Joshua Sunday

• B.Sc., M.Sc., Ph.D.
• Professor at University of Jos

The derivation of numerical schemes for the solution of Lane-Emden equations requires meticulous consideration because they are highly nonlinear in nature; have singularity behaviors at the origin and in some cases, their exact solutions are known for only a few parameter ranges. This explains the reason for the failure of some existing methods in...

It is a known fact that in most cases, to integrate an oscillatory problem, higher order A-stable methods are often needed. This is because such problems are characterized by stiffness, chaos and damping, thus making them tedious to solve. However, in this research, an accuracy-preserving relatively lower order Block Hybrid Algorithm (BHA) is propo...

Second-order oscillatory problems have been found to be applicable in studying various phenomena in science and engineering; this is because these problems have the capabilities of replicating different aspects of the real world. In this research, a new hybrid method shall be formulated for the simulations of second-order oscillatory problems with...

In this paper, Fredholm integro-differential equations are solved using the derivative of the Lucas polynomials in matrix form. The equation is first transformed into systems of nonlinear algebraic equations using the Lucas polynomials. The unknown parameters required for approximating the solution of Fredholm integro-differential equations are the...

This study focuses on the formulation of optimised variable step-size algorithm (OVSSA) for the solution of dynamical systems. In order to minimise the number of iterations and number of steps taken as well as improve the accuracy of the proposed algorithm, an embedded strategy was deployed in formulating the method using variable stepsize mode. O...

In this paper, we present a ninth–order block hybrid method for the numerical solution of stiff and non–stiff systems of first–order differential equations. The method is based on an interpolation and collocation approach which results in a single continuous formulation; from which eight discrete schemes that make the block method were obtained. A...

Fredholm integro-differential equations play a crucial role in mathematical modelling across various disciplines, including physics, biology, and finance. In this paper, Fredholm integro-differential equations are solved using the derivative of the Lucas polynomials in matrix form. The equation is first transformed into systems of nonlinear algebra...

The research introduces a new four-point integrator (FPI) for solving second-order differential equations characterized by oscillatory behavior. The proposed FPI is developed through a continuous scheme within a linear multistep framework, utilizing two off-step points to enhance efficiency. Unlike conventional methods that reduce higher-order equa...

The research paper formulates a variable step block backward differentiation formula (VSBBDF) for solving nonlinear fuzzy differential equations (FDEs). Developed to address uncertainties within differential equations by using fuzzy environments, VSBBDF offers a flexible approach to solve equations with triangular fuzzy numbers. This method incorpo...

Attracted by the importance of ordinary differential equations in many physical situations like, engineering, business and health care in particular, an effective and successful numerical algorithm is needed in order to explain many of the ambiguities about the phenomena in many fields of human endeavor. In this study, an interpolation and collocat...

Third-order problems have been found to model real-life phenomena such as thick film fluid flow, boundary layer problems, and nonlinear Genesio problems, to name a few. This research focuses on the development of an algorithm using a basis function that combines an exponential function with a power series. This algorithm, called the Exponential Fun...

In this article, we consider Lucas series for = 2, 5, 6 in numerical solutions of linear and nonlinear Volterra integral equations of the second kind. The series is used to transform the equation into a system of nonlinear algebraic equations, and the unknown parameters are determined. The application of this method has shown that the Lucas series...

In this paper, we compare the performances of two Butcher-based block hybrid methods for the numerical integration of initial value problems. We compare the condition numbers of the linear system of equations arising from both methods and the absolute errors of the solution obtained. The results of the numerical experiments illustrate that the bett...

A major challenge in simulating chemical reaction processes is integrating the stiff systems of Ordinary Differential Equations (ODEs) describing the chemical reactions due to stiffness. Thus, it would be of interest to search systematically for stiff solvers that are close to optimal for such problems. This paper presents an implicit 3-Point Block...

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 sol...

Over the years, researches have shown that fixed (constant) step size methods have been efficient in integrating 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 accuracy to be attained. This is because such systems have solution...

