower step numbers k = 3 and 2 integrated with hybrid extended future points when compared with the exact solutions and other existing methods at fixed step size .

Stability Analysis of Thin Power-Law Fluid Film Flowing down a Moving Plane in a Vertical Direction

This paper analyses the stability of thin power-law fluid flowing down a moving plane in a vertical direction by using the long-wave perturbation method. Linear and nonlinear stability are discussed. The linear stable region increases as the downward speed increases and the power-law index increases. More accurate results are obtained on the coefficients of the nonlinear generalized kinematic equation in the power-law part. The regions of sub-critical instability and absolute stability are expanded when the downward movement of plane increases, or the power-law index increases, and meanwhile the parts of supercritical stability and explosive supercritical instability are compressed. By substituting the power-law index and moving speed into the generalized nonlinear kinematic equation of the power-law film on the free surface, the results can be applied to estimate the stability of the thin film flow in the field of engineering.

Variable Step Block Hybrid Method for Stiff Chemical Kinetics Problems

Integration of a larger stiff system of initial value problems emerging from chemical kinetics models requires a method that is both efficient and accurate, with a large absolute stability region. To determine the solutions of the stiff chemical kinetics ordinary differential equations that help in explaining chemically reactive flows, a numerical integration methodology known as the 3-point variable step block hybrid method has been devised. An appropriate time step is automatically chosen to give accurate results. To check the efficiency of the new method, the numerical integration of a few renowned stiff chemical problems is evaluated such as Belousov–Zhabotinskii reaction and Hires, which are widely used in numerical studies. The results generated are then compared with the MATLAB stiff solver, ode15s.