14 results on '"Appadu, A. R."'
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2. Some Novel Numerical Schemes for 1-D Korteweg-de-Vries Burger's Equation.
- Author
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Appadu, A. R.
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KORTEWEG-de Vries equation , *BURGERS' equation , *FINITE difference method , *BOUNDARY value problems , *COLLOCATION methods - Abstract
In this paper, we devise three finite difference methods in order to solve a 1-D Korteweg-de-Vries Burger's (KdVB) equation with specified initial and boundary conditions. The test case is taken from [6] and the exact solution is known for this problem. We obtain the stability region and compare the performance of our three methods with the sinc-collocation method [6] by computing L∞, L2 errors at some different times and also obtain absolute errors and relative errors at some spatial nodes. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
3. Comparative Study of some Numerical Methods to Solve a 3D Advection-Diffusion Equation.
- Author
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Appadu, A. R., Djoko, J. K., and Gidey, H. H.
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FINITE difference method , *ADVECTION-diffusion equations , *BOUNDARY value problems , *CRANK-nicolson method , *FINITE differences - Abstract
In this work, three finite difference methods have been used to solve a three dimensional advection-diffusion equation with given initial and boundary conditions. The three methods are fourth order finite difference method, Crank-Nicolson and Implicit Chapeau Function. We compare the performance of the methods by computing L2 error, L∞ error and some performance indices such as mass distribution ratio (MDR), mass conservation ratio (MCR), total mass and R² which is a measure of total variation in particle distribution. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
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4. Comparative Study of Three Numerical Schemes for Contaminant Transport with Kinetic Langmuir Sorption.
- Author
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Appadu, A. R.
- Subjects
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LANGMUIR isotherms , *EULER equations , *FINITE differences , *COMPUTER simulation , *NUMERICAL analysis - Abstract
In this paper, we consider a contaminant transport model with Langmuir sorption under nonequilibrium conditions which is described by two coupled equations. Three numerical methods are used namely; Euler scheme forward in space, Euler scheme backward in space and nonstandard finite difference (NSFD) schemes. We compare the performance of the three methods by computing some errors: L1 error, dispersion and dissipation errors for both coarse and fine grids and short and long propagation times. We also perform a spectral analysis of the dispersive and dissipative properties of the three methods for the transport model. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
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5. Analysis of the Unconditionally Positive Finite Difference Scheme for Advection-Diffusion-Reaction Equations with Different Regimes.
- Author
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Appadu, A. R.
- Subjects
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FINITE difference method , *DIFFUSION , *ADVECTION , *ENERGY dissipation , *NUMERICAL analysis - Abstract
An unconditionally positive definite scheme has been derived in [1] to approximate a linear advection-diffusion-reaction equation which models exponential travelling waves and the coefficients of advective, diffusive and reactive terms have been chosen as one. The scheme has been baptised as Unconditionally Positive Finite Difference (UPFD). In this work, we use the UPFD scheme to solve the advection-diffusion-reaction problem in [1] and we also extend our study to three other important regimes involved in this model. The temporal step size is varied while fixing the spatial step size. We compute some errors namely; L1 error, dispersion, dissipation errors. We also study the variation of the modulus of the exact amplification factor, modulus of amplification factor of the scheme and relative phase error, all vs the phase angle for the four different regimes. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
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6. A Computational Study of Some Numerical Schemes for a Test Case with Steep Boundary Layers.
- Author
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Appadu, A. R., Djoko, J. K., and Gidey, H. H.
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TRANSPORT equation , *BOUNDARY layer (Aerodynamics) , *BOUNDARY value problems , *INITIAL value problems , *FINITE difference method , *ENERGY dissipation , *REYNOLDS number - Abstract
In this paper, three numerical methods have been used to solve a 1-D Convection-Diffusion equation with specified initial and boundary conditions. The methods used are the third order upwind scheme [1], fourth order upwind scheme [1] and a Non-Standard Finite Difference (NSFD) scheme [4]. The problem we considered has steep boundary layers near x = 1 [3] and this is a challenging test case as many schemes are plagued by nonphysical oscillation near steep boundaries. We compute the L2 and L∞ errors, dissipation and dispersion errors when the three numerical schemes are used and observe that the NSFD is much better than the other two schemes for both coarse and fine grids and also at low and high Reynolds numbers. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
7. Numerical Solution of the 1D Advection-Diffusion Equation Using Standard and Nonstandard Finite Difference Schemes.
- Author
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Appadu, A. R.