In this article, a variable step size strategy is adopted in formulating a new variable step hybrid block method (VSHBM) for the solution of the Kepler problem, which is known to be a rigid and stiff differential equation. To derive the VSHBM, the step size ratio r is left the same, halved, or doubled in order to optimize the total number of steps,...

In this research article, a pair of optimized two-step second derivative methods is derived and implemented on stiff systems. The influence of equidistant and non-equidistant hybrid points spacing on the performance of the methods derived is investigated. Firstly, the methods are derived using interpolation and collocation of a finite power series...

Over the years, the systematic search for stiff model solvers that are near-optimal has attracted the attention of many researchers. An attempt has been made in this research to formulate an implicit Four-Point Hybrid Block Integrator (FPHBI) for the simulations of some renowned rigid stiff models. The integrator is formulated by using the Lagrange...

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...

Multistep integration methods are being extensively used in the simulations of high dimensional systems due to their lower computational cost. The block methods were developed with the intent of obtaining numerical results on numerous points at a time and improving computational efficiency. Hybrid block methods for instance are specifically used in...

In this research paper, a pair of three-step hybrid block methods is derived for the solutions of linear and nonlinear first-order systems. The derivation is carried out with the aid of collocation and interpolation technique and the adoption of power series as basis function. The first and second three-step hybrid block methods are derived by inco...

The Duffing oscillator (damped or undamped) is one of the most significant and classical nonlinear ordinary differential equations in view of its diverse applications in science and engineering. The Duffing oscillator is an equation that has a cubic stiffness term regardless of the type of damping or excitation. Over the years, different methods ha...

In this paper, a collocation approach for solving initial value problem of ordinary differential equations (ODEs) of the first order is presented. This approach consists of reducing the problem to a set of linear multi-step algebraic equations by approximating the ODE with a shifted Legendre polynomial basis function to determine the unknown consta...

In this paper, an alternative method shall be presented for the approximation of periodic Abel's Differential Equation (ADE) of the first kind. The periodic ADE that shall be considered here are those that do not have a closed form (exact) solution (even though the solution of such equations is known to exist). First, the Theorems of shall be emplo...

In this paper, a matrix approach to the direct computation method for solving Fredholm integro-differential equations (FIDEs) of the second kind with degenerate kernels is presented. Our approach consists of reducing the problem to a set of linear algebraic equations by approximating the kernel with a finite sum of products and determining the unkn...

In this paper, a one-step hybrid block method is formulated for the numerical approximation of Fuzzy Differential Equations (FDEs). In deriving the method, the techniques of interpolation and collocation were adopted using the sum of the first seven terms of Legendre polynomial as basis function. The convergence and stability properties of the meth...

A reformulated Non-Standard Finite Difference Method (NSFDM) for the solution of Autonomous Dynamical Differential Equations (ADDEs) is proposed in this research. To effectively carry out the reformulation, three major steps were adopted. These include the reconstruction of the numerator function, the reconstruction of the denominator function and...

In this paper, an efficient hybrid algorithm shall be formulated for the computation of second-order Fredholm integro-differential equations. In developing the algorithm using the method of interpolation and collocation, power series was adopted as the basis function with the integration carried out within a one-step interval. The algorithm derived...

Autonomous Dynamical Differential Equations (ADDEs) with one, two and three fixed points have been found to be applicable in various fields of human endeavor. There is therefore need to find approximate solutions to such differential equations since some of them do not have solutions in closed form. In view of this, we are motivated to develop an a...

In this paper, a one-step algorithm is derived and implemented for third order oscillatory problems. The derivation is carried out using the procedure of collocation and interpolation of power series basis function within a one-step integration interval   1 ,  n n x x .The paper also analyzed some basic properties of the algorithm derived. The r...

A special form of Non-Standard Finite Difference Method (NSFDM) called the Exact Finite Difference Scheme (EFDS) for the computation of second-order Fredholm Integro-differential equation shall be constructed in this research. In carrying out the construction of the method, it was assumed that at any point within the interval of integration, the ap...

