stability analysis for fractional order advection reaction diffusion system abstract in this paper we present an alternative representation of the advection reaction diffusion model involving fractional order derivatives with mittag leffler kernel the study includes three main aspects existence and uniqueness of solutions hyers ulam stability and numerical simulations for the existence and uniqueness of solutions we use fixed point approach also we presents the hyers ulam stability for the numerical simulations a new numerical scheme that involve lagrange interpolation laplace transform and forward euler technique is considered numerical simulations were obtained for some specific parameters

stability and error estimates of local discontinuous galerkin methods with implicit explicit time marching for advection diffusion problems the main purpose of this paper is to analyze the stability and error estimates of the local discontinuous galerkin ldg methods coupled with carefully chosen implicit explicit imex runge kutta time discretization up to third order accuracy for solving one dimensional linear advection diffusion equations in the time discretization the advection term is treated explicitly and the diffusion term implicitly there are three highlights of this work the first is that we establish an important relationship between the gradient and interface jump of the numerical solution with the independent numerical solution of the gradient in the ldg methods the second is that by aid of the aforementioned relationship and the energy method we show that the imex ldg schemes are unconditionally stable for the linear problems in the sense that the time step tau is only required to be upper bounded by a constant which depends on the ratio of the diffusion and the square of the advection coefficients and is independent

the use of cornu spirals in drawing planar curves of controlled curvature abstract cornu spirals or clothoids have been used in highway design for many years in the past the spirals have been found manually by draftsmen the purpose of this paper is to show that it is practical to find these spirals with a micro computer the design curve will be made up of arcs of circles and segments of cornu spirals joined in such a way that the curvature is continuous throughout and takes its largest values on the arcs of circles thus the radii of the circles used will limit and control the curvature of the whole design curve

adapted numerical methods for advection reaction diffusion problems generating periodic wavefronts abstract the paper presents an adapted numerical integration for advection reaction diffusion problems the numerical scheme exploiting the a priori knowledge of the qualitative behaviour of the solution gains advantages in terms of efficiency and accuracy with respect to classic schemes already known in literature the adaptation is here carried out through the so called trigonometrical fitting technique for the discretization in space giving rise to a system of odes whose vector field contains both stiff and non stiff terms due to this mixed nature of the vector field an implicit explicit imex method is here employed for the integration in time based on the first order forward backward euler method the coefficients of the method here introduced rely on unknown parameters which have to be properly estimated in this work such an estimate is performed by minimizing the leading term of the local truncation error the effectiveness of this problem oriented approach is shown through a rigorous theoretical analysis and some numerical experiments

on the evolution of periodic plane waves in reaction diffusion systems of l w type lambda omega systems are a class of simple reaction diffusion equations with a limit cycle in the reaction kinetics the author considers the solution of the system given by lambda r lambda 0 r p omega r omega 0 r p on a semi infinite spatial domain with initial data decaying exponentially across the domain numerical evidence is presented showing that this initial condition induces a wave front moving across the domain with periodic plane waves behind the front these periodic waves can move in either direction depending on the parameter values the author uses intuitive criteria to derive an expression for the speed of the advancing front and by reducing the system to ordinary differential equations for similarity solutions the amplitude and speed of the periodic plane waves are determined the speed of these periodic waves varies continuously with the initial data perturbation theory is then used to obtain an analytical approximation to the solutions these anal

a generalisation of the pythagorean hodograph quintic spiral an existing planar pythagorean hodograph ph quintic spiral with zero curvature at one end is generalised to form a two parameter family of ph quintic spirals the additional degree of freedom allows specification of an ending angle of tangent and curvature its shape and curvature compares well to that of the clothoid which has applications in areas such as highway design and robot path planning it is thus a viable nonuniform rational b spline alternative for such applications it is also useful for other computer aided design or computer aided geometric design applications in which drawing curves with controlled curvature e g fair curves is important

convergence of the viscosity method for isentropic gas dynamics a convergence theorem for the method of artificial viscosity applied to the isentropic equations of gas dynamics is established convergence of a subsequence in the strong topology is proved without uniform estimates on the derivatives using the theory of compensated compactness and an analysis of progressing entropy waves

exponentially fitted two step hybrid methods for y f x y it is the purpose of this paper to derive two step hybrid methods for y f x y with oscillatory or periodic solutions specially tuned to the behaviour of the solution through the usage of the exponential fitting technique the construction of two step exponentially fitted hybrid methods is shown and their properties are discussed some numerical experiments confirming the theoretical expectations are provided