- Subjects
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NUMERICAL analysis , *ADVECTION-diffusion equations , *NONSTANDARD mathematical analysis , *FINITE differences , *SCHEMES (Algebraic geometry) , *NUMERICAL solutions to partial differential equations - Abstract
Three numerical methods have been used to solve the one-dimensional advection-diffusion equation with constant coefficients. This partial differential equation is dissipative but not dispersive. We consider the Lax-Wendroff scheme which is explicit, the Crank-Nicolson scheme which is implicit, and a nonstandard finite difference scheme (Mickens 1991). We solve a 1D numerical experiment with specified initial and boundary conditions, for which the exact solution is known using all these three schemes using some different values for the space and time step sizes denoted by h and k, respectively, for which the Reynolds number is 2 or 4. Some errors are computed, namely, the error rate with respect to the L1 norm, dispersion, and dissipation errors. We have both dissipative and dispersive errors, and this indicates that the methods generate artificial dispersion, though the partial differential considered is not dispersive. It is seen that the Lax-Wendroff and NSFD are quite good methods to approximate the 1D advectiondiffusion equation at some values of k and h. Two optimisation techniques are then implemented to find the optimal values of k when h = 0.02 for the Lax-Wendroff and NSFD schemes, and this is validated by numerical experiments. [ABSTRACT FROM AUTHOR]
- Published
- 2013
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8. Optimized Weighted Essentially Nonoscillatory Third-Order Schemes for Hyperbolic Conservation Laws.
- Author
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Appadu, A. R. and Peer, A. A. I.
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HYPERBOLIC functions , *CONSERVATION laws (Mathematics) , *SPECTRAL theory , *DISCRETIZATION methods , *APPROXIMATION theory , *MATHEMATICAL optimization , *TEMPORAL integration - Abstract
We describe briefly howa third-order Weighted Essentially Nonoscillatory (WENO) scheme is derived by coupling aWENO spatial discretization scheme with a temporal integration scheme. The scheme is termed WENO3. We perform a spectral analysis of its dispersive and dissipative properties when used to approximate the 1D linear advection equation and use a technique of optimisation to find the optimal cfl number of the scheme. We carry out some numerical experiments dealing with wave propagation based on the 1D linear advection and 1D Burger's equation at some different cfl numbers and show that the optimal cfl does indeed cause less dispersion, less dissipation, and lower L1 errors. Lastly, we test numerically the order of convergence of the WENO3 scheme. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
9. Time-Splitting Procedures for the Numerical Solution of the 2D Advection-Diffusion Equation.
- Author
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Appadu, A. R. and Gidey, H. H.
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ADVECTION-diffusion equations , *ENERGY dissipation , *PARTIAL differential equations , *BOUNDARY value problems , *ERROR rates , *MATHEMATICAL optimization - Abstract
We perform a spectral analysis of the dispersive and dissipative properties of two time-splitting procedures, namely, locally onedimensional (LOD) Lax-Wendroff and LOD (1, 5) [9] for the numerical solution of the 2D advection-diffusion equation. We solve a 2D numerical experiment described by an advection-diffusion partial differential equation with specified initial and boundary conditions for which the exact solution is known. Some errors are computed, namely, the error rate with respect to the L1 norm, dispersion and dissipation errors. Lastly, an optimization technique is implemented to find the optimal value of temporal step size that minimizes the dispersion error for both schemes when the spatial step is chosen as 0.025, and this is validated by numerical experiments. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
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10. The Technique of MIEELDLD in Computational Aeroacoustics.
- Author
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Appadu, A. R.
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COMPUTATIONAL fluid dynamics , *AEROACOUSTICS , *COMPUTER simulation , *ENERGY dissipation , *ADVECTION-diffusion equations , *PARAMETER estimation , *ERROR analysis in mathematics - Abstract
The numerical simulation of aeroacoustic phenomena requires high-order accurate numerical schemes with low dispersion and low dissipation errors. A technique has recently been devised in a Computational Fluid Dynamics framework which enables optimal parameters to be chosen so as to better control the grade and balance of dispersion and dissipation in numerical schemes (Appadu and Dauhoo, 2011; Appadu, 2012a; Appadu, 2012b; Appadu, 2012c). This technique has been baptised as the Minimized Integrated Exponential Error for Low Dispersion and Low Dissipation (MIEELDLD) and has successfully been applied to numerical schemes discretising the 1-D, 2-D, and 3-D advection equations. In this paper, we extend the technique of MIEELDLD to the field of computational aeroacoustics and have been able to construct high-order methodswith Low Dispersion and Low Dissipation properties which approximate the 1-D linear advection equation. Modifications to the spatial discretization schemes designed by Tam and Webb (1993), Lockard et al. (1995), Zingg et al. (1996), Zhuang and Chen (2002), and Bogey and Bailly (2004) have been obtained, and also a modification to the temporal scheme developed by Tam et al. (1993) has been obtained. These novel methods obtained using MIEELDLD have in general better dispersive properties as compared to the existing optimised methods. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
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11. On Semi-Classical Orthogonal Polynomials Associated with a Modified Sextic Freud-Type Weight.
- Author
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Kelil, Abey S. and Appadu, Appanah R.
