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NonLocal Partial Differential Equations for Engineering and Biology: Mathematical Modeling and AnalysisThis book presents new developments in nonlocal mathematical modeling and mathematical analysis on the behavior of solutions with novel technical tools. Theoretical backgrounds in mechanics, thermodynamics, game theory, and theoretical biology are examined in details. It starts off with a review and summary of the basic ideas of mathematical modeling frequently used in the sciences and engineering. The authors then employ a number of models in bioscience and material science to demonstrate applications, and provide recent advanced studies, both on deterministic nonlocal partial differential equations and on some of their stochastic counterparts used in engineering. Mathematical models applied in engineering, chemistry, and biology are subject to conservation laws. For instance, decrease or increase in thermodynamic quantities and nonlocal partial differential equations, associated with the conserved physical quantities as parameters. These present novel mathematical objects are engaged with rich mathematical structures, in accordance with the interactions between species or individuals, selforganization, pattern formation, hysteresis. These models are based on various laws of physics, such as mechanics of continuum, electromagnetic theory, and thermodynamics. This is why many areas of mathematics, calculus of variation, dynamical systems, integrable systems, blowup analysis, and energy methods are indispensable in understanding and analyzing these phenomena. This book aims for researchers and upper grades students in mathematics, engineering, physics, economics, and biology.

A nonpolynomial collocation method for fractional terminal value problemsIn this paper we propose a nonpolynomial collocation method for solving a class of terminal (or boundary) value problems for differential equations of fractional order α, 0 < α < 1. The approach used is based on the equivalence between a problem of this type and a Fredholm integral equation of a particular form. Taking into account the asymptotic behaviour of the solution of this problem, we propose a nonpolynomial collocation method on a uniform mesh. We study the order of convergence of the proposed algorithm and a result on optimal order of convergence is obtained. In order to illustrate the theoretical results and the performance of the method we present several numerical examples.

A note on finite difference methods for nonlinear fractional differential equations with nonuniform meshesWe consider finite difference methods for solving nonlinear fractional differential equations in the Caputo fractional derivative sense with nonuniform meshes. Under the assumption that the Caputo derivative of the solution of the fractional differential equation is suitably smooth, Li et al. \lq \lq Finite difference methods with nonuniform meshes for nonlinear fractional differential equations\rq\rq, Journal of Computational Physics, 316(2016), 614631, obtained the error estimates of finite difference methods with nonuniform meshes. However the Caputo derivative of the solution of the fractional differential equation in general has a weak singularity near the initial time. In this paper, we obtain the error estimates of finite difference methods with nonuniform meshes when the Caputo fractional derivative of the solution of the fractional differential equation has lower smoothness. The convergence result shows clearly how the regularity of the Caputo fractional derivative of the solution affect the order of convergence of the finite difference methods. Numerical results are presented that confirm the sharpness of the error analysis.

A Note on the WellPosedness of Terminal Value Problems for Fractional Differential Equations.This note is intended to clarify some im portant points about the wellposedness of terminal value problems for fractional di erential equations. It follows the recent publication of a paper by Cong and Tuan in this jour nal in which a counterexample calls into question the earlier results in a paper by this note's authors. Here, we show in the light of these new insights that a wide class of terminal value problems of fractional differential equations is well posed and we identify those cases where the wellposedness question must be regarded as open.

A novel highorder algorithm for the numerical estimation of fractional differential equationsThis paper uses polynomial interpolation to design a novel highorder algorithm for the numerical estimation of fractional differential equations. The RiemannLiouville fractional derivative is expressed by using the Hadamard finitepart integral and the piecewise cubic interpolation polynomial is utilized to approximate the integral. The detailed error analysis is presented and it is established that the convergence order of the algorithm is O(h4−a). Asymptotic expansion of the error for the presented algorithm is also investigated. Some numerical examples are provided and compared with the exact solution to show that the numerical results are in well agreement with the theoretical ones and also to illustrate the accuracy and efficiency of the proposed algorithm.

A novel ‘bottomup’ synthesis of few and multilayer graphene platelets with partial oxidation via cavitationThe transient cavitation of diaromatic components such as 1methylnaphthalene has been used to produce graphene platelets in a ‘bottomup’ synthesis via the high temperature (>5000 K) conditions that are generated inside collapsing bubbles. Acoustic cavitation produced yields of 5.7×10−11 kgJ−1 at a production rate of 2.2×10−9 kgs−1. This can be improved by generating cavitation hydrodynamically, thus making commercial scale production viable. Hydrodynamic cavitation produced platelets with larger lateral dimensions (≥2 μm) than those formed by acoustic cavitation (10–200 nm). The partially oxidised nature of the platelets enables their covalent chemical functionalisation, which was achieved by combining suitable molecules in the reaction medium to affect a onepot formation and functionalisation of graphene

Numerical analysis for distributed order differential equationsIn this paper we present and analyse a numerical method for the solution of a distributed order differential equation.

Numerical analysis of a singular integral equationThis preprint discusses the numerical analysis of an integral equation to which convential analytical and numerical theory does not apply.

