rational chebyshev tau method for solving volterra s population model an approximate method for solving volterra s population model for population growth of a species in a closed system is proposed volterra s model is a nonlinear integro differential equation where the integral term represents the effect of toxin the approach is based on a rational chebyshev tau method the volterra s population model is first converted to a nonlinear ordinary differential equation the operational matrices of derivative and product of rational chebyshev functions are presented these matrices together with the tau method are then utilized to reduce the solution of the volterra s model to the solution of a system of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the technique and a comparison is made with existing results

distributed source modeling and direction of arrival estimation techniques in direction of arrival doa estimation the direction of a signal is usually assumed to be a point if the direction of a signal is distributed due to some environmental phenomenon however doa estimation methods based on the point source assumption may result in poor performance we consider doa estimation when the signal sources are distributed parametric and nonparametric models are proposed and estimation methods are considered under these models in addition the asymptotic distribution of estimation errors is obtained to show the models statistical properties

nonlinear coupling near a degenerate hopf bautin bifurcation a nonlinearly coupled system of bistable fixed point and limit cycle differential equations is analyzed the nonlinear equations arise from the first several terms in the normal form expansion near a bautin bifurcation existence and stability of in phase and out of phase periodic solutions to a pair of identical systems are explored existence uniqueness and stability of traveling wave solutions from a stable rest state to a stable periodic solution are proved for the associated evolution convolution equation numerical simulations suggest some interesting patterns in regimes where waves no longer exist the results are shown to hold for a nonreduced conductance based model

periodic travelling wave selection by dirichlet boundary conditions in oscillatory reaction diffusion systems periodic travelling waves are a fundamental solution form in oscillatory reaction diffusion equations here i discuss the generation of periodic travelling waves in a reaction diffusion system of the generic lambda omega form i present numerical results suggesting that when this system is solved on a semi infinite domain subject to dirichlet boundary conditions in which the variables are fixed at zero periodic travelling waves develop in the domain the amplitude and speed of these waves are independent of the initial conditions which i generate randomly in numerical simulations using a combination of numerical and analytical methods i investigate the mechanism of periodic travelling wave selection by looking for an appropriate similarity solution i reduce the problem to an ode system using this i derive a formula for the selected speed and amplitude as a function of parameters finally i discuss applications of this work to ecology

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

two boundedness and monotonicity preserving methods for a generalized fisher kpp equation two finite difference methods are proposed for a generalized fisher kpp equation these methods are capable of preserving the skew symmetry of the equation these methods preserve the positivity boundedness and monotonicity of solutions these methods are solvable these methods are convergent a semi explicit finite difference method and an implicit finite difference method are proposed for a generalized fisher kpp equation with two space variables it is proved that these two methods could preserve the skew symmetry of this equation moreover under a condition on the step sizes the semi explicit method is capable of preserving the positivity the boundedness and the spatial and temporal monotonicity of initial approximations and the implicit method is able to preserve these properties with no restriction on the step sizes the stability and convergence of these two methods are also analyzed respectively finally some numerical simulations are provided to verify the validity of our analytical results

wavelet based iterative methods for a class of 2d partial integro differential equations abstract in this paper an iterative method based on quasilinearization is presented to solve a class of two dimensional partial integro differential equations that arise in nuclear reactor models and population models two different approaches based on haar and legendre wavelets are studied to develop numerical methods in the first approach time domain is approximated with the help of forward finite difference approach in the second approach both time as well as space domains are approximated by wavelets appropriate examples are solved using these methods and the obtained results are compared with the methods available in the recent literature

monotone iterates for solving nonlinear integro parabolic equations of volterra type the paper deals with numerical solving of nonlinear integro parabolic problems based on the method of upper and lower solutions a monotone iterative method is constructed existence and uniqueness of a solution to the nonlinear difference scheme are established an analysis of convergence rates of the monotone iterative method is given construction of initial upper and lower solutions is discussed numerical experiments are presented

classroom note numerical and analytical solutions of volterra s population model to illustrate various mathematical methods which may be used to solve problems of engineering mathematical physics and mathematical biology volterra s model for population growth of a species in a closed system is solved using several methods familiar to junior or senior level students in applied mathematics volterra s model is a first order integro ordinary differential equation where the integral term represents the effect of toxin accumulation on the species the solution methods used are i numerical methods for solving a first order initial value problem supplemented with numerical integration ii numerical methods for solving a coupled system of two first order initial value problems and iii phase plane analysis a singular perturbation solution previously presented is also outlined while conclusions drawn using the four methods are correlated the student may analyze and solve the problem using any of the methods independently of the others

