##### Construction of Functions for Fractional Derivatives using Matlab

Olagunju Adeyemi Sunday, Joseph Folake Lois

Journal of Advances in Mathematics and Computer Science, Page 1-10
DOI: 10.9734/jamcs/2021/v36i630368

MATLAB is a high level programming tool for technical computing, its application cuts across different sphere of science, engineering, finance, communication, music etc. With the current increase in the use of non-integer order derivatives, there is a need to have tools that handle them for effective applications. In this paper, we present a brief comparative review of 2 expressions of fractional derivative. MATLAB functions for approximating Riemann-Liouville and Caputo fractional derivatives are presented alongside. Numerical simulations with test examples are implemented and results compared. To effectively handle non-polynomial function, Taylor series expansion is employed to convert the function into a form that can be easily handled.

##### Numerical Approximations of ODEs Initial Value Problem; A Case Study of Gluconic Acid Fermentation by Pseudomonas ovalis

N. A. Ihoeghian, B. O. John

Journal of Advances in Mathematics and Computer Science, Page 11-23
DOI: 10.9734/jamcs/2021/v36i630369

Across different sections of life, physical and chemical sciences, differential equations which could be ordinary differential equations (ODEs) or partial differential equations (PDEs) are used to model the various systems as observed. Some types of ODEs, and a few PDEs are solvable by analytical methods with much difficulties. However, the great majority of ODEs, especially the non-linear ones and those that involve large sets of simultaneous differential equations, do not have analytical solutions but require the application of numerical techniques.  This work focused on exemplifying numerical approximations (Adams-Bashforth-Moulton, Bogacki-Shampine, Euler) of ODEs Initial value Problem in its simplest approach using a case study of gluconic acid frementation by Psuedonomas Ovalis. The performance of the methods was checked by comparing their accuracy.  The accuracy was detrermined by the size of the discretization error estimated from the difference between analytical solution and numerical approximations. The results obtained are in good agreement with the exact solution. This work affirms that numerical methods give approximate solutions with less rigorous work and time as there is room for flexibility in terms of using different step sizes with the Euler solver as most accurate.

##### Stability Analysis of Perturbed Linear Non-integer Differential Systems

Ubong D. Akpan

Journal of Advances in Mathematics and Computer Science, Page 24-29
DOI: 10.9734/jamcs/2021/v36i630370

In this work, the effect of perturbation on linear fractional differential system is studied. The analysis is done using Riemann-Liouville derivative and the conclusion extended to using Caputo derivative since the result is similar. Conditions for determining the stability and asymptotic stability of perturbed linear fractional differential system are given.

##### Generalized Fibonacci Numbers with Indices in Arithmetic Progression and Sum of Their Squares: The Sum Formula ∑nk=0 xkW2mk+j

Y¨uksel Soykan

Journal of Advances in Mathematics and Computer Science, Page 30-62
DOI: 10.9734/jamcs/2021/v36i630371

In this paper, closed forms of the sum formulas ∑n k=0 xkWmk 2 +j for generalized Fibonacci numbers are presented. As special cases, we give sum formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas numbers. We present the proofs to indicate how these formulas, in general, were discovered. Of course, all the listed formulas may be proved by induction, but that method of proof gives no clue about their discovery.

##### On the Analysis of Damped Gyroscopic Systems Using Lyapunov Direct Method

Ubong D. Akpan

Journal of Advances in Mathematics and Computer Science, Page 63-74
DOI: 10.9734/jamcs/2021/v36i630372

In this work, the stability properties of damped gyroscopic systems have been studied using Lyapunov direct method. These systems are generally stable because of the presence of gyroscopic effect. Conditions for determining the stability of the damped gyroscopic systems have been developed. Solution bounds of amplitude and velocity have been obtained for both homogeneous and inhomogeneous cases. An example is given to show how the stability conditions are applied to systems to determine its stability status.

##### Exact Solution of Space-Time Fractional Partial Differential Equations by Adomian Decomposition Method

Vidya N. Bhadgaonkar, Bhausaheb R. Sontakke

Journal of Advances in Mathematics and Computer Science, Page 75-87
DOI: 10.9734/jamcs/2021/v36i630373

The intention behind this paper is to achieve exact solution of one dimensional nonlinear fractional partial differential equation(NFPDE) by using Adomian decomposition method(ADM) with suitable initial value. These equations arise in gas dynamic model and heat conduction model. The results show that ADM is powerful, straightforward and relevant to solve NFPDE. To represent usefulness of present technique, solutions of some differential equations in physical models and their graphical representation are done by MATLAB software.

##### Common Fixed Point Theorems for Compatible Maps in Generalized Fuzzy Metric Spaces

M. Jeyaraman, S. Sowndrarajan, A. Ramachandran

Journal of Advances in Mathematics and Computer Science, Page 88-96
DOI: 10.9734/jamcs/2021/v36i630374

In this paper, we consider generalized fuzzy metric spaces and provide existence and uniqueness fixed point results. First, we use compatible maps of type (β) to prove fixed point results, then we introduce weakly compatible maps to approximate common fixed point results by using an implicit relation.

##### Ultimate Boundedness of Discrete-Time Uncertain Neural Networks with Leakage and Time-Varying Delays

Jiajun Hua, Danhua He

Journal of Advances in Mathematics and Computer Science, Page 97-109
DOI: 10.9734/jamcs/2021/v36i630377

In this paper, by using the general discrete Halanay inequalities, the techniques of inequalities and some other properties, we study the ultimate boundedness of a class of the discrete-time uncertain neural network systems and obtain several sufficient conditions to ensure the ultimate boundedness of discrete-time uncertain neural networks with leakage and time-varying delays. Finally numerical examples are given to verify the correctness of the conclusion.