velocity induced numerical solutions of reaction diffusion systems on continuously growing domains reaction diffusion systems have been widely studied in developmental biology chemistry and more recently in financial mathematics most of these systems comprise nonlinear reaction terms which makes it difficult to find closed form solutions it therefore becomes convenient to look for numerical solutions finite difference finite element finite volume and spectral methods are typical examples of the numerical methods used most of these methods are locally based schemes we examine the implications of mesh structure on numerically computed solutions of a well studied reaction diffusion model system on two dimensional fixed and growing domains the incorporation of domain growth creates an additional parameter the grid point velocity and this greatly influences the selection of certain symmetric solutions for the adi finite difference scheme when a uniform square mesh structure is used domain growth coupled with grid point velocity on a uniform square mesh stabilises certain patterns which are however very sensitive to any kind of perturbation in mesh structure we compare our results to those obtained by use of finite elements on unstructured triangular elements

the diffusion brusselator equation abstract a solution is found for the reaction diffusion equation called the diffusion brusselator equation 1 using the decomposition method

some new solitonary solutions of the modified benjamin bona mahony equation abstract in this paper we use the exp function method to construct some new soliton solutions of the benjamin bona mahony and modified benjamin bona mahony equations these equations have important and fundamental applications in mathematical physics and engineering sciences the exp function method is used to find the soliton solution of a wide class of nonlinear evolution equations with symbolic computation this method provides the concise and straightforward solution in a very easier way the results obtained in this paper can be viewed as a refinement and improvement of the previously known results

construction of optimal order nonlinear solvers using inverse interpolation abstract there is a vast literature on finding simple roots of nonlinear equations by iterative methods these methods can be classified by order by the information used or by efficiency there are very few optimal methods that is methods of order 2 m requiring m 1 function evaluations per iteration here we give a general way to construct such methods by using inverse interpolation and any optimal two point method the presented optimal multipoint methods are tested on numerical examples and compared to existing methods of the same order of convergence

a new family of optimal eighth order methods with dynamics for nonlinear equations we propose a simple yet efficient family of three point iterative methods for solving nonlinear equations each method of the family requires four evaluations namely three functions and one derivative per full iteration and possesses eighth order of convergence thus the family is optimal in the sense of kung traub conjecture and has the efficiency index 1 682 which is better than that of newton s and many other higher order methods various numerical examples are considered to check the performance and to verify the theoretical results computational results including the elapsed cpu time confirm the efficient and robust character of proposed technique moreover the presented basins of attraction also confirm better performance of the methods as compared to other established methods in literature

endoproperties of modules and local duality abstract let r be any ring and n i i n i be a direct sum of finitely presented left r modules n i suppose that d n and d n i are the local duals of n and n i for each i i we prove that the lattice of endosubmodules of n is anti isomorphic to the lattices of matrix subgroups of d n and of m i i d n i as consequences n is endoartinian if and only if m or d n is endonoetherian and n is endonoetherian if and only if m or d n is pure injective we obtain in particular that if r is a krull schmidt ring and m is an indecomposable pure injective endonoetherian right r module which is the source of a left almost split morphism in mod r then m is endofinite as an application a ring r is of finite representation type if and only if every pure injective right r module is endonoetherian

a family of three point methods of optimal order for solving nonlinear equations a family of three point iterative methods for solving nonlinear equations is constructed using a suitable parametric function and two arbitrary real parameters it is proved that these methods have the convergence order eight requiring only four function evaluations per iteration in this way it is demonstrated that the proposed class of methods supports the kung traub hypothesis 1974 3 on the upper bound 2 n of the order of multipoint methods based on n 1 function evaluations consequently this class of root solvers possesses very high computational efficiency numerical examples are included to demonstrate exceptional convergence speed with only few function evaluations