legendre wavelets operational method for the numerical solutions of nonlinear volterra integro differential equations system in this paper legendre wavelet method is developed to approximate the solutions of system of nonlinear volterra integro differential equations the properties of legendre wavelets are first presented the properties of legendre wavelets are used to reduce the system of integral equations to a system of algebraic equations which can be solved numerically by newton s method also the results obtained by present method have been compared with that of by b spline wavelet method illustrative examples have been discussed to demonstrate the validity and applicability of the present method

the effects of obstacle size on periodic travelling waves in oscillatory reaction diffusion equations many natural populations undergo multi year cycles and field studies have shown that these can be organized into periodic travelling waves ptws mathematical studies have shown that large scale landscape obstacles represent a natural mechanism for wave generation here we investigate how the amplitude and wavelength of the selected waves depend on the obstacle size we firstly consider a large circular obstacle in an infinite domain for a reaction diffusion system of l u type we use perturbation theory to derive a leading order approximation to the wave generated by the obstacle this shows the dependence of the wave properties on both parameter values and obstacle size we find that the limiting values of the amplitude and wavelength are approached algebraically with distance from the obstacle edge rather than exponentially in the case of a flat boundary we use our results to predict the properties of waves generated by a large circular obstacle for an oscillatory predator prey system via a reduction of the predator prey model to normal form close to hopf bifurcation our predictions compare well with numerical simulations we also discuss the implications of these results for wave stability and briefly investigate the effects of obstacles with elliptical geometries

bearing estimation for a distributed source of nonconstant modulus ounds and analysis the paper considers the problem of estimating the bearing of a single far field source surrounded by local scatterers using passive sensor array measurements an associated source bearing estimation problem is formulated and the crame spl acute r rao lower bound is evaluated in addition a comprehensive analysis is performed of the maximum likelihood estimates that due to mismodeling assume a constant modulus source and the degradation in performance is quantified as a function of the source s empirical variance it is shown that for a limited price in terms of mean square error the constant modulus maximum likelihood estimator can replace the optimal nonconstant modulus estimator

legendre wavelets based numerical algorithm for simulation of multidimensional benjamin bona mahony burgers and sobolev equations abstract in this work a numerical algorithm is developed with the help of legendre wavelets and quasilinearization method for simulation of multidimensional benjamin bona mahony burgers bbmb and sobolev equations initially non linear equation is linearized by quasilinearization method and then legendre wavelets are used for both space and temporal discretization the algorithm is developed without the aid of finite difference method to preserve the accuracy of the legendre wavelets in numerical experiments time dependent 1d 2d and 3d problems are considered to check the accuracy and efficiency of the algorithm a comparison is made with the analytical and numerical solutions available in the literature it is concluded that the present algorithm without finite difference based legendre method gives better results than finite difference based haar wavelet methods for time dependent 2d bbmb and 2d sobolev equations also numerical experiments reveal that the proposed algorithm produces more accurate results even with comparatively coarser grids

spatial noise stabilizes periodic wave patterns in oscillatory systems on finite domains invasions in oscillatory systems generate in their wake spatiotemporal oscillations consisting of either periodic wavetrains or irregular oscillations that appear to be spatiotemporal chaos we have shown previously that when a finite domain with zero flux boundary conditions has been fully invaded the spatiotemporal oscillations persist in the irregular case but die out in a systematic way for periodic traveling waves in this paper we consider the effect of environmental inhomogeneities on this persistence we use numerical simulations of several predator prey systems to study the effect of random spatial variation of the kinetic parameters on the die out of regular oscillations and the long time persistence of irregular oscillations we find no effect on the latter but remarkably a moderate spatial variation in parameters leads to the persistence of regular oscillations via the formation of target patterns in order to study this target pattern production analytically we turn to systems numerical simulations confirm analagous behavior in this generic oscillatory system we then repeat this numerical study using piecewise linear spatial variation of parameters rather than random variation which also gives formation of target patterns under certain circum stances which we discuss we study this in detail by deriving an analytical approximation to the targets formed when the parameter 0 varies in a simple piecewise linear manner across the do main using perturbation theory we end by discussing the applications of our results in ecology and chemistry

self organized pulse generator abstract numerical simulations of a reaction diffusion system of the bonhoeffer van der pol type will be presented the system admits a coesistence of a stable stationary solution and a stable limit cycle which leads to the generation of a self organized pulse generator pattern dynamics of the system are explored in one and two dimensional spaces

