100 research outputs found
Accurate Spectral Algorithms for Solving Variable-order Fractional Percolation Equations
A high accurate spectral algorithm for one-dimensional variable-order fractional percolation equations (VO-FPEs) is considered.We propose a shifted Legendre Gauss-Lobatto collocation (SL-GLC) method in conjunction with shifted Chebyshev Gauss-Radau collocation (SC-GR-C) method to solve the proposed problem. Firstly, the solution and its space fractional derivatives are expanded as shifted Legendre polynomials series. Then, we determine the expansion coefficients by reducing the VO-FPEs and its conditions to a system of ordinary differential equations (SODEs) in time. The numerical approximation of SODEs is achieved by means of the SC-GR-C method. The under-studyās problem subjected to the Dirichlet or non-local boundary conditions is presented and compared with the results in literature, which reveals wonderful results
Space-Time Spectral Collocation Algorithm for the Variable-Order Galilei Invariant Advection Diffusion Equations with a Nonlinear Source Term
This paper presents a space-time spectral collocation technique for solving the variable-order Galilei invariant advection diffusion equation with a nonlinear source term (VO-NGIADE). We develop a collocation scheme to approximate VONGIADE by means of the shifted Jacobi-Gauss-Lobatto collocation (SJ-GL-C) and shifted Jacobi-Gauss-Radau collocation (SJ-GR-C) methods. We successfully extend the proposed technique to solve the two-dimensional space VO-NGIADE. The discussed numerical tests illustrate the capability and high accuracy of the proposed methodologies
Computational and numerical analysis of differential equations using spectral based collocation method.
Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.In this thesis, we develop accurate and computationally eļ¬cient spectral collocation-based methods,
both modiļ¬ed and new, and apply them to solve diļ¬erential equations. Spectral collocation-based
methods are the most commonly used methods for approximating smooth solutions of diļ¬erential
equations deļ¬ned over simple geometries. Procedurally, these methods entail transforming the gov
erning diļ¬erential equation(s) into a system of linear algebraic equations that can be solved directly.
Owing to the complexity of expanding the numerical algorithms to higher dimensions, as reported
in the literature, researchers often transform their models to reduce the number of variables or
narrow them down to problems with fewer dimensions. Such a process is accomplished by making
a series of assumptions that limit the scope of the study. To address this deļ¬ciency, the present
study explores the development of numerical algorithms for solving ordinary and partial diļ¬erential
equations deļ¬ned over simple geometries. The solutions of the diļ¬erential equations considered are
approximated using interpolating polynomials that satisfy the given diļ¬erential equation at se
lected distinct collocation points preferably the Chebyshev-Gauss-Lobatto points. The size of the
computational domain is particularly emphasized as it plays a key role in determining the number
of grid points that are used; a feature that dictates the accuracy and the computational expense of
the spectral method. To solve diļ¬erential equations deļ¬ned on large computational domains much
eļ¬ort is devoted to the development and application of new multidomain approaches, based on
decomposing large spatial domain(s) into a sequence of overlapping subintervals and a large time
interval into equal non-overlapping subintervals. The rigorous analysis of the numerical results con
ļ¬rms the superiority of these multiple domain techniques in terms of accuracy and computational
eļ¬ciency over the single domain approach when applied to problems deļ¬ned over large domains.
The structure of the thesis indicates a smooth sequence of constructing spectral collocation method
algorithms for problems across diļ¬erent dimensions. The process of switching between dimensions
is explained by presenting the work in chronological order from a simple one-dimensional problem
to more complex higher-dimensional problems. The preliminary chapter explores solutions of or
dinary diļ¬erential equations. Subsequent chapters then build on solutions to partial diļ¬erential
i
equations in order of increasing computational complexity. The transition between intermediate
dimensions is demonstrated and reinforced while highlighting the computational complexities in
volved. Discussions of the numerical methods terminate with development and application of a
new method namely; the trivariate spectral collocation method for solving two-dimensional initial
boundary value problems. Finally, the new error bound theorems on polynomial interpolation are
presented with rigorous proofs in each chapter to benchmark the adoption of the diļ¬erent numerical
algorithms. The numerical results of the study conļ¬rm that incorporating domain decomposition
techniques in spectral collocation methods work eļ¬ectively for all dimensions, as we report highly
accurate results obtained in a computationally eļ¬cient manner for problems deļ¬ned on large do
mains. The ļ¬ndings of this study thus lay a solid foundation to overcome major challenges that
numerical analysts might encounter
Spatio-temporal dynamics in pipe flow
When fluid flows through a channel, pipe or duct, there are two basic forms of motion:
smooth laminar flow and disordered turbulent motion. The transition between these two
states is a fundamental and open problem which has been studied for over 125 years. What
has received far less attention are the intermittent dynamics which possess qualities of
both turbulent and laminar regimes. The purpose of this thesis is therefore to investigate
large-scale intermittent states through extensive numerical simulations in the hopes of further
understanding the transition to turbulence in pipe flow.
We begin by reviewing the spectral-element code Semtex which is used to perform the
simulations. We discuss modifications to this code to impose a constant flowrate to the flow
through a pipe and to improve the computational efficiency on certain multicore architectures.