planar g 2 transition with a fair pythagorean hodograph quintic curve recently planar cubic and pythagorean hodograph quintic transition curves that are suitable for g2 blending were developed they are suitable for blending e g rounding corners or for smooth transition between two curves e g two circular arcs it was shown that a single cubic segment can be used as a transition curve with the guarantee that an s shaped transition curve will have no curvature extrema and a c shaped transition curve will have a single curvature extremum the results for the cubic curve are now extended to pythagorean hodograph quintic curves a pythagorean hodograph curve has the attractive properties that its arclength is a polynomial of its parameter and its offset is rational a quintic is the lowest degree pythagorean hodograph curve that may have an inflection point pythagorean hodograph curves with no curvature extrema for an s shaped transition and a single curvature extremum for a c shaped transition are suitable for the design of fair curves e g in highway design or for blending in cad applications

planar interpolation with a pair of rational spirals spirals are curves of one signed monotone increasing or decreasing curvature spiral segments have the advantage that the minimum and maximum curvatures are at their endpoints two situations where the planar two point g 2 hermite interpolation problem can be solved with a pair of rational spiral segments are outlined here

existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in eulerian and lagrangian coordinates we prove the existence and compactness stability of entropy solutions for the hyperbolic systems of conservation laws corresponding to the isentropic gas dynamics where the pressure and density are related by a law for any 1 our results considerably extend and simplify the program initiated by diperna and provide a complete existence proof our methods are based on the compensated compactness and the kinetic formulation of systems of conservation laws 1996 john wiley sons inc

smooth polynomial approximation of spiral arcs constructing fair curve segments using parametric polynomials is difficult due to the oscillatory nature of polynomials even nurbs curves can exhibit unsatisfactory curvature profiles curve segments with monotonic curvature profiles for example spiral arcs exist but are intrinsically non polynomial in nature and thus difficult to integrate into existing cad systems a method of constructing an approximation to a generalised cornu spiral gcs arc using non rational quintic bezier curves matching end points end slopes and end curvatures is presented by defining an objective function based on the relative error between the curvature profiles of the gcs and its bezier approximation a curve segment is constructed that has a monotonic curvature profile within a specified tolerance

curvature extrema of planar parametric polynomial cubic curves parametric polynomial cubic curve segments are widely used in computer aided design and computer aided geometric design applications because their flexibility makes them suitable for use in the interactive design of curves and surfaces shape properties of these segments have been studied and results on the occurrence of cusps and whether the segment is s or c shaped are available in the literature their critical points however are not as well known this paper presents results on the number and location of curvature extrema of these segments all possible numbers of curvature extrema are found and it is demonstrated that all possible curvature extrema can be obtained numerically for any planar parametric cubic curve these results are useful in the study of the fairness of curves designed with parametric polynomial cubic curve segments

an arc spline approximation to a clothoid the clothoid is a spiral used in highway and railway route design clothoids are transcendental functions and so have been approximated by polynomials by power series and continued fractions and by rational functions here the clothoid is approximated by an arc spline the chief advantage in doing so is that arc splines are very easy to lay out and to offset examples show that the approximation is of extremely high accuracy it is proved that if the arc spline has n arcs then the error in the approximation is of order o 1 n2

generalized fresnel integrals and fractal properties of related spirals abstract we obtain a new asymptotic expansion of generalized fresnel integrals x t 0 t cos q s d s for large t where q s s p when s and p 1 the terms of the expansion are defined via a simple iterative algorithm using this we show that the box dimension of the related q clothoid also called the generalized euler or cornu spiral is equal to d 2 p 2 p 1 furthermore this curve is minkowski measurable and we compute its d dimensional minkowski content we also find oscillatory dimension of fresnel integrals by studying the corresponding chirps

approximating cornu spirals by arc splines while modeling planar parametric curves it is desirable to approximate them efficiently by arc splines with fewest segments within the prescribed tolerance planar arc splines are ideal for describing the path of a numerically controlled cutting machine because they are easy to offset in this paper we derive a vector solution to interpolate a g1 continuous biarc curve between a given pair of points with specified start joint and end tangents using this solution as the underlying idea we present an efficient algorithm for approximating a cornu spiral with a sequence of arcs within the prescribed maximum deviation the method is simple and generic and can be applied to other types of planar parametric curves as well