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

ostrowski type methods for solving systems of nonlinear equations abstract four generalized algorithms builded up from ostrowski s method for solving systems of nonlinear equations are written and analyzed a development of an inverse first order divided difference operator for functions of several variables is presented as well as a direct computation of the local order of convergence for these variants of ostrowski s method furthermore a sequence that approximates the order of convergence is generated for the examples and it confirms in a numerical way that the order of the methods is well deduced

a non iterative method for solving non linear equations abstract in this paper using a hyperbolic tangent function tanh x 0 we develop a non iterative method to estimate a root of an equation f x 0 the problem of finding root is transformed to evaluating an integral and thus we need not take account of choosing initial guess the larger the value of the better the approximation to the root alternatively we employ the signum function sgn x instead of the hyperbolic tangent function which results in an exact formula for the root availability of the present method is shown by some numerical examples for which the traditional newton s method is not appropriate

numerical study of quantized vortex interaction in complex ginzburg landau equation on bounded domains in this paper we study numerically quantized vortex dynamics and their interaction in the two dimensional complex ginzburg landau equation cgle with a dimensionless parameter e 0 on bounded domains under either dirichlet or homogeneous neumann boundary condition we begin with a review of the reduced dynamical laws rdls for time evolution of quantized vortex centers in cgle and show how to solve these nonlinear ordinary differential equations numerically then we present efficient and accurate numerical methods for solving the cgle on either a rectangular or a disk domain under either dirichlet or homogeneous neumann boundary condition based on these efficient and accurate numerical methods for cgle and the rdls we explore rich and complicated quantized vortex dynamics and interaction of cgle with different e and under different initial physical setups including single vortex vortex pair vortex dipole and vortex lattice compare them with those obtained from the corresponding rdls and identify the cases where the rdls agree qualitatively and or quantitatively as well as fail to agree with those from cgle on vortex interaction finally we also obtain numerically different patterns of the steady states for quantized vortex lattices in the cgle dynamics on bounded domains

a new eighth order iterative method for solving nonlinear equations abstract in this paper we present an improvement of the fourth order newton type method for solving a nonlinear equation the new newton type method is shown to converge of the order eight per iteration the new method requires three evaluations of the function and one evaluation of its first derivative and therefore the new method has the efficiency index of 8 4 which is better than the well known newton type methods of lower order we shall examine the effectiveness of the new eighth order newton type method by approximating the simple root of a given nonlinear equation numerical comparisons are made with several other existing methods to show the performance of the presented method

a local point interpolation method for static and dynamic analysis of thin beams abstract the local point interpolation method lpim is a newly developed truly meshless method based on the idea of meshless local petrov galerkin mlpg approach in this paper a new lpim formulation is proposed to deal with fourth order boundary value and initial value problems for static and dynamic analysis stability free vibration and forced vibration of beams local weak forms are developed using weighted residual method locally in order to introduce the derivatives of the field variable into the interpolation scheme a technique is proposed to construct polynomial interpolation with kronecker delta function property based only on a group of arbitrarily distributed points because the shape functions so obtained possess delta function property the essential boundary conditions can be implemented with ease as in the conventional finite element method fem the validity and efficiency of the present lpim formulation are demonstrated through numerical examples of beams under various loads and boundary conditions

some sixth order variants of ostrowski root finding methods abstract in this paper we present some sixth order class of modified ostrowski s methods for solving nonlinear equations per iteration each class member requires three function and one first derivative evaluations and is shown to be at least sixth order convergent several numerical examples are given to illustrate the performance of some of the presented methods

numerical solution of transient heat conduction problems using improved meshless local petrov galerkin method an improved meshless local petrov galerkin mlpg method is presented and applied to calculate the two dimensional unsteady state heat conduction problems in this method the moving kriging interpolation is employed instead of the traditional mls approximation to construct the mlpg shape functions which possess kronecker delta function property and thus make it easy to implement essential boundary conditions and then the heaviside step function is used as the test function over a local sub domain since no mesh is needed either for integration of the local weak form or for construction of the shape functions the presently developed mlpg method is a truly meshless method several examples are performed to illustrate the accuracy and efficiency of the present method a good agreement can be found among the proposed analytical and finite element methods

surfaces generated by moving least squares methods an analysis of moving least squares m l s methods for smoothing and interpolating scattered data is presented in particular theorems are proved concerning the smoothness of interpolants and the description of m l s processes as projection methods some properties of compositions of the m l s projector with projectors associated with finiteelement schemes are also considered the analysis is accompanied by examples of univariate and bivariate problems