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LINEAR differential equations , *DIFFERENTIAL-difference equations , *ORTHOGONAL polynomials , *POLYNOMIAL operators , *POLYNOMIALS - Abstract
Polynomials that are orthogonal with respect to a perturbation of the Freud weight function by some parameter, known to be modified Freudian orthogonal polynomials, are considered. In this contribution, we investigate certain properties of semi-classical modified Freud-type polynomials in which their corresponding semi-classical weight function is a more general deformation of the classical scaled sextic Freud weight | x | α exp (− c x 6) , c > 0 , α > − 1 . Certain characterizing properties of these polynomials such as moments, recurrence coefficients, holonomic equations that they satisfy, and certain non-linear differential-recurrence equations satisfied by the recurrence coefficients, using compatibility conditions for ladder operators for these orthogonal polynomials, are investigated. Differential-difference equations were also obtained via Shohat's quasi-orthogonality approach and also second-order linear ODEs (with rational coefficients) satisfied by these polynomials. Modified Freudian polynomials can also be obtained via Chihara's symmetrization process from the generalized Airy-type polynomials. The obtained linear differential equation plays an essential role in the electrostatic interpretation for the distribution of zeros of the corresponding Freudian polynomials. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
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12. On the numerical solution of Fisher's equation with coefficient of diffusion term much smaller than coefficient of reaction term.
- Author
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Agbavon, K. M., Appadu, A. R., and Khumalo, M.
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HEAT equation , *PARTIAL differential equations , *DIFFUSION coefficients , *ALGEBRAIC equations , *FINITE differences , *EXPONENTIAL functions - Abstract
Li et al. (SIAM J. Sci. Comput. 20:719–738, 1998) used the moving mesh partial differential equation (MMPDE) to solve a scaled Fisher's equation and the initial condition consisting of an exponential function. The results obtained are not accurate because MMPDE is based on a familiar arc-length or curvature monitor function. Qiu and Sloan (J. Comput. Phys. 146:726–746, 1998) constructed a suitable monitor function called modified monitor function and used it with the moving mesh differential algebraic equation (MMDAE) method to solve the same problem of scaled Fisher's equation and obtained better results. In this work, we use the forward in time central space (FTCS) scheme and the nonstandard finite difference (NSFD) scheme, and we find that the temporal step size must be very small to obtain accurate results. This causes the computational time to be long if the domain is large. We use two techniques to modify these two schemes either by introducing artificial viscosity or using the approach of Ruxun et al. (Int. J. Numer. Methods Fluids 31:523–533, 1999). These techniques are efficient and give accurate results with a larger temporal step size. We prove that these four methods are consistent for partial differential equations, and we also obtain the region of stability. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
13. An explicit nonstandard finite difference scheme for the FitzHugh–Nagumo equations.
- Author
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Chapwanya, M., Jejeniwa, O. A., Appadu, A. R., and Lubuma, J. M.-S.
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LINEAR differential equations , *ORDINARY differential equations , *NONLINEAR differential equations , *PARTIAL differential equations , *FINITE differences , *EQUATIONS - Abstract
In this work, we consider numerical solutions of the FitzHugh–Nagumo system of equations describing the propagation of electrical signals in nerve axons. The system consists of two coupled equations: a nonlinear partial differential equation and a linear ordinary differential equation. We begin with a review of the qualitative properties of the nonlinear space independent system of equations. The subequation approach is applied to derive dynamically consistent schemes for the submodels. This is followed by a consistent and systematic merging of the subschemes to give three explicit nonstandard finite difference schemes in the limit of fast extinction and slow recovery. A qualitative study of the schemes together with the error analysis is presented. Numerical simulations are given to support the theoretical results and verify the efficiency of the proposed schemes. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
14. On Certain Properties and Applications of the Perturbed Meixner–Pollaczek Weight.
- Author
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Kelil, Abey S., Jooste, Alta S., Appadu, Appanah R., Costabile, Francesco Aldo, Gualtieri, Maria I., and Napoli, Anna
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ORTHOGONAL polynomials , *QUANTUM wells , *FISHER information , *INTEGRAL representations , *POLYNOMIALS , *DIFFERENCE equations , *JACOBI polynomials , *GAUSSIAN quadrature formulas - Abstract
This paper deals with monic orthogonal polynomials orthogonal with a perturbation of classical Meixner–Pollaczek measure. These polynomials, called Perturbed Meixner–Pollaczek polynomials, are described by their weight function emanating from an exponential deformation of the classical Meixner–Pollaczek measure. In this contribution, we investigate certain properties such as moments of finite order, some new recursive relations, concise formulations, differential-recurrence relations, integral representation and some properties of the zeros (quasi-orthogonality, monotonicity and convexity of the extreme zeros) of the corresponding perturbed polynomials. Some auxiliary results for Meixner–Pollaczek polynomials are revisited. Some applications such as Fisher's information, Toda-type relations associated with these polynomials, Gauss–Meixner–Pollaczek quadrature as well as their role in quantum oscillators are also reproduced. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
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