42 research outputs found
Analytical solutions to nonlinear differential equations arising in physical problems
Nonlinear partial differential equations are difficult to solve, with many of the approximate solutions in the literature being numerical in nature. In this work, we apply the Homotopy Analysis Method to give approximate analytical solutions to nonlinear ordinary and partial differential equations. The main goal is to apply different linear operators, which can be chosen, to solve nonlinear problems. In the first three chapters, we study ordinary differential equations (ODEs) with one or two linear operators. As we progress, we apply the method to partial differential equations (PDEs) and use several linear operators. The results are all purely analytical, meaning these are approximate solutions that we can evaluate at points and take their derivatives. Another main focus is error analysis, where we test how good our approximations are. The method will always produce approximations, but we use residual errors on the domain of the problem to find a measure of error. In the last two chapters, we apply similarity transforms to PDEs to transform them into ODEs. We then use the Homotopy Analysis Method on one, but are able to find exact solutions to both equations
Approximate Analytical Solution of Advection-Dispersion Equation By Means of OHAM.
This work deals with the analytical solution of advection dispersion equation arising in solute transport along unsteady groundwater flow in finite aquifer. A time dependent input source concentration is considered at the origin of the aquifer and it is assumed that the concentration gradient is zero at the other end of the aquifer. The optimal homotopy analysis method (OHAM) is used to obtain numerical and graphical representation
Analytic Analysis for Oil Recovery During Cocurrent Imbibition in Inclined Homogeneous Porous Medium
This paper focuses on the analysis of cocurrent imbibition phenomenon which occurs during secondary oil recovery process.In cocurrent imbibition, a strongly wetting phase(water) displaces a non-wetting phase(oil) spontaneously under the influence of capillary forces such that the oil moves in the same direction to the water. We use an optimal homotopy analysis method to derive an approximate analytical expression for saturation of water when the viscosity of the non-wetting phase is non-negligible
Sharp estimates for the global attractor of scalar reaction-diffusion equations with a Wentzell boundary condition
In this paper, we derive optimal upper and lower bounds on the dimension of
the attractor AW for scalar reaction-diffusion equations with a Wentzell
(dynamic) boundary condition. We are also interested in obtaining explicit
bounds about the constants involved in our asymptotic estimates, and to compare
these bounds to previously known estimates for the dimension of the global
attractor AK; K \in {D;N; P}, of reactiondiffusion equations subject to
Dirichlet, Neumann and periodic boundary conditions. The explicit estimates we
obtain show that the dimension of the global attractor AW is of different order
than the dimension of AK; for each K \in {D;N; P} ; in all space dimensions
that are greater or equal than three.Comment: to appear in J. Nonlinear Scienc
A Laguerre spectral method for quadratic optimal control of nonlinear systems in a semi-infinite interval
This paper presents a Laguerre homotopy method for quadratic optimal control problems in semi-infinite intervals (LaHOC), with particular interests given to nonlinear interconnected large-scale dynamic systems. In LaHOC, the spectral homotopy analysis method is used to derive an iterative solver for the nonlinear two-point boundary value problem derived from Pontryagin\u27s maximum principle. A proof of local convergence of the LaHOC is provided. Numerical comparisons are made between the LaHOC, Matlab BVP5C generated results and results from the literature for two nonlinear optimal control problems. The results show that LaHOC is superior in both accuracy and efficiency
Hopf bifurcation from fronts in the Cahn-Hilliard equation
We study Hopf bifurcation from traveling-front solutions in the Cahn-Hilliard
equation. The primary front is induced by a moving source term. Models of this
form have been used to study a variety of physical phenomena, including pattern
formation in chemical deposition and precipitation processes. Technically, we
study bifurcation in the presence of essential spectrum. We contribute a simple
and direct functional analytic method and determine bifurcation coefficients
explicitly. Our approach uses exponential weights to recover Fredholm
properties and spectral flow ideas to compute Fredholm indices. Simple mass
conservation helps compensate for negative indices. We also construct an
explicit, prototypical example, prove the existence of a bifurcating front, and
determine the direction of bifurcation
Computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model
We present a computer-assisted proof of heteroclinic connections in the
one-dimensional Ohta-Kawasaki model of diblock copolymers. The model is a
fourth-order parabolic partial differential equation subject to homogeneous
Neumann boundary conditions, which contains as a special case the celebrated
Cahn-Hilliard equation. While the attractor structure of the latter model is
completely understood for one-dimensional domains, the diblock copolymer
extension exhibits considerably richer long-term dynamical behavior, which
includes a high level of multistability. In this paper, we establish the
existence of certain heteroclinic connections between the homogeneous
equilibrium state, which represents a perfect copolymer mixture, and all local
and global energy minimizers. In this way, we show that not every solution
originating near the homogeneous state will converge to the global energy
minimizer, but rather is trapped by a stable state with higher energy. This
phenomenon can not be observed in the one-dimensional Cahn-Hillard equation,
where generic solutions are attracted by a global minimizer
Semi-Analytical Solutions of Non-linear Differential Equations Arising in Science and Engineering
Systems of coupled non-linear differential equations arise in science and engineering are inherently nonlinear and difficult to find exact solutions. However, in the late nineties, Liao introduced Optimal Homotopy Analysis Method (OHAM), and it allows us to construct accurate approximations to the systems of coupled nonlinear differential equations. The drawback of OHAM is, we must first choose the proper auxiliary linear operator and then solve the linear higher-order deformation equation by spending lots of CPU time. However, in the latest innovation of Liao\u27s Method of Directly Defining inverse Mapping (MDDiM) which he introduced to solve a single nonlinear ordinary differential equation has great freedom to define the inverse linear map directly. In this way, one can solve higher order deformation equations quickly, and it is unnecessary to calculate an inverse linear operator. Our primary goal is to extend MDDiM to solve systems of coupled nonlinear ordinary differential equations. In the first chapter, we will introduce MDDiM and briefly discuss the advantages of MDDiM Over OHAM. In the second chapter, we will study a nonlinear coupled system using OHAM. Next three chapters, we will apply MDDiM to coupled non-linear systems arise in mechanical engineering to study fluid flow and heat transfer. In chapter six we will apply this novel method to study coupled non-linear systems in epidemiology to investigate how diseases spread throughout time. In the last chapter, we will discuss our conclusions and will propose some future work. Another main focus is to compare MDDiM with OHAM
Phase-field models for thin elastic structures: Willmore's energy and topological constraints
In this dissertation, I develop a phase-field approach to minimising a geometric energy functional in the class of connected structures confined to a small container. The functional under consideration is Willmore's energy, which depends on the mean curvature and area measure of a surface and thus allows for a formulation in terms of varifold geometry. In this setting, I prove existence of a minimiser and a very low level of regularity from simple energy bounds.
In the second part, I describe a phase-field approach to the minimisation problem and provide a sample implementation along with an algorithmic description to demonstrate that the technique can be applied in practice. The diffuse Willmore functional in this setting goes back to De Giorgi and the novel element of my approach is the design of a penalty term which can control a topological quantity of the varifold limit in terms of phase-field functions. Besides the design of this functional, I present new results on the convergence of phase-fields away from a lower-dimensional subset which are needed in the proof, but interesting in their own right for future applications. In particular, they give a quantitative justification for heuristically identifying the zero level set of a phase field with a sharp interface limit, along with a precise description of cases when this may be admissible only up to a small additional set.
The results are optimal in the sense that no further topological quantities can be controlled in this setting, as is also demonstrated. Besides independent geometric interest, the research is motivated by an application to certain biological membranes