implicit explicit methods for time dependent partial differential equations implicit explicit imex schemes have been widely used especially in conjunction with spectral methods for the time integration of spatially discretized partial differential equations pdes of diffusion convection type typically an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term reaction diffusion problems can also be approximated in this manner in this work we systematically analyze the performance of such schemes propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods for the prototype linear advection diffusion equation a stability analysis for first second third and fourth order multistep imex schemes is performed stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified numerical experiments demonstrate that weak decay

numerical solution of reaction diffusion systems of λ ω type by trigonometrically fitted methods the numerical solution of reaction diffusion equations of type which are known to possess a one parameter family of periodic plane wave solutions is object of this paper due to the periodic character of such solutions a special purpose numerical integration is here proposed based on adapted finite differences the adaptation occurs at the level of the problem by a suitable spatial semi discretization based on trigonometrically fitted finite differences numerical experiments confirming the effectiveness of the approach are given

the vacuum case in diperna s paper abstract lemma 4 1 in comm math phys 91 1983 by ron diperna pertaining to the vacuum case for an existence proof of polytropic gas dynamics using compensated compactness is incomplete as given here we give a quick fix of the lemma plus some generalization

clothoid spline transition spirals highway and railway designers use clothoid splines planar g2 curves consisting of straight line segments circular arcs and clothoid segments as center lines in route location this paper considers the problem of finding a clothoid spline transition spiral which joins two given points and matches given curvatures and unit tangents at the two points conditions are given for the existence and uniqueness of the clothoid spline transition spirals and algorithms for finding them are outlined

rational approximations for the fresnel integrals a class of simple rational polynomial approximations for the fresnel integrals is given with maximum errors from 1 7 x 10 3 down to 4 x 10 8 the domain 0 oo is not subdivided the format is particularly convenient for programmable hand calculators and microcomputer subroutines the best known algorithms for computer evaluation of the fresnel integrals 1 c x is x j exp ip t2 dt 0 are those of hastings 1 41 and of boersma 2 the former is of very limited precision the latter is relatively cumbersome and is subject to nonobvious underflow subtraction errors as is direct evaluation of the standard maclaurin and asymptotic series an extensive set of rational polynomial approximations has been given by cody 3 these are efficient for very high precision work but tedious for intermediate levels of precision because the domain is subdivided into five intervals our approach follows hastings but recasts the formulas into the polar form 2 c x 2ri x sinf1pr ajk x x2 i 3 s x 2 rim x cos 7t ajk x x2 where the rim and ajk functions are rational approximations of the form i m2 4 r 01 l c1xi e dix i o i 0 i k s ajk e a x e b x o i o the domain of x is positive real numbers from zero to infinity we consider error functions of the form 6 sr rim ro 7 sa 7tro ak ao where r0 and ao are exact values these functions represent orthogonal errors in the plane of the cornu spiral with the coefficient inserted in 7 to make the magnitudes received may 2 1983 revised january 9 1984 1980 mathematics subject classification primary 65d20 secondary 33a20 41a50 78a50

exponentially fitted methods applied to fourth order boundary value problems fourth order boundary value problems are solved by means of exponentially fitted methods of different orders these methods which depend on a parameter can be constructed following a six step flow chart of ixaru and vanden berghe special attention is paid to the expression for the error term and to the choice of the parameter in order to make the error as small as possible some numerical examples are given to illustrate the practical implementation issues of these methods

exponentially fitted two step runge kutta methods construction and parameter selection abstract we derive exponentially fitted two step runge kutta methods for the numerical solution of y f x y specially tuned to the behaviour of the solution such methods have nonconstant coefficients which depend on a parameter to be suitably estimated the construction of the methods is shown and a strategy of parameter selection is presented some numerical experiments are provided to confirm the theoretical expectations

generation of log aesthetic curves using adaptive runge kutta methods log aesthetic curve lac has been explored extensively by many researchers since 2005 at first gaussian kronrod has been proposed to evaluate lac as the formulation of lac involves double integration recently incomplete gamma function igf has been proposed to represent lac analytically which decreases the computation time up to 13 times this paper embarks on the representation of lac using adaptive runge kutta methods to decrease the lac computation time the famous adaptive methods such as runge kutta fehlberg dormand prince sarafyan and kutta merson are employed to evaluate lac so that a desired accuracy can be achieved this paper ends with a detailed investigation on performance metric of igf and adaptive rk methods to compute lac these methods will be compared in terms of computation time and truncation error numerical results indicate that the computation time of lac can be greatly improved and at the same time preserving the lac s family