on a nonlinear diffusion system arising in reactor dynamics abstract in this paper the existence uniqueness and asymptotic behavior of the solutions of a nonlinear diffusion system u t lu f t x u h u t x occurring in reactor dynamics is studied the problem is approached on the bases of the monotone method and the notion of upper and lower solutions

solving parabolic integro differential equations by an explicit integration method vasudeva murthy a s and j g verwer solving parabolic integro differential equations by an explicit integration method journal of computational and applied mathematics 39 1992 121 132 the subject of this note is the numerical integration in time of nonclassical parabolic initial boundary value problems which involve nonlocal integral terms over the spatial domain the integral terms may appear in the boundary conditions and or in the governing partial differential equation itself these terms generally complicate the application of standard methods our purpose here is to draw attention to an explicit method the runge kutta chebyshev method whose application remains straightforward in many cases of practical interest we discuss two numerical examples to illustrate this a comparison is made respectively with the implicit bdf code dassl and with the keller box scheme

evans functions and bifurcations of nonlinear waves of some nonlinear reaction diffusion equations abstract the main purposes of this paper are to accomplish the existence stability instability and bifurcation of the nonlinear waves of the nonlinear system of reaction diffusion equations u t u x x h u u w w t e u w and to establish the existence stability instability and bifurcation of the nonlinear waves of the nonlinear scalar reaction diffusion equation u t u x x h u u under different conditions on the model constants to establish the bifurcation for the system we will study the existence and instability of a standing pulse solution if 0 2 1 the existence and stability of two standing wave fronts if 2 1 and 2 e 1 the existence and instability of two standing wave fronts if 2 1 and 0 2 e 1 the existence and instability of an upside down standing pulse solution if 0 1 2 1 to establish the bifurcation for the scalar equation we will study the existence and stability of a traveling wave front as well as the existence and instability of a standing pulse solution if 0 2 the existence and stability of two standing wave fronts if 2 the existence and stability of a traveling wave front as well as the existence and instability of an upside down standing pulse solution if 0 2 by the way we will also study the existence and stability of a traveling wave back of the nonlinear scalar reaction diffusion equation u t u x x h u u w 0 where w 0 2 0 is a positive constant if 0 2 to achieve the main goals we will make complete use of the special structures of the model equations and we will construct evans functions and apply them to study the eigenvalues and eigenfunctions of several eigenvalue problems associated with several linear differential operators it turns out that a complex number 0 is an eigenvalue of the linear differential operator if and only if 0 is a zero of the evans function the stability instability and bifurcations of the nonlinear waves follow from the zeros of the evans functions a very important motivation to study the existence stability instability and bifurcations of the nonlinear waves is to study the existence and stability instability of infinitely many fast slow multiple traveling pulse solutions of the nonlinear system of reaction diffusion equations the existence and stability of infinitely many fast multiple traveling pulse solutions are of great interests in mathematical neuroscience

population growth in a closed system volterra s model for population growth in a closed system includes an integral term to indicate accumulated toxicity in addition to the usual terms of the logistic equation solutions to this equation are studied by means of singular perturbation techniques two major types of systems are discovered those with populations immediately sensitive to toxins and those that succumb to toxins only after long times

numerical solution of two dimensional elliptic pdes with nonlocal boundary conditions in the present paper two numerical methods are analyzed for the solution of two dimensional poisson equation with two different types of nonlocal boundary conditions the first numerical method is a collocation method based on haar wavelet whereas the second numerical method is a meshless method based on different types of radial basis functions rbfs a two point boundary condition and an integral boundary condition are the two types of nonlocal boundary conditions considered in the present work for the collocation method based on haar wavelet a new approach is formulated which involves the approximation of a fourth order mixed derivative by a haar expansion which is integrated subsequently to get wavelet approximation of the solution for the meshless method based on rbfs the algorithm is implemented using two different splitting schemes with and without shape parameter splitting for numerical solution of the model the comparative analysis of the meshless methods with and without shape parameter splitting scheme is performed between themselves as well as with the haar wavelet accuracy and efficiency wise performance is confirmed through application of the algorithms on the benchmark tests

