a new method for solving interval and fuzzy equations linear case a new approach to the solution of interval and fuzzy equations based on the generalized procedure of interval extension called interval extended zero method is proposed the central for the proposed approach is the treatment of the interval zero as an interval centered around zero it is shown that such proposition is not of heuristic nature but is the direct consequence of the interval subtraction operation some methodological problems concerned with this definition of interval zero are discussed it is shown that the resulting solution of interval linear equations based on the proposed method may be naturally treated as a fuzzy number an important advantage of a new method is that it substantially decreases the excess width effect on the other hand we show that it can be used as a reliable practical tool for solving linear interval and fuzzy equations as well as the systems of them the features of the method are illustrated by the example of the solution of the well known leontief input output problem in the interval setting

grammian solutions and pfaffianization of a non isospectral and variable coefficient kadomtsev petviashvili equation abstract in this paper we first present the grammian determinant solutions to the non isospectral and variable coefficient kadomtsev petviashvili vckp equation then by using the pfaffianization procedure of hirota and ohta a new non isospectral and variable coefficient integrable coupled system is generated moreover gramm type pfaffian solutions of the pfaffianized system are proposed

fast rate of formation of dead core for the heat equation with strong absorption and applications to fast blow up we consider the dead core problem for the semilinear heat equation with strong absorption ut uxx up with 0 p 1 and positive boundary values we investigate the dead core rate i e the rate at which the solution reaches its first zero surprisingly we find that the dead core rate is faster than the one given by the corresponding ode this stands in sharp contrast with known results for the related extinction quenching and blow up problems moreover we find that the dead core rate is actually quite unstable the ode rate can be recovered if the absorption term is replaced by a t x up for a suitable bounded uniformly positive function a t x

the fuzzy mathematics of finance this paper develops fuzzy analogues of the elementary compound interest problems in the mathematics of finance we find the fuzzy present value and fuzzy future value of fuzzy cash amounts using fuzzy interest rates over n periods where n may be crisp or fuzzy this paper also develops a method of comparing fuzzy net cash flows in order to rank fuzzy investment alternatives from best to worst

mathematical analysis of a chemical reaction with lumped temperature and strong absorption abstract we consider a mathematical model for the evolution of a single reactant and of the temperature in an isothermal catalyst the temperature is assumed to be spatially homogeneous and the absorption term is supposed to be nonlipschitzian existence and uniqueness of a regular solution is proved together with some bounds the steady state problem is also investigated and information about the dead core of the reactant are given

a model of porous catalyst accounting for incipiently non isothermal effects abstract an approximate model accounting for incipiently non isothermal effects is derived from a well known model of porous catalyst for appropriate realistic limiting values of the parameters in this limit the original model is a singularly perturbed m d reaction diffusion system and the approximate model is given by the m d heat equation with nonlinear boundary condition coupled with infinitely many ifm 2 1 d semilinear parabolic equations one for each point of the boundary of the spatial domain some limiting cases are still considered in the approximate model that lead to further simplifications

positon solutions of the sine gordon equation the concept of positons is introduced for the sine gordon equation exploiting its covariance under darboux transformations positon solutions represent potentials in the associated linear system corresponding to a vanishing reflection coefficient and a transmission coefficient identically one the mutual interaction of positons and the interaction of positons with solitons is considered

a singular reaction diffusion system modelling prey predator interactions invasion and co extinction waves abstract we consider a singular reaction diffusion system arising in modelling prey predator interactions in a fragile environment since the underlying odes system exhibits a complex dynamics including possible finite time quenching one first provides a suitable notion of global travelling wave weak solution then our study focusses on the existence of travelling waves solutions for predator invasion in such environments we devise a regularized problem to prove the existence of travelling wave solutions for predator invasion followed by a possible co extinction tail for both species under suitable assumptions on the diffusion coefficients and on species growth rates we show that travelling wave solutions are actually positive on a half line and identically zero elsewhere such a property arising for every admissible wave speeds

the n soliton solutions of the sine gordon equation with self consistent sources abstract the hierarchy of the sine gordon equation with self consistent sources is derived by using the eigenfunctions of recursion operator the bilinear form of the sine gordon equation with self consistent sources is given and the n soliton solutions are obtained through hirota method and wronskian technique respectively some novel determinantal identities are presented to treat the nonlinear term in the time evolution and finish the wronskian verifications