fast switch and spline scheme for accurate inversion of nonlinear functions the new first choice solution to kepler s equation abstract numerically obtaining the inverse of a function is a common task for many scientific problems often solved using a newton iteration method here we describe an alternative scheme based on switching variables followed by spline interpolation which can be applied to monotonic functions under very general conditions to optimize the algorithm we designed a specific ultra fast spline routine we also derive analytically the theoretical errors of the method and test it on examples that are of interest in physics in particular we compute the real branch of lambert s w y function which is defined as the inverse of xexp x and we solve kepler s equation in all cases our predictions for the theoretical errors are in excellent agreement with our numerical results and are smaller than what could be expected from the general error analysis of spline interpolation by many orders of magnitude namely by an astonishing 3 10 22 factor for the computation of w in the range w y 0 10 and by a factor 2 10 4 for kepler s problem in our tests this scheme is much faster than newton raphson method by a factor in the range 10 4 10 3 for the execution time in the examples when the values of the inverse function over an entire interval or for a large number of points are requested for kepler s equation and tolerance 10 6 rad the algorithm outperforms newton s method for all values of the number of points n 2

certain improvements of newton s method with fourth order convergence in this paper we present two new schemes one is third order and the other is fourth order these are improvements of second order methods for solving nonlinear equations and are based on the method of undetermined coefficients we show that the fourth order method is more efficient than the fifth order method due to kou et al j kou y li x wang some modifications of newton s method with fifth order covergence j comput appl math 209 2007 146 152 numerical examples are given to support that the methods thus obtained can compete with other iterative methods

uniform attractor of non autonomous three component reversible gray scott system we discuss the long time behavior of solutions for the non autonomous three component reversible gray scott system with dirichlet boundary condition on a bounded domain of space dimension n 3 when the external force is translation compact in appropriate space we obtain the existence and the structure of the uniform attractor for the processes associated to the system by constructing skew product flow on the extended phase space also we establish that there exists a unique bounded asymptotically stable solution to the system when the external forces are properly small

on a general class of multipoint root finding methods of high computational efficiency a general class of n point iterative methods for solving nonlinear equations is constructed by combining methods of newton s type and an arbitrary two point method of the fourth order of convergence it is proved that these methods have the convergence order 2 n requiring only n 1 function evaluations per iteration in this way it is demonstrated that the proposed class of methods supports the kung traub hypothesis 1974 on the upper bound 2 n of the order of multipoint methods based on n 1 function evaluations consequently this class possess as high as possible computational efficiency in the sense of the kung traub hypothesis numerical examples are included to demonstrate exceptional convergence speed with only few function evaluations

a meshless local moving kriging method for two dimensional solids abstract an improved meshless local petrov galerkin method mlpg for stress analysis of two dimensional solids is presented in this paper the mlpg method based on the moving least squares approximation is one of the recent meshless approaches however accurate imposition of essential boundary conditions in the mlpg method often presents difficulties because the mlpg shape functions does not possess the kronecker delta property in order to eliminate this shortcoming this approach uses the moving kriging interpolation instead of the traditional moving least square approximation to construct the mlpg shape functions and then the heaviside step function is used as the test function over a local sub domain in this method the essential boundary conditions can be enforced as the fem no domain integration is needed and only regular boundary integration is involved in addition the sensitivity of several important parameters of the present method is mainly studied and discussed comparing with the original meshless local petrov galerkin method the present method has simpler numerical procedures and lower computation cost the effectiveness of the present method for two dimensional solids problem is investigated by numerical examples in this paper