continuous approximate solution of ordinary differential equations and their systems abstract the solution of initial value problems in ordinary differential equations are continuously differentiable functions yet all effective methods presently used for the numerical solution of these problems either in scalar or vector form are discrete processes they yield a finite set of approximate values of the solution the process derived in this paper yields a continuously differentiable function a polynomial which gives fourth order runge kutta approximations at all interior points of each interval of application and a fifth order approximation at each end point like conventional fifth order runge kutta algorithms the process requires only six evaluations of the given directional function per application or step it is endowed with an imbedding property for step size control and provides valuable information about committed local truncation errors it may be called a continuous imbedded runge kutta process of maximum order five it is valid for both scalar and vector cases

existence of solutions to hyperbolic conservation laws with a source existence of solutions to three different systems of eqs 1 1 1 2 and 1 3 coming from physically relevant models is shown each needing a different proof which are given in sects 2 3 and 4 the unifying theme is the presence of source terms and the general method of proof is vanishing viscosity together with compensated compactness for system 1 2 entropy entropy flux pairs of lax type are constructed and estimates from singular perturbation theory of odes are used for 1 1 and 1 3 weak entropy entropy flux pairs are constructed following the compensated compactness framework set up by diperna 4

error analysis of imex runge kutta methods derived from differential algebraic systems in this paper we present an error analysis of the imex runge kutta methods when applied to stiff problems containing a nonstiff term and a stiff term characterized by a small stiffness parameter varepsilon in this analysis we expand the global error in powers of varepsilon and show that the coefficients of the error are the global errors of the imex runge kutta method applied to a differential algebraic system interesting convergence results of these errors and of the remainder of the expansion allow us to determine sharp error bounds for stiff problems as a representative example of stiff problems we have chosen the van der pol equation we illustrate that the theoretical prediction is confirmed by the numerical test specifically an order reduction phenomenon is observed when the problem becomes increasingly stiff in particular making several assumptions we try to improve global error estimates of several imex runge kutta methods existing in the literature

the effects of unequal diffusion coefficients on periodic travelling waves in oscillatory reaction diffusion systems abstract many oscillatory biological systems show periodic travelling waves these are often modelled using coupled reaction diffusion equations however the effects of different movement rates diffusion coefficients of the interacting components on the predictions of these equations are largely unknown here we investigate the ways in which varying the diffusion coefficients in such equations alters the wave speed time period wavelength amplitude and stability of periodic wave solutions we focus on two sets of kinetics that are commonly used in ecological applications lambda omega equations which are the normal form of an oscillatory coupled reaction diffusion system close to a supercritical hopf bifurcation and a standard predator prey model our results show that changing the ratio of the diffusion coefficients can significantly alter the shape of the one parameter family of periodic travelling wave solutions the position of the boundary between stable and unstable waves also depends on the ratio of the diffusion coefficients in all cases stability changes through an eckhaus sideband instability these effects are always symmetrical in the two diffusion coefficients for the lambda omega equations but are asymmetric in the predator prey equations especially when the limit cycle of the kinetics is of large amplitude in particular there are two separate regions of stable waves in the travelling wave family for some parameter values in the predator prey scenario our results also show the existence of a one parameter family of travelling waves but not necessarily a hopf bifurcation for all values of the diffusion coefficients simulations of the full partial differential equations reveals that varying the ratio of the diffusion coefficients can significantly change the properties of periodic travelling waves that arise from particular wave generation mechanisms and our analysis of the travelling wave families assists in the understanding of these effects

planar g 2 transition curves composed of cubic bezier spiral segments in curve and surface design it is often desirable to have a planar transition curve composed of at most two spiral segments between two circles the purpose may be practical e g in highway design or aesthetic cubic bezier curves are commonly used in curve and surface design because they are of low degree are easily evaluated and allow inflection points this paper generalizes earlier results on planar cubic bezier spiral segments which were proposed as transition curve elements and examines techniques for curve design using the new results

g1 interpolation with a single cornu spiral segment cornu spiral segments are used in applications such as the geometric design of highways and railways robot path planning and shape completion for some applications e g shape completion in computer vision it is important to use a single visually pleasing curve segment to smoothly fill a gap even though the gap may not be filled in a curvature continuous manner an improved method for doing so using a cornu spiral segment is discussed the method is generally suitable for any application where it is required to smoothly fit a curve between two given points when the corresponding tangent directions but not the curvatures are also given or known