Transylvanian Review

 Journal

 On The Numerical Exploration Of Burgers-Fisher Equation Using Robust Implicit-Explicit Schemes

 Owolabi, K.M. And Patidar, K.C.

 2016

Burgers equation has been used, time and again to model some of the physical phenomena encountered in science and technology such as in shock waves, acoustic transmission, traffic flow, supersonic flow and propagation of waves in thermoelastic medium. In this paper, we design, analyze some robust numerical integrators to solve Burgers-Fisher equation and its variants. We split the problems considered in this paper into linear and nonlinear parts. Implicit-explicit (IMEX) schemes involve the use of an explicit multistep method to advance the nonlinear part of the problem and an implicit scheme to advance the linear part. The use of existing time-stepping IMEX schemes coupled with central finite difference in space for the solution of generalized Burgers-Fisher equation is presented in this paper. Two important family of schemes considered are the IMEX linear multistep methods and IMEX predictor-corrector schemes, both obtained as a result of the transformation of general linear multistep schemes to suite the present work are used for the integration in time. Various patterns and feature are obtained when numerically simulating the popular Burgers-Fisher equation. We illustrate the tradeoff between the time step and accuracy, with a gold-standard run (computed with IMEXLM1, MEXLM2, IMEXPC1, IMEXPC2 and ?t = 10?4), the relative errors are displayed as a function of the time-step. In the numerical results, we observe that the convergence of the schemes is mostly influenced by the choice of the parameters because it is very difficult to decide which scheme has the best convergence. All the codes run faster under two seconds. Mathematical analysis and numerical methods based on IMEX schemes presented in this work on Burgers-Fisher equation can easily be extended to solve nonlinear reaction-diffusion equations in two or more dimensional space. 

 Advanced Modeling And Optimization

 Journal

 Linear Multistep Method Of Order-six For The Integration Linear And Nonlinear Initial Value Problems Of ODEs

 Owolabi, K.M.

 2016

In this work, a sixth-order linear multistep method (LMM) is constructed for the numerical integration of linear and nonlinear second order initial value problems of ordinary dierential equations. This method, which depend on certain algebraic parameters is developed following an extension of higher order derivatives and step length. The analysis of the basic properties of our method is examined and found to be zero-stable, symmetric and consistent. Error and step-length control is carried out by using Richardson extrapolation procedure. Extensive numerical results demonstrate increased accuracy with the same computational effort when compared with similar sixth and higher order formulas. 

 Chaos, Solitons And Fractals

 Journal

 Mathematical Analysis And Numerical Simulation Of Patterns In Fractional And Classical Reaction-diffusion Systems

 Owolabi, K.M.

 2016

The aim of this paper is to examine pattern formation in the sub—and super-diffusive scenarios and compare it with that of classical or standard diffusive processes in two-component fractional reaction- diffusion systems that modeled a predator-prey dynamics. The focus of the work concentrates on the use of two separate mathematical techniques, we formulate a Fourier spectral discretization method as an efficient alternative technique to solve fractional reaction-diffusion problems in higher-dimensional space, and later advance the resulting systems of ODEs in time with the adaptive exponential time-differencing solver. Obviously, the fractional Fourier approach is able to achieve spectral convergence up to machine precision regardless of the fractional order ?, owing to the fact that our approach is able to give full diagonal representation of the fractional operator. The complexity of the dynamics in this system is theo- retically discussed and graphically displayed with some examples and numerical simulations in one, two and three dimensions. 

 THE EUROPEAN PHYSICAL JOURNAL PLUS

 Journal

 Numerical Solution Of Fractional-in-space Nonlinear Schrodinger Equation With The Riesz Fractional Derivative

 Owolabi, K.M. And Atangana, A.

 2016

In this paper, dynamics of time-dependent fractional-in-space nonlinear Schr¨odinger equation with harmonic potential V (x), x ? R in one, two and three dimensions have been considered. We approximate the Riesz fractional derivative with the Fourier pseudo-spectral method and advance the resulting equation in time with both Strang splitting and exponential time-differencing methods. The Riesz derivative introduced in this paper is found to be so convenient to be applied in models that are connected with applied science, physics, and engineering. We must also report that the Riesz derivative introduced in this work will serve as a complementary operator to the commonly used Caputo or Riemann-Liouville derivatives in the higher-dimensional case. In the numerical experiments, one expects the travelling wave to evolve from such an initial function on an infinite computational domain (??,?), which we truncate at some large, but finite values L. It is important that the value of L is chosen large enough to give enough room for the wave function to propagate. We observe a different distribution of complex wave functions for the focusing and defocusing cases. 

 International Journal Of Bioinformatics And Biomedical Engineering

 Journal

 Robust Numerical Solution Of The Time-Dependent Problems With Blow-Up

 Owolabi, K.M.