partial coxeter functors and right pure semisimple hereditary rings an important problem in the representation theory of artinian rings is to obtain a charcterization of rings of finite representation type see 2 7 16 201 we recall that a ring r is of finite representation type if r is both left and right artinian and there is only a finite number of isomorphism classes of indecomposable finitely generated r modules the main tools for studying the representation theory of hereditary artinian rings are partial coxeter functors and coxeter functors see 4 8 10 141 they are successfuly applied in the investigation of hereditary rings of finite representation type 8 9 12 141 in the present paper we study partial coxeter functors for hereditary artinian rings and we use them as a tool for studying right pure semisimple hereditary rings which are close to rings of finite representation type we recall that for any ring r the right pure global dimension r p gl dim r and the left pure global dimension l p gl dim r are defined 19 21 261 there is an interesting problem how to characterize rings r with r p gl dim r 0 which are exactly those right artinian rings r all of whose right r modules are algebraically compact 19 20 261 or equivalently all of whose right r modules are direct sums of modules of finite length a ring r with r p gl dim r 0 is called right pure semisimple 25 271 it is well known that any ring of finite representation type is both left and right pure semisimple 12 20 23 251 and the converse implication also holds true 2 15 201 however it is still an open question if a right pure semisimple ring r is of finite representation type or equivalently if r p gl dim r 0 implies l p gl dim r 0 see 29 a positive solution of this problem is given by auslander 3 for artin algebras he also discusses the problem for arbitrary left artinian rings see also 1181 we recall that a 195 0021 8693 81 070195 24 02 00 0

a new tool to study real dynamics in this paper the author presents a graphical tool that allows to study the real dynamics of iterative methods whose iterations depends on one parameter in an easy and compact way this tool gives the information as previous tools such as feigenbaum diagrams and lyapunov exponents for every initial point the convergence plane can be used inter alia to find the elements of a family that have good convergence properties to see how the basins of attraction changes along the elements of the family to study two point methods such as secant method or even to study two parameter families of iterative methods to show the applicability of the tool an example of the dynamics of the damped newton s method applied to a cubic polynomial is presented in this paper

turing patterns in three dimensions we investigate three dimensional turing patterns in two component reaction diffusion systems the fitzhugh nagumo equation the brusselator and the gray scott model are solved numerically in three dimensions several interconnected structures of domains as well as lamellar hexagonal and spherical domains are obtained as stable motionless equilibrium patterns the relative stability of these structures is studied analytically based on the reduction approximation the relation with the microphase separated structures in block copolymers is also discussed

the improved complex variable element free galerkin method for two dimensional schrodinger equation abstract a numerical study of two dimensional schrodinger equation is carried out using the improved complex variable element free galerkin icvefg method the icvefg method involves employment of the improved complex variable moving least squares icvmls in the element free galerkin efg procedure for numerical approximation the icvmls is used to construct trial functions for the two dimensional schrodinger equation in the form of one dimensional basis function that effectively reduces the number of unknown coefficients in this study the applicability of the icvefg method is examined through a number of numerical example problems convergence studies are carried out for these example problems by varying the number of nodes to ascertain convergent results are achieved as the number of nodes increases the stability and accuracy of the icvefg method are validated by comparing the computed results with the exact solutions

numerical study of three dimensional turing patterns using a meshless method based on moving kriging element free galerkin efg approach in this paper a numerical procedure is presented for solving a class of three dimensional turing system first we discrete the spatial direction using element free galerkin efg method based on the shape functions of moving kriging interpolation then to achieve a high order accuracy we use the fourth order exponential time differencing runge kutta etdrk4 method using this discretization for the temporal dimension we obtain an explicit scheme and do not need to solve nonlinear system of equations the efg method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the efg method test and trial functions are moving least squares approximation mls shape functions since the shape functions of moving least squares mls approximation do not have kronecker delta property we cannot implement the essential boundary condition directly also building shape functions of mls approximation is a time consuming procedure because of the mentioned reasons we employ the shape functions of moving kriging interpolation technique which have the mentioned property and less cpu time is required for building them for testing this method on three dimensional pdes we select some equations and system of pdes such as allen cahn gray scott ginzburg landau brusselator models predator prey model with additional food supply to predator several test problems are solved and numerical simulations are reported which confirm the efficiency of the proposed scheme

a multi parameter family of three step eighth order iterative methods locating a simple root a multi parameter family of three step eighth order iterative methods free from second derivatives are proposed in this paper to find a simple root of nonlinear algebraic equations convergence analysis as well as numerical experiments confirms the eighth order convergence and asymptotic error constants

the endomorphism ring of a localized coherent functor abstract letcbe a commutative artinian ring and an artinc algebra the category of coherent additive functorsa mod ab on the finitely presented right modules will be denoted by ab this category is equivalent to the free abelian category over the ring if s 0 ab is the serre subcategory of the finite length objects of ab anda ab it is proved that the endomorphism ring endab s 0 a s 0of the localized objecta s 0is a locally artinc algebra this is used to show that the krull gabriel dimension of the category ab cannot equal 1 in particular this holds for finite rings