An A-stable backward difference second derivative linear multistep method for solving stiff ordinary differential equation via multi-step interpolation and collocation methods has be studied in this research. In deriving the methods, we use the power series as a basis function to obtain the main continuous and discrete. The analysis of the basic pr...

In this paper, a new Non-Standard Finite Difference Method (NSFDM) is constructed for the computation of a special class of nonlinear differential equations called the Duffing oscillators. The nonstandard method is formulated by approximating the nonlinear terms nonlocally in the Duffing oscillator and also by reconstructing the denominator functio...

One of the sets of differential equations that find applications in real life is the oscillatory problems. In this paper, the oscillation criteria and computation of third order oscillatory differential equations are studied. The conditions for a third order differential equation to have oscillatory solutions on the interval I = [t 0 , ∞) shall be...

One of the most efficient ways to model the propagation of epistemic uncertainties (in dynamical environments/systems) encountered in applied sciences, engineering and even social sciences is to employ Fuzzy Differential Equations (FDEs). The FDEs are special type of Interval Differential Equations (IDEs). The IDEs are differential equations used t...

In this research work, the development of two-step implicit second derivative block methods for the solution of initial value problems of general second order ordinary differential equations is studied. In the derivation of the method, power series is adopted as basis function to obtain the main continuous scheme through collocation and interpolati...

In this paper, we propose an implicit two-step third derivative hybrid block method with one off-step point for the direct solution of second order Ordinary Differential Equations. We adopted the method of interpolation and collocation of power series approximate solution to generate the continuous hybrid linear multistep method, which was evaluate...

A Hybrid Backward Differentiation Formula (HBDF) of uniform order ten is proposed for the solution of second order stiff Initial Value Problems (IVPs) is studied in this article. The approach adopted for the derivation of backward differentiation formulae involves interpolation and collocation at appropriate selected points. The proposed order ten...

The concept of systems theory has been applied in various disciplines to analyze systems in such disciplines. In this research, systems theory was employed to model, analyze and study the natures of some problems in mass-spring systems. Mass-spring systems are second order linear differential equations that have variety of applications in science a...

A first derivative method for the integration of first-order oscillatory problems is derived in this research. The integration is carried out within a two-step interval. The method of collocation and interpolation of power series basis function was adopted in deriving the method. The method derived was tested on some sampled oscillatory differentia...

In this research, an efficient two-step algorithm is derived for the solution of nonlinear first order differential equations. The derivation is carried out with the aid of collocation and interpolation of power series basis function. The reliability and applicability of the two-step algorithm derived was established by solving some nonlinear diffe...

In this research, an efficient one-third-step algorithm is derived for the solution of nonlinear first order differential equations. Derivation is carried out with the aid of collocation and interpolation of power series basis function. Reliability and applicability of the one third-step algorithm derived was established by solving some nonlinear d...

Damping is an influence within or upon an oscillatory system that has the effect of reducing, restricting or preventing its oscillations. A one-step sixth-order computational method is proposed in this paper for the solution of second order free undamped and free damped motions in mass-spring systems. The method of interpolation and collocation of...

Description of circuits using differential equations is very convenient for the electrical circuits' behavioral analysis. In this paper, a one-step fifth-order computational method is proposed for the solution of second order differential equations using the Hermite polynomial as a basis function. The computational method was then applied on two re...

In this paper, we present the derivation and implementation of a new quarter-step computational hybrid block method for first-order modeled differential equations. The block method was developed using Laguerre polynomial of degree five as our basis function via interpolation and collocation techniques. We went further to apply the quater-step metho...

In this paper, we develop a block method using Chebyshev polynomial basis function and use it to produce discrete methods which are simultaneously applied as numerical integrators by assembling them into a block method. The paper further investigates the properties of the block method and found it to be zero-stable, consistent and convergent. We al...

This paper presents a new approach to constructing Non-Standard Finite Difference Method (NSFDM) for the solution of autonomous Ordinary Differential Equations (ODEs). The need for this method came up due to the shortcomings of the standard methods; in which the qualitative properties of the exact solutions are not usually transferred to the numeri...