 2015

Numerical solutions of nonlinear time-dependent partial differential equations with blow-up are considered in this paper. Such systems of PDEs are categorized into linear and nonlinear parts to allow the use of two classic mathematical ideas in space and time. The main focus in this paper is to discretized in space with higher order finite difference approximation and integrate the resulting nonlinear ordinary differential equations with an adaptive fourth-order exponential time differencing Runge-Kutta (ETDRK4) scheme. Stability analysis of the scheme is also examined. This paper is primarily concerned with the use of the ETDRK4 method to simulate some of the blow-up phenomena in nonlinear parabolic equations that are largely encountered in a number of physical situations, like chemical reaction-diffusion, electrical heating, fluid flow and population growth. It is expected that the time at which blow-up occurs will reflect in the numerical results. 

 International Journal Of Bioinformatics And Biomedical Engineering

 Journal

 Numerical Solution Of The Generalized Burgers-Huxley Equation By Exponential Time Differencing Scheme

 Owolabi, K.M.

 2015

Numerical solutions of nonlinear partial differential equations, such as the generalized and extended Burgers-Huxley equations which combine effects of advection, diffusion, dispersion and nonlinear transfer are considered in this paper. Such system can be divided into linear and nonlinear parts, which allow the use of two numerical approaches. Higher order finite difference schemes are employed for the spatial discretization, the resulting nonlinear system of ordinary differential equation is advanced with the modified fourth-order exponential time differencing Runge-Kutta (ETDRK4) method designed to generate the scheme with a smaller truncation error and better stability properties. The stability region of this scheme is shown and computed via its amplification factor. Numerical simulations with comparisons are presented to address any queries that may arise. 

 International Journal Of Statistics And Mathematics

 Journal

 Second Or Fourth-order Finite Difference Operators, Which One Is Most Effective

 Owolabi, K.M.

 2014

This paper presents higher-order finite difference (FD) formulas for the spatial approximation of the time-dependent reaction-diffusion problems with a clear justification through examples, “why fourth-order FD formula is preferred to its second-order counterpart” that has been widely used in literature. As a consequence, methods for the solution of initial and boundary value PDEs, such as the method of lines (MOL), is of broad interest in science and engineering. This procedure begins with discretizing the spatial derivatives in the PDE with algebraic approximations. The key idea of MOL is to replace the spatial derivatives in the PDE with the algebraic approximations. Once this procedure is done, the spatial derivatives are no longer stated explicitly in terms of the spatial independent variables. In other words, only one independent variable is remaining, the resulting semi-discrete problem has now become a system of coupled ordinary differential equations (ODEs) in time. Thus, we can apply any integration algorithm for the initial value ODEs to compute an approximate numerical solution to the PDE. Analysis of the basic properties of these schemes such as the order of accuracy, convergence, consistency, stability and symmetry are well examined. 

 International Journal Of Nonlinear Science And Numerical Simulations

 Journal

 Numerical Solution Of Singular Patterns In One-dimensional Gray-Scott-like Models

 Owolabi, K.M. And Patidar, K.C.

 2014

In this paper, numerical simulations of coupled one-dimensional Gray-Scott model for pulse splitting process, self-replicating patterns and unsteady oscillatory fronts associated with autocatalytic reaction-diffusion equations as well as homoclinic stripe patterns, selfreplicating pulse and other chaotic dynamics in Gierer- Meinhardt equations [12] are investigated. Our major approach is the use of higher order exponential time differencing Runge-Kutta (ETDRK) scheme that was earlier proposed by Cox and Matthews [5], which was later presented as a result of instability in a modified form by Krogstad [24] to solve stiff semi-linear problems. The semi-linear problems under consideration in this context are split into linear, which harbors the stiffest part of the dynamical system and nonlinear part that varies slowly than the linear part. For the spatial discretization, we employ higher-order symmetric finite difference scheme and solve the resulting system of ODEs with higher-order ETDRK method. Numerical examples are given to illustrate the accuracy and implementation of the methods, results and error comparisons with other standard schemes are well presented. 

 Missouri Journal Mathematical Sciences

 Journal

 A Symmetric K-step Method For Direct Integration Of Second Order Initial Value Problems Of Ordinary Differential Equations

 Owolabi, K.M.

 2013

The design and implementation analysis of a K-step lin- ear multistep method for direct integration of second order initial value problems of ordinary differential equations without reformu- lation into first order systems is discussed. The derivation of the method and analysis of its basic properties are adopted from the Taylor series expansion and Dahlquist stability model test methods. The result when examined with step-number k = 6 shows that the scheme is symmetric, consistent, zero-stable, and convergent. 

 Far East Journal Of Mathematical Sciences

 Journal

 A Contradictory Log Case Problem Of Homogeneous Second Order Ordinary Differential Equation, Power Series Approach

 Owolabi, K.M.

 2011

The power series method of solving homogeneous linear ordinary differential equations with analytic coefficients could and in practice be applied to various examples. To illustrate the different ways of surmounting the obstacles connected with the different forms which the roots of the indicial equation can take for the simple equations for which transformations are available, the power series method is comparatively tedious. But for the general equation with analytic coefficients, for which constant coefficient reducing transformation is not available, the power series method remains the most effective method of attack.