development of a novel meshless local kriging lokriging method for structural dynamic analysis this paper presents a novel meshless approach to study the dynamic responses of structures it is called local kriging lokriging method where the kriging interpolation is employed to construct the field approximation function and meshless shape functions and the local weak form of partial differential governing equations is derived by the weighted residual technique since the shape functions constructed by this interpolation possess the delta function property even for a set of randomly distributed points the essential boundary condition can be easily imposed in implementation of local weak form the spline function with high continuity is used as the weight function no mesh is required either for integration of the local weak form or for construction of the shape functions thus the present lokriging method is a truly meshless method several numerical examinations for the free forced vibrational analysis of structures and the dynamic performance of microelectromechanical systems mems device are carried out to demonstrate the accuracy and efficiency of the lokriging method the numerical results show that the presently developed method is quite easy to implement very accurate and highly efficient for the problems considered

bifurcations and exact solutions of ac driven complex ginzburg landau equation in this paper we study the exact solutions of ac driven complex ginzburg landau equation under the given parameter conditions we apply the method of dynamical systems to analyze the dynamical behavior of the stationary solutions and their bifurcations depending on the parameters of systems all bounded exact solutions are obtained to guarantee the existence of those solutions the constraint parameter conditions are given

an optimal family of eighth order simple root finders with weight functions dependent on function to function ratios and their dynamics underlying extraneous fixed points we extend in this paper an optimal family of three step eighth order methods developed by duni etal 2011 with higher order weight functions employed in the second and third sub steps and investigate their dynamics under the relevant extraneous fixed points among which purely imaginary ones are specially treated for the analysis of the rich dynamics their theoretical and computational properties are fully investigated along with a main theorem describing the order of convergence and the asymptotic error constant as well as proper choices of special cases a wide variety of relevant numerical examples are illustrated to confirm the underlying theoretical development in addition this paper investigates the dynamics of selected existing optimal eighth order iterative maps with the help of illustrative basins of attraction for various polynomials

exponential time differencing for stiff systems we develop a class of numerical methods for stiff systems based on the method of exponential time differencing we describe schemes with second and higher order accuracy introduce new runge kutta versions of these schemes and extend the method to show how it may be applied to systems whose linear part is nondiagonal we test the method against other common schemes including integrating factor and linearly implicit methods and show how it is more accurate in a number of applications we apply the method to both dissipative and dispersive partial differential equations after illustrating its behavior using forced ordinary differential equations with stiff linear parts

auslander reiten components over pure semisimple hereditary rings abstract let r be a hereditary indecomposable left pure semisimple ring we show that r has finite representation type if and only if a certain finitely presented module is endofinite namely the tilting and cotilting module w studied in l angeleri hugel 2007 2 we then apply the tilting and the cotilting functors to study the endomorphism ring of w and its auslander reiten components finally we transfer this information to the category of right r modules

some variants of ostrowski s method with seventh order convergence in this paper we present a class of new variants of ostrowski s method with order of convergence seven per iteration the new methods require three evaluations of the function and one evaluation of its first derivative and therefore this class of methods has the efficiency index equal to 1 627 numerical tests verifying the theory are given and multistep iterations based on the present methods are developed

a family of optimal three point methods for solving nonlinear equations using two parametric functions abstract using an interactive approach which combines symbolic computation and taylor s series a wide family of three point iterative methods for solving nonlinear equations is constructed these methods use two suitable parametric functions at the second and third step and reach the eighth order of convergence consuming only four function evaluations per iteration this means that the proposed family supports the kung traub hypothesis 1974 on the upper bound 2 m of the order of multipoint methods based on m 1 function evaluations providing very high computational efficiency different methods are obtained by taking specific parametric functions the presented numerical examples demonstrate exceptional convergence speed with only few function evaluations