on using a cuts to evaluate fuzzy equations abstract consider an equation y f x1 xn where the extension principle is used to find y f x 1 x n when we subtitute fuzzy numbers x i for the xi 1 i n if x i denotes an cut of x i then we argue that for only certain simple equations will one obtain y equal to f x 1 x n where f is evaluated using interval arithmetic

fuzzy interval methods in investment risk appraisal abstract standard financial techniques neglect extreme situations and regards large market shifts as too unlikely to matter this approach may account for what occurs most of the time in the market but the picture it presents does not reflect the reality as major events happen in the residual time and investors are surprised by unexpected market movements an alternative fuzzy approach permits fluctuations well beyond the probability type of uncertainty and allows one to make fewer assumptions about the data distribution and market behaviour we suggest a fuzzy criterion and subsequently derive a measure of the risk associated with each investment opportunity and an estimate of the projects robustness towards market uncertainty the procedure is applied to 35 uk companies traded on the london stock exchange finally neural networks capabilities of approximating the fuzzy appraisal function are investigated as an initial step towards building a soft investment classifier based on the developed alternative ranking technique

negaton and positon solutions of the kdv and mkdv hierarchy we give a systematic classification and a detailed discussion of the structure motion and scattering of the recently discovered negaton and positon solutions of the korteweg de vries hierarchy there are two distinct types of negaton solutions which we label and where n 1 is the order of the wronskian used in the derivation for negatons the number of singularities and zeros is finite and they show very interesting time dependence the general motion is in the positive x direction except for certain negatons which exhibit one oscillation around the origin in contrast there is just one type of positon solution which we label for positons one gets a finite number of singularities for n odd but an infinite number for even values of n the general motion of positons is in the negative x direction with periodic oscillations negatons and positons retain their identities in a scattering process and their phase shifts are discussed we obtain a simple explanation of all phase shifts by generalizing the notions of mass and centre of mass to singular solutions finally it is shown that negaton and positon solutions of the kdv hierarchy can be used to obtain corresponding new solutions of the modified kdv hierarchy

neural net solutions to fuzzy problems the quadratic equation abstract this paper continues our research on using neural nets to solve fuzzy equations after outlining our general method of employing neural nets to solve fuzzy problems we concentrate on solving the fuzzy quadratic equation we show how neural nets can produce both real and complex fuzzy number solutions

using fuzzy set theoretic techniques to identify preference rules from interactions in the linear model an empirical study abstract this paper seeks to establish a parametric linkage between fuzzy set theoretic techniques and commonly used preference formation rules in psychology and marketing such a linkage helps to benefit both fields we accomplish this objective by using a linear model with interaction term which nests many common preference protocols conjunction fuzzy and disjunction fuzzy or counterbalance fuzzy xor and linear compensatory the resulting linear model with interactions can be employed when one has no a priori hypothesis about the individual s preference formation rule involved to determine the most likely preference rule or to test more formally the adequacy of a given rule one illustrative application studies two attribute decisions in six product categories and demonstrates differences in preference formation processes by product category a second application demonstrates how fuzzy logical operators can be applied to situations involving more than two attributes

quenching phenomena in strongly degenerate diffusion equations with strong absorption abstract we examine the degenerate parabolic equation u t u p u xx u q u 0 with boundary value u 1 in a bounded real interval where p 1 and q 1 p 1 when investigating the evolution of the dead core set u t 0 we find that the two regimes q p and q 1 p show completely different behavior finally we rule out the possibility of quenching in infinite time under certain assumptions on p q and the size of

the asymptotic justification of a nonlocal 1 d model arising in porous catalyst theory an asymptotic model of isothermal catalyst is obtained from a well known model of porous catalyst for appropriate realistic limiting values of some nondimensional parameters in this limit the original model is a singularly perturbedm d reaction diffusion system the asymptotic model consists of an ordinary differential equation coupled with a semilinear parabolic equation on a semi infinite one dimensional interval

soliton solutions for a negative order non isospectral akns equation abstract in this paper the bilinear form of a negative order non isospectral akns equation is given the soliton solutions are obtained through hirota s direct method and the wronskian technique respectively

