Y Skwame’s research while affiliated with Adamawa State University and other places

This paper presents a novel block hybrid method designed to improve accuracy and efficiency in solving differential equations, with specific applications in epidemiological and biological models. The new method were derived using a power series polynomial via interpolation and collocation procedure. The basic properties of the new method was analyzed numerically and it is obvious that the method is of uniform order nine, consistency, zero-stability, convergent. We further obtain the absolute stability through the stability polynomial showing to be A-stable. Numerical experiments demonstrate the method’s effectiveness, with smaller absolute errors than existing methods across various models, including the SIR model and growth models in population dynamics. The results affirm the potential of the new method for high-precision applications in epidemiology, biology and related fields, marking an advancement in differential equation modeling.

Real-world problems, particularly in the sciences and engineering, are often analyzed using differential equations to understand physical phenomena. Many situations involve rates of change of independent variables, represented by derivatives, which lead to differential equations. Solving higher-order ordinary differential equations typically involves reducing them to systems of first-order equations, but this approach has challenges. To overcome these and enhance numerical methods, a novel one-step block method with eight partitions was developed for the direct solution of higher-order initial value problems. This method will target issues in physics, biology, chemistry and economics. The new method was formulated using the linear block approach and numerical analysis was ensure essential and sufficient conditions. The new method addresses second-order problems like simple harmonic motion, third-order issues such as oscillatory differential equations, and fourth-order problems like thin film dynamics. The new method demonstrates faster convergence and improved accuracy compared to existing solutions for second, third, and fourth-order oscillatory differential equations.

The numerical application of third derivative on third order initial value problem of ordinary differential equations is consider in this paper. The method is derived by collocating and interpolating the approximate solution in power series, while Taylor series is used to generate the independent solution at selected grid and off grid points. The basic analysis of the method were established and it was found to be consistent, zero-stable and convergent. The developed method is then applied to solve some third order initial value problems of ODEs, and the result computed shows that the derived method is more accurate than some existing methods considered in this paper. We further plotted the solution graph of each problems and it is obvious that the numerical solution convergence toward the exact solution. Introduction Many problems in Physics, Chemistry, and Engineering science are demonstrated mathematically by third-order boundary value problems or initial value problems. These boundary value problems can be found in different areas of applied mathematics and physics as, in the deflection of a curved beam having a constant or varying cross section, a three-layer beam, electromagnetic waves, or gravity driven flows. Third-order boundary value problems were discussed in many papers in recent years Ejaz and Mustafa [1] , Areo and Omojola [2] , Adeyeye and Omar [3] , Sunday [4] , Fsasis [5] , Omar and Adeyeye [6] , Awoyemi, Kayode and Adeghe [7] , etc. Ordinary differential equations are widely applied to model real life situations particularly involving engineering problems. The third order initial value problems of ordinary differential equations of the form

Our focus in this research is to developed block method for solving higher order ordinary differential equation using power series on implicit one-step. In order to achieve the aim and objective of this research, we used interpolation, collocation and evaluate a power series approximation at some chosen grid and off-grid points to generate an implicit continuous hybrid one-step method. As requirement of any numerical analyst, the properties of one-step block method was done and results showed that it is consistent, convergent, zero stable and with region of absolutely stable. The method was tested with numerical examples solved using the existing methods and our method was found to give better results when compared with the existing method. Obviously, the solution graphs show the convergence of the method with exact solutions.

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.

This research examines the general step k − 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. Keywords: Accuracy; computational burden; higher order IVPs; linear block approach.

The introduction of new linear block method for the direct simulation of fourth order IVPs has been developed in this article. The reason for adopting direct simulation of fourth order initial value problems is to address some setbacks in reduction method. When developing the method, we adopted the linear block approach through a one step method. We have validated the accuracy of the method on some fourth order initial value problems without reduction process, and the results are better than the conventional method. The numerical experiments were given and the results obtained were found to be better in accuracy than the existing methods in literature.

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 polynomial as basis function. The properties of the integrator which include order, consistency, and convergence were analyzed. Further analysis showed that the proposed integrator has an A-stability region. The A-stability nature of the integrator makes it more robust and fitted for the simulation of stiff models. To test the computational reliability of the new integrator, few well-known technical stiff models such as the pharmacokinetics, Robertson and Van der Pol models were solved. The results generated were then compared with those of some existing methods including the MATLAB solid solvent, ode 15s. From the results generated, the new implicit FPHBI performed better than the ones with which we compared our results with.

In this article, the direct simulation of third order linear problems on single step block method has been proposed. In order to overcoming the setbacks in reduction method, direct method has been proposed using power series to reduce computational burden that occur in the reduction method. Numerical properties for the block method are established and the method developed is consistent, convergent and zero-stable. To validate the accuracy of the block method, certain numerical test problems were considered, the results shown that the accuracy of our method are more accurate over the existing method in literature.

The numerical applications of implicit second derivative on second order initial value problems of ordinary differential equations was studied in this paper. Special attention was given when deriving the method via interpolation and collocation procedure as a basic function, using the power series method. The analysis of the method was examined, and it was found to be consistent, zero-stable and convergent. The new method was tested on some real-life problems without reduction to the equivalents system of first order ODEs. And our method performs better when compared with the existing methods in terms of accuracy. We further sketched the solution graph of our method and it is obvious that the new method convergence toward the exact solution.