We develop a two-step hybrid block method for the solution of stiff and oscillatory first-order Ordinary Differential Equations (ODEs) using the Laguerre polynomial as our basis function via interpolation and colloca-tion techniques. The paper further investigates the basic properties of the method and found it to be zero-stable, consistent and con...

This paper considered the development of numerical integrator for the solution of second order initial value problems. The method was derived through the interpolation and collocation of the basis polynomial which is combination of power series and exponential function to derive a continuous linear multistep method. The method was implemented in bl...

This paper presents the derivation and implementation of a block integrator for the solution of stiff and oscillatory first-order initial value problems of Ordinary Differential Equations (ODEs). The integrator was derived by collocation and interpolation of the combination of power series and exponential function to generate a continuous implicit...

This paper examines the derivation and implementation of a self-starting four-step fifth order block integrator for direct integration of stiff and oscillatory first-order ordinary differential equations using interpolation and collocation procedures. The method was developed by collocation and interpolation of the combination of power series and e...

This paper considers the development of one step four, hybrid block method for the solution of first order initial value problems of ordinary differential equations. The method was developed by collocation and interpolation of power series approximate solution to generate a continuous implicit linear multistep method. Both the predictor and correct...

In this paper, we consider the development of an extended block integrator for the solution of stiff and oscillatory first-order Ordinary Differential Equations (ODEs) using interpolation and collocation techniques. The integrator was developed by collocation and interpolation of the combination of power series and exponential function to generate...

In this paper, we present a continuous block integrator for direct integration of stiff and oscillatory first-order ordinary differential equations using interpolation and collocation techniques. The approximate solution used in the derivation is a combination of power series and exponential function. The paper further investigates the properties o...

We proposed a continuous blocks method for the solution of second order initial value problems with constant step size in this paper. The method was developed by interpolation and collocation of power series approximate solution to generate a continuous linear multistep method; this is evaluated for the independent solution to give a continuous blo...

In this paper, we developed a new continuous block method using the approach of collocation of the differential system and interpolation of the power series approximate solution. A constant step length within a half step interval of integration was adopted. We evaluated at grid and off grid points to get a continuous linear multistep method. The co...

We construct a self-starting Simpson's type block hybrid method (BHM) consisting of very closely accurate members each of order p=q+2 as a block. The higher order members of each were obtained by increasing the number k in the multi-step collocation (MC) used to derive the k-step continuous formula (5k6) through the aid of MAPLE software program....

This research paper examines a computational approach to solving the Verhulst–Pearl Model (often called the Logistic Model). The paper also examines the applications of such a model in some related fields. The aim of this research paper is to put forward the integrator derived by Odekunle, Adesanya and Sunday (2012B) as an alternative tool for solv...

We know that for any numerical method to be efficient and computational reliable, it must be convergent, consistent, and stable. This paper adopted the method of interpolation of the approximate solution and collocation ofits differential system at grid and off grid points to yield a continuous linear multistep method with a constant step size. The...

This research paper presents the development, analysis, and implementation of a new numerical integrator capable of solving first order initial value problems in ordinary differential equations. The algorithm developed is based on a local representation of the theoretical solution () y x to the initial value problem by a nonlinear interpolating fun...

This research paper examines the derivation and implementation of a new 4-point block method for direct integration of first-order ordinary differential equations using interpolation and collocation techniques. The approximate solution is a combination of power series and exponential function. The paper further investigates the properties of the ne...

We develop a continuous linear multistep method using interpolation and collocation of the approximate solution for the solution of first order ordinary differential equation with a constant step-size. The approximate solution is combination of power series and exponential function. The independent solution was then derived by adopting block integr...

For any numerical integrator to be efficient, ingenious and computationally reliable, it is expected that it be convergent, consistent and stable. In this paper, we develop a new numerical integrator which is particularly well suited for solving initial value problems in ordinary differential equations. The algorithm developed is based on a local r...

This research paper examines the dynamical model of love as in the case of Romeo and Juliet. The model is in the form of Coupled Ordinary Differential Equation (ODE) describing the time-dependent variation of the love and hate displayed by individuals in a relationship. An additional work is done by applying a new numerical integrator developed by...