We then move on to examine the reverse transition from turbulence to laminar flow in a long,
125 diameter periodic pipe, which unlike the forward transition does not depend on finiteamplitude
perturbations to the flow and thus captures the natural dynamics contained within
the transition. The Reynolds number Re is reduced from Re = 2,800 to Re = 2,250 over
a long timescale, and by investigating the resultant spatio-temporal dynamics we discover
that the transition can be characterised by three fundamentally different states separated by
two Reynolds numbers. Below Rec <= 2,300, turbulence takes the form of equilibrium puffs
which eventually decay. Above Rei = 2,600, flow remains uniformly turbulent throughout
the domain. Between these two values, the dynamics are an intermitent mixture of both
turbulent and laminar regimes which take the form of unsteady alternating laminar-turbulent
bands.
Finally, we concentrate on finding a more exact value for Rec, which marks the onset
of sustained turbulence in pipe flow. We examine the process through which isolated
turbulent puffs split and find that, like decay, this process is stochastic and memoryless.
By drawing comparisons with other simple stochastically driven systems ā in particular,
directed percolation ā we compare the timescales for decay and splitting, and ascertain that
Rec = 2,040 +- 10
Stability analysis of a second-order difference scheme for the time-fractional mixed sub-diffusion and diffusion-wave equation
This study investigates a class of initial-boundary value problems pertaining
to the time-fractional mixed sub-diffusion and diffusion-wave equation (SDDWE).
To facilitate the development of a numerical method and analysis, the original
problem is transformed into a new integro-differential model which includes the
Caputo derivatives and the Riemann-Liouville fractional integrals with orders
belonging to (0,1). By providing an a priori estimate of the solution, we have
established the existence and uniqueness of a numerical solution for the
problem. We propose a second-order method to approximate the fractional
Riemann-Liouville integral and employ an L2 type formula to approximate the
Caputo derivative. This results in a method with a temporal accuracy of
second-order for approximating the considered model. The proof of the
unconditional stability of the proposed difference scheme is established.
Moreover, we demonstrate the proposed method's potential to construct and
analyze a second-order L2-type numerical scheme for a broader class of the
time-fractional mixed SDDWEs with multi-term time-fractional derivatives.
Numerical results are presented to assess the accuracy of the method and
validate the theoretical findings
Numerical approximations of fractional differential equations: a Chebyshev pseudo-spectral approach.
Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.This study lies at the interface of fractional calculus and numerical methods. Recent
studies suggest that fractional differential and integral operators are well suited to model
physical phenomena with intrinsic memory retention and anomalous behaviour. The global
property of fractional operators presents difficulties in fnding either closed-form solutions
or accurate numerical solutions to fractional differential equations. In rare cases, when
analytical solutions are available, they often exist only in terms of complex integrals and
special functions, or as infinite series. Similarly, obtaining an accurate numerical solution
to arbitrary order differential equation is often computationally demanding. Fractional
operators are non-local, and so it is practicable that when approximating fractional
operators, non-local methods should be preferred. One such non-local method is the
spectral method. In this thesis, we solve problems that arise in the
ow of non-Newtonian
fluids modelled with fractional differential operators. The recurrent theme in this thesis
is the development, testing and presentation of tractable, accurate and computationally
efficient numerical schemes for various classes of fractional differential equations. The
numerical schemes are built around the pseudo{spectral collocation method and shifted
Chebyshev polynomials of the first kind. The literature shows that pseudo-spectral
methods converge geometrically, are accurate and computationally efficient. The objective
of this thesis is to show, among other results, that these features are true when the method
is applied to a variety of fractional differential equations. A survey of the literature
shows that many studies in which pseudo-spectral methods are used to numerically
approximate the solutions of fractional differential equations often to do this by expanding
the solution in terms of certain orthogonal polynomials and then simultaneously solving
for the coefficients of expansion. In this study, however, the orthogonality condition of
the Chebyshev polynomials of the first kind and the Chebyshev-Gauss-Lobatto quadrature
are used to numerically find the coefficients of the series expansions. This approach is
then applied to solve various fractional differential equations, which include, but are not
limited to time{space fractional differential equations, two{sided fractional differential
equations and distributed order differential equations. A theoretical framework is provided
for the convergence of the numerical schemes of each of the aforementioned classes of
fractional differential equations. The overall results, which include theoretical analysis
and numerical simulations, demonstrate that the numerical method performs well in
comparison to existing studies and is appropriate for any class of arbitrary order differential
equations. The schemes are easy to implement and computationally efficient
Theoretical analysis (Convergence and stability) of a difference approximation for multiterm time fractional convection diffusion-wave equations with delay
In this paper, we introduce a high order numerical approximation method for convection diffusion wave equations armed with a multiterm time fractional Caputo operator and a nonlinear fixed time delay. A temporal second-order scheme which is behaving linearly is derived and analyzed for the problem under consideration based on a combination of the formula of L2 ā 1Ļ and the order reduction technique. By means of the discrete energy method, convergence and stability of the proposed compact difference scheme are estimated unconditionally. A numerical example is provided to illustrate the theoretical results. Ā© 2020 by the authors. Licensee MDPI, Basel, Switzerland.The first author wishes to acknowledge the support of RFBR Grant 19-01-00019
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