eighth order family of iterative methods for nonlinear equations and their basins of attraction abstract we present a new family of eighth order methods for solving nonlinear equations the order of convergence of the considered methods is proved and corresponding asymptotic error constants are expressed in terms of four parameters numerical examples obtained using mathematica with high precision arithmetic demonstrate convergence and efficacy of our family of methods for some combinations of parameter values the new eighth order methods produce very good results on tested examples compared to the results produced by some of the eighth order methods existing in the related literature an exploration of the relevant dynamics of the proposed methods is presented along with illustrative basins of attraction for various polynomials

interpolatory multipoint methods with memory for solving nonlinear equations abstract a general way to construct multipoint methods for solving nonlinear equations by using inverse interpolation is presented the proposed methods belong to the class of multipoint methods with memory in particular a new two point method with memory with the order 5 17 2 4 562 is derived computational efficiency of the presented methods is analyzed and their comparison with existing methods with and without memory is performed on numerical examples it is shown that a special choice of initial approximations provides a considerably great accuracy of root approximations obtained by the proposed interpolatory iterative methods

an unconditionally stable hybrid numerical method for solving the allen cahn equation we present an unconditionally stable second order hybrid numerical method for solving the allen cahn equation representing a model for antiphase domain coarsening in a binary mixture the proposed method is based on operator splitting techniques the allen cahn equation was divided into a linear and a nonlinear equation first the linear equation was discretized using a crank nicolson scheme and the resulting discrete system of equations was solved by a fast solver such as a multigrid method the nonlinear equation was then solved analytically due to the availability of a closed form solution various numerical experiments are presented to confirm the accuracy efficiency and stability of the proposed method in particular we show that the scheme is unconditionally stable and second order accurate in both time and space

new family of seventh order methods for nonlinear equations abstract a family of seventh order iterative methods for the solution of nonlinear equations is presented the new methods are based on king s fourth order methods and without using the second derivatives per iteration the new methods require three evaluations of the function and one evaluation of its first derivative therefore this family of methods has efficiency index equal to 1 627 numerical comparisons are made with several other existing methods to show the performance of the presented methods

a new family of eighth order iterative methods for solving nonlinear equations a family of eighth order iterative methods with four evaluations for the solution of nonlinear equations is presented kung and traub conjectured that an iteration method without memory based on n evaluations could achieve optimal convergence order 2 n 1 the new family of eighth order methods agrees with the conjecture of kung traub for the case n 4 therefore this family of methods has efficiency index equal to 1 682 numerical comparisons are made with several other existing methods to show the performance of the presented methods

an improvement to ostrowski root finding method an improvement to the iterative method based on the ostrowski one to compute nonlinear equation solutions which increases the local order of convergence is suggested the adaptation of a strategy presented here gives a new iteration function with an additional evaluation of the function it also shows a smaller cost if we use adaptive multi precision arithmetic the numerical results computed using this system with a floating point system representing 200 decimal digits support this theory

a meshless local kriging method for large deformation analyses in this paper a well developed meshless numerical technique local kriging lokriging method is used to develop the lokriging formulation for the large deformation problems which are geometrically nonlinear this is the first work for the geometrically nonlinear analyses by this meshless local weak form method in the lokriging method the kriging interpolation is employed to construct the meshless shape functions the spline function with high continuity is used as the weight function the discrete equations of the large deformation analysis for two dimensional solids are obtained using the local weak forms and based on the total lagrangian tl approach the presently developed lokriging method is a truly meshless method as it does not require the mesh either for the construction of the shape functions or for the integration of the local weak form several numerical examples of the two dimensional geometrically nonlinear analysis are presented to illustrate the performance of the present lokriging method the present nonlinear lokriging formulation is also used for the large deformation analysis of the microelectromechanical systems mems device the numerical results are compared with the experimental results very good results are obtained all these applications have demonstrated that the present lokriging method is very effective for the large deformation analyses because it avoids all mesh distortion issues