negaton and positon solutions of the soliton equation with self consistent sources the korteweg de vries kdv equation with self consistent sources kdves is used as a model to illustrate this method we present a generalized binary darboux transformation gbdt with an arbitrary time dependent function for the kdves as well as the formula for n times repeated gbdt this gbdt provides non auto backlund transformation between two kdv equations with different degrees of sources and enables us to construct more general solutions with n arbitrary t dependent functions by taking the special t function we obtain multisoliton multipositon multinegaton multisoliton positon multinegaton positon and multisoliton negaton solutions of the kdves

the best prices of three mutually complementary merchandises in the fuzzy sense abstract let the demand functions of three mutually complementary merchandises x 1 x 2 x 3 be x 1 a 1 a 2 p 1 a 3 p 2 a 4 p 3 x 2 b 1 b 2 p 1 b 3 p 2 b 4 p 3 x 3 c 1 c 2 p 1 c 3 p 2 c 4 p 3 0 p 1 a 1 a 2 0 p 2 b 1 b 3 0 p 3 c 1 c 4 with a j 0 b j 0 c j 0 j 1 2 3 4 known the total revenue is r p 1 p 2 p 3 x 1 p 1 x 2 p 2 x 3 p 3 the monopolists can find the best prices p 1 p 2 p 3 for x 1 x 2 x 3 that make r p 1 p 2 p 3 reach its maximum in this paper we deal with a perfect competitive market and find out the best prices in the fuzzy sense to get the maximum revenue

fuzzy hierarchical analysis revisited abstract we present a new method of finding the fuzzy weights in fuzzy hierarchical analysis which is the direct fuzzification of the original method used by saaty in the analytic hierarchy process we test our new procedure in two cases where there are formulas for the crisp weights an example is presented where there are five criteria and three alternatives

positons negatons and complexitons of the mkdv equation with non uniformity terms abstract the n soliton solution of the mkdv equation with non uniformity terms is obtained through hirota method and wronskian technique we can also derive its positons negatons and complexitons by a matrix extension of the wronskian formulation

nonpropagating solitons of the variable coefficient and nonisospectral korteweg de vries equation the method of inverse scattering is extended to solve the initial value problem of a variable coefficient and nonisospectral korteweg de vries equation with time varying boundary condition one and two soliton solutions are examined in detail by an appropriate decomposition soliton interactions and asymptotic behaviors are investigated oscillating and asymptotically standing two soliton solutions are discussed

symbolic computation backlund transformation and analytic n soliton like solution for a nonlinear schrodinger equation with nonuniformity term from certain space laboratory plasmas abstract with symbolic computation a bilinear backlund transformation is presented for a nonlinear schrodinger equation with nonuniformity term from certain space laboratory plasmas and correspondingly the one soliton like solution is derived from the backlund transformation simultaneously the n soliton like solution in double wronskian form is also given besides the authors verify that the n 1 and n soliton like solutions satisfy the backlund transformation the results obtained in this paper might be valuable for the study of the nonuniform media

fuzzy initial value problem for n th order linear differential equations we present two methods of solving nth order linear differential equation that have fuzzy initial conditions for the second order constant coefficient case we show that 1 the first method always gives a solution and 2 the second solution quite often does not exist and when it does exist it may be equal to the solution determined using the first method we apply the first method to three example problems including an electrical circuit and a vibrating mass

branches of positive and free boundary solutions for some singular quasilinear elliptic problems we study the existence and multiplicity of solutions strictly positive or nonnegative having a free boundary the boundary of the set where the solution vanishes of some one dimensional quasilinear problems of eigenvalue type with possibly singular nonlinear terms

investigation on a nonisospectral fifth order korteweg de vries equation generalized from fluids in this paper a nonisospectral fifth order korteweg de vries equation generalized from fluids is investigated with symbolic computation such equation is transformed into its bilinear form through a proposed dependent variable transformation with one more parameter than those in the existing literature n soliton solutions backlund transformation and lax pair in the explicit forms are constructed based on the above results the characteristic line method is applied to discuss the features of the solitons for the nonisospectral problem i e the controllable solitonic velocities and widths four types of solitonic structures with the different solitonic velocities widths amplitudes and backgrounds are also illustrated

elementary fuzzy calculus abstract in 2 dubois and prade introduced the notion of fuzzy numbers and defined the basic operations of addition subtraction multiplication and division a slightly modified definition of fuzzy numbers was presented in 4 and in that paper a metric was defined for this family of fuzzy sets another less restrictive definition of fuzzy numbers can be found in 1 in the present paper we consider fuzzy numbers from a somewhat different perspective basically we shall view fuzzy numbers in a