parametric localization of distributed sources most array processing algorithms are based on the assumption that the signals are generated by point sources this is a mathematical constraint that is not satisfied in many applications in this paper we consider situations where the sources are distributed in space with a parametric angular cross correlation kernel we propose an algorithm that estimates the parameters of this model using a generalization of the music algorithm the method involves maximizing a cost function that depends on a matrix array manifold and the noise eigenvectors we study two particular cases coherent and incoherent spatial source distributions the spatial correlation function for a uniformly distributed signal is derived from this we find the array gain and show that in contrast to point sources it does not increase linearly with the number of sources we compare our method to the conventional point source music algorithm the simulation studies show that the new method outperforms the music algorithm by reducing the estimation bias and the standard deviation for scenarios with distributed sources it is also shown that the threshold signal to noise ratio required for resolving two closely spaced distributed sources is considerably smaller for the new method

detection and localization in colored noise via generalized least squares a method for detection and localization of multiple signals in spatially colored noise by an arbitrary passive sensor array is presented the method also enables exploitation of prior knowledge that the signals are uncorrelated so as to improve the performance and to allow detection and localization even if the number of signals exceeds the number of sensors the estimation based on the generalized least squares criterion is both consistent and asymptotically efficient the detection is performed via the minimum description length mdl principle and is proved to be consistent simulation results confirming the theoretical results are included

music maximum likelihood and cramer rao bound the performance of the music and ml methods is studied and their statistical efficiency is analyzed the cramer rao bound crb for the estimation problems is derived and some useful properties of the crb covariance matrix are established the relationship between the music and ml estimators is investigated as well a numerical study is reported of the statistical efficiency of the music estimator for the problem of finding the directions of two plane waves using a uniform linear array an exact description of the results is included

approximate maximum likelihood estimators for array processing in multiplicative noise environments we consider the problem of localizing a source by means of a sensor array when the received signal is corrupted by multiplicative noise this scenario is encountered for example in communications owing to the presence of local scatterers in the vicinity of the mobile or due to wavefronts that propagate through random inhomogeneous media since the exact maximum likelihood ml estimator is computationally intensive two approximate solutions are proposed originating from the analysis of the high and low signal to noise ratio snr cases respectively first starting with the no additive noise case a very simple approximate ml aml sub 1 estimator is derived the performance of the aml sub 1 estimator in the presence of additive noise is studied and a theoretical expression for its asymptotic variance is derived its performance is shown to be close to the cramer rao bound crb for moderate to high snr next the low snr case is considered and the corresponding aml sub 2 solution is derived it is shown that the approximate ml criterion can be concentrated with respect to both the multiplicative and additive noise powers leaving out a two dimensional 2 d minimization problem instead of a four dimensional 4 d problem required by the exact ml numerical results illustrate the performance of the estimators and confirm the validity of the theoretical analysis

on a class of non linear parabolic control systems with memory effects this work establishes the local exact null controllability of non linear parabolic integrodifferential equations with non linearities under the integral sign in a bounded domain and with homogeneous dirichlet boundary conditions the control is assumed to be distributed along a subdomain of using a carleman type inequality we first prove the global exact null controllability of a linearized equation then by a fixed point method we obtain the main result for the non linear case this result asserts that when the non linearity is of class c 1 and globally lipschitz the system is null controllable

transition fronts and localized structures in bistable reaction diffusion equations models of chemical reactions that undergo subcritical hopf bifurcations and have a regime of bistability are shown to exhibit some interesting behavior when diffusively coupled waves joining a stable steady state to a periodic wavetrain are constructed for low and high thresholds for intermediate thresholds localized oscillatory regions are found these can interact at a distance and behave like weakly coupled oscillators these patterns are found in both solvable and realistic models for chemical oscillations

numerical solving nonlinear integro parabolic equations by the monotone weighted average method the paper deals with numerical solving nonlinear integro parabolic problems of volterra type based on the weighted average method a monotone iterative method is presented construction of initial upper and lower solutions is given existence and uniqueness of a solution to the nonlinear difference scheme are established an analysis of convergence rates of the monotone iterative method is given numerical experiments are presented

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

wavelet based quasilinearization method for semi linear parabolic initial boundary value problems finite difference free wavelets based quasilinearization methods developed for spibvp fisher huxley and newell whitehead segel equations solved very efficiently comparison study provided for quasilinearization based on haar and legendre wavelets proposed approach performs better than the methods in the recent literature in this paper numerical methods based on quasilinearization and haar and legendre wavelets to solve a class of semi linear parabolic initial boundary value problem spibvp have been presented the haar and legendre wavelet methods have been successfully combined with quasilinearization to solve spibvp efficiently the presented numerical scheme has been illustrated using appropriate examples including fisher equation and the obtained results show that the proposed numerical scheme is robust and easy to apply