a semi analytical fourier spectral method for the allen cahn equation in recent years fourier spectral methods have been widely used as a powerful tool for solving phase field equations to improve its effectiveness many researchers have employed stabilized semi implicit fourier spectral sifs methods which allow a much larger time step than a usual explicit scheme our mathematical analysis and numerical experiments however suggest that an effective time step is smaller than a time step specified in the sifs schemes in consequence the sifs scheme is inaccurate for a considerably large time step and may lead to incorrect morphologies in phase separation processes in order to remove the time step constraint and guarantee the accuracy in time for a sufficiently large time step we present a first and a second order semi analytical fourier spectral safs methods for solving the allen cahn equation the core idea of the methods is to decompose the original equation into linear and nonlinear subequations which have closed form solutions in the fourier and physical spaces respectively both the first and the second order methods are unconditionally stable and numerical experiments demonstrate that our proposed methods are more accurate than the stabilized semi implicit fourier spectral method

on the sparsity of representations of rings of pure global dimension zero it is shown that the rings r all of whose left modules are direct sums of finitely generated modules satisfy the following finiteness condition for each positive integer n there are only finitely many isomorphism types of a indecomposable left r modules of length n b finitely presented indecomposable right r modules of length n c indecomposable right r modules having minimal projective resolutions with n relations moreover our techniques yield a very elementary proof for the fact that the presence of the above decomposability hypothesis for both left and right r modules entails finite representation type

second order implicit explicit scheme for the gray scott model a second order scheme for the gray scott gs model used to describe the pattern formation is studied the linear part of the gs equation for the time derivative and the viscous terms is discretized implicitly while the other or nonlinear part of the gs equation explicitly galerkin finite element approximation methods are presented and analyzed as well as methods for solving the resulting system of equations the optimal l 2 norm error estimates are derived numerical experiments are presented

exact homoclinic and heteroclinic solutions of the gray scott model for autocatalysis in this paper we obtain explicit nontrivial stationary patterns in the one dimensional gray scott model for cubic autocatalysis involved in the reaction are two chemicals a and b whose diffusion coefficients are denoted by da and db respectively the chemical a is fed into the system at a rate kf reacts with the catalyst b at a rate k1 and the catalyst decays at a rate k2 if these parameters obey the relation kf da k2 db then for appropriate values of the rate constants we present explicit expressions for two families of pulses and one kink we also show that if is only satisfied approximately these families of pulses are preserved and there exists a smooth branch of kinks leading from the explicit one obtained when is satisfied we determine the local behavior of this branch near the explicit kink

endofinite modules and pure semisimple rings let r be a right pure semisimple ring i e a ring r such that every right r module is a direct sum of finitely generated modules it is proved that r is of finite representation type if and only if every finitely presented indecomposable right r module is endofinite if and only if every finitely presented right r module has a left artinian endomorphism ring as applications we obtain an alternative proof of the pure semisimplicity conjecture for pi rings and new criteria for a right pure semisimple ring to be of finite representation type

a family of modified ostrowski methods with accelerated sixth order convergence based on ostrowski fourth order multipoint method we derive a one parameter family of sixth order methods for solving equations each member of the family requires three evaluations of the given function and one evaluation of its derivative per iteration numerical examples are presented and performance is compared with ostrowski method

generating function method for constructing new iterations abstract in this paper we propose a generating function method for constructing new two and three point iterations with p 3 p 8 order of convergence this approach allows us to derive a new family of the optimal order iterative methods that include well known methods as special cases the necessary and sufficient conditions for pth order convergence of the proposed iterations are given in terms of parameters n and n we also propose some generating functions for n and n we give the extension of a class of optimal fourth order jarratt s type iterations with a 2 3 we develop a unified representation of all optimal eighth order methods several numerical results are given to demonstrate the efficiency and the performance of the presented methods and compare them with some other existing methods

indecomposable modules over pure semisimple hereditary rings abstract if r is a hereditary left artinian ring then r is left pure semisimple if and only if the family r ind of all finitely generated indecomposable left r modules has a unique ext injective partition r ind u this partition is used to give a complete description of the distribution of all indecomposable modules over a left pure semisimple hereditary indecomposable ring r of infinite representation type more precisely r ind is the disjoint union of the countable set of all preinjective modules and the finite set of all preprojective modules and countable sets of auslander reiten components of the form k u k for all limit ordinals constructed from the ext injective partition of r ind in particular we show that an indecomposable left r module m is not the source of a left almost split morphism in r mod if and only if m belongs to u where is an infinite limit ordinal and the direct sum of modules in u is not endofinite for each infinite limit ordinal moreover the endomorphism ring of each indecomposable left r module is a division ring