Numerical analysis of a twoparameter fractional telegraph equationIn this paper we consider the twoparameter fractional telegraph equation of the form $$\, ^CD_{t_0^+}^{\alpha+1} u(t,x) + \, ^CD_{x_0^+}^{\beta+1} u (t,x) \, ^CD_{t_0^+}^{\alpha}u (t,x)u(t,x)=0.$$ Here $\, ^CD_{t_0^+}^{\alpha}$, $\, ^CD_{t_0^+}^{\alpha+1}$, $\, ^CD_{x_0^+}^{\beta+1}$ are operators of the Caputotype fractional derivative, where $0\leq \alpha < 1$ and $0 \leq \beta < 1$. The existence and uniqueness of the equations are proved by using the Banach fixed point theorem. A numerical method is introduced to solve this fractional telegraph equation and stability conditions for the numerical method are obtained. Numerical examples are given in the final section of the paper.

Numerical analysis of some integral equations with singularitiesIn this thesis we consider new approaches to the numerical solution of a class of Volterra integral equations, which contain a kernel with singularity of nonstandard type. The kernel is singular in both arguments at the origin, resulting in multiple solutions, one of which is differentiable at the origin. We consider numerical methods to approximate any of the (infinitely many) solutions of the equation. We go on to show that the use of product integration over a short primary interval, combined with the careful use of extrapolation to improve the order, may be linked to any suitable standard method away from the origin. The resulting splitinterval algorithm is shown to be reliable and flexible, capable of achieving good accuracy, with convergence to the one particular smooth solution.

Numerical approaches to bifurcations in solutions to integrodifferential equationsThis conference paper discusses the qualitative behaviour of numerical approximations of a carefully chosen class of integrodifferential equations of the Volterra type. The results are illustrated with some numerical experiments.

Numerical approaches to delay equations with small solutionsThis book chapter discusses the use of numerical schemes to find whether dalay differential equations have small solutions. Two questions  can the onset of small solutions be predicted for a wider range of delay differential equations in a similar way and how should one chose the appropriate numerical method for the investigation  are discussed.

Numerical Approximation of Stochastic TimeFractional DiffusionWe develop and analyze a numerical method for stochastic timefractional diffusion driven by additive fractionally integrated Gaussian noise. The model involves two nonlocal terms in time, i.e., a Caputo fractional derivative of order $\alpha\in(0,1)$, and fractionally integrated Gaussian noise (with a RiemannLiouville fractional integral of order $\gamma \in[0,1]$ in the front). The numerical scheme approximates the model in space by the standard Galerkin method with continuous piecewise linear finite elements and in time by the classical Gr\"unwaldLetnikov method, and the noise by the $L^2$projection. Sharp strong and weak convergence rates are established, using suitable nonsmooth data error estimates for the deterministic counterpart. One and twodimensional numerical results are presented to support the theoretical findings.

Numerical approximation of the Stochastic CahnHilliard Equation near the Sharp Interface LimitAbstract. We consider the stochastic CahnHilliard equation with additive noise term that scales with the interfacial width parameter ε. We verify strong error estimates for a gradient flow structureinheriting timeimplicit discretization, where ε only enters polynomially; the proof is based on highermoment estimates for iterates, and a (discrete) spectral estimate for its deterministic counterpart. For γ sufficiently large, convergence in probability of iterates towards the deterministic HeleShaw/MullinsSekerka problem in the sharpinterface limit ε → 0 is shown. These convergence results are partly generalized to a fully discrete finite element based discretization. We complement the theoretical results by computational studies to provide practical evidence concerning the effect of noise (depending on its ’strength’ γ) on the geometric evolution in the sharpinterface limit. For this purpose we compare the simulations with those from a fully discrete finite element numerical scheme for the (stochastic) MullinsSekerka problem. The computational results indicate that the limit for γ ≥ 1 is the deterministic problem, and for γ = 0 we obtain agreement with a (new) stochastic version of the MullinsSekerka problem.

A Numerical Feasibility Study of Kinetic Energy Harvesting from Lower Limb ProstheticsWith the advancement trend of lower limb prosthetics headed towards bionics (active ankle and knee) and smart prosthetics (gait and condition monitoring), there is an increasing integration of various sensors (microelectromechanical system (MEMS) accelerometers, gyroscopes, magnetometers, strain gauges, pressure sensors, etc.), microcontrollers and wireless systems, and power drives including motors and actuators. All of these active elements require electrical power. However, inclusion of a heavy and bulky battery risks to undo the lightweight advancements achieved by the strong and flexible composite materials in the past decades. Kinetic energy harvesting holds the promise to recharge a small onboard battery in order to sustain the active systems without sacrificing weight and size. However, careful design is required in order not to overburden the user from parasitic effects. This paper presents a feasibility study using measured gait data and numerical simulation in order to predict the available recoverable power. The numerical simulations suggest that, depending on the axis, up to 10s mW average electrical power is recoverable for a walking gait and up to 100s mW average electrical power is achievable during a running gait. This takes into account parasitic losses and only capturing a fraction of the gait cycle to not adversely burden the user. The predicted recoverable power levels are ample to selfsustain wireless communication and smart sensing functionalities to support smart prosthetics, as well as extend the battery life for active actuators in bionic systems. The results here serve as a theoretical foundation to design and develop towards regenerative smart bionic prosthetics.

Numerical investigation of noise induced changes to the solution behaviour of the discrete FitzHughNagumo equationIn this work we introduce and analyse a stochastic functional equation, which contains both delayed and advanced arguments. This equation results from adding a stochastic term to the discrete FitzHughNagumo equation which arises in mathematical models of nerve conduction. A numerical method is introduced to compute approximate solutions and some numerical experiments are carried out to investigate their dynamical behaviour and compare them with the solutions of the corresponding deterministic equation.