three step iterative methods with eighth order convergence for solving nonlinear equations a family of eighth order iterative methods for the solution of nonlinear equations is presented the new family of eighth order methods is based on king s fourth order methods and the family of sixth order iteration methods developed by chun et al per iteration the new methods require three evaluations of the function and one evaluation of its first derivative therefore this family of methods has the efficiency index which equals 1 682 kung and traub conjectured that a multipoint iteration without memory based on n evaluations could achieve optimal convergence order 2 n 1 thus we provide a new example which agrees with the conjecture of kung traub for n 4 numerical comparisons are made to show the performance of the presented methods

basin attractors for various methods abstract there are many methods for the solution of a nonlinear algebraic equation the methods are classified by the order informational efficiency and efficiency index here we consider other criteria namely the basin of attraction of the method and its dependence on the order we discuss several methods of various orders and present the basin of attraction for several examples it can be seen that not all higher order methods were created equal newton s halley s murakami s and neta johnson s methods are consistently better than the others in two of the examples neta s 16th order scheme was also as good

eighth order methods with high efficiency index for solving nonlinear equations in this paper we derive a new family of eighth order methods for solving simple roots of nonlinear equations by using weight function methods per iteration these methods require three evaluations of the function and one evaluation of its first derivative which implies that the efficiency indexes are 1 682 numerical comparisons are made to show the performance of the derived methods as shown in the illustration examples

three step iterative methods with optimal eighth order convergence in this paper based on ostrowski s method a new family of eighth order methods for solving nonlinear equations is derived in terms of computational cost each iteration of these methods requires three evaluations of the function and one evaluation of its first derivative so that their efficiency indices are 1 682 which is optimal according to kung and traub s conjecture numerical comparisons are made to show the performance of the new family

a key module over pure semisimple hereditary rings abstract let r be a hereditary indecomposable left pure semisimple ring inspired by i reiten c m ringel infinite dimensional representations of canonical algebras canad j math 58 2006 180 224 we investigate the perfect cotorsion pair c d in r mod generated by the preinjective component q we show that there is a finitely generated product complete tilting and cotilting left r module w such that c cogen w and d gen w the finite subcategory w of r mod given by the indecomposable summands of w stores important information on r for example if we assume r of infinite representation type then by b zimmermann huisgen strong preinjective partitions and representation type of artinian rings proc amer math soc 109 1990 309 322 there are non preinjective indecomposable modules occurring as direct summands of products of preinjective modules and it turns out that w is precisely the class of such modules moreover we prove that r has finite representation type if and only if every module in w is source of a left almost split map in r mod finally we address the question when w is endofinite

on optimal fourth order iterative methods free from second derivative and their dynamics in this paper new fourth order optimal root finding methods for solving nonlinear equations are proposed the classical jarratt s family of fourth order methods are obtained as special cases we then present results which describe the conjugacy classes and dynamics of the presented optimal method for complex polynomials of degree two and three the basins of attraction of existing optimal methods and our method are presented and compared to illustrate their performance

turing instability conditions for growing domains with divergence free mesh velocity abstract questions have been asked from a theoretical point of view about the effects of domain growth on turing a m turing the chemical basis of morphogenesis phil trans roy soc lond b 237 1952 37 72 diffusion driven instability analysis for reaction diffusion systems on continuously growing domains the increase in computational power and tools has given the scientific community an almost immediate vast amount of computational data on pattern transitions peak splitting peak insertion period doubling circular patterns etc on growing domains but with little knowledge of the theoretical justifications of the role played by domain growth in pattern formation this paper derives the conditions for turing diffusion driven instability on growing domains whose domain velocity is divergence free in particular we establish the result that the conditions for diffusion driven instability growing domains whose domain velocity is divergence free are equivalent to those derived on a fixed domain for this type of growth the advection terms have no effect on the standard diffusion driven instability conditions under these conditions travelling wave solutions are obtained this result has implications in the selection of parameter values as these are selected such that they belong to the turing space our results do not however generalise to arbitrary growth protocols such as exponential linear or logistic growths since the divergence of the mesh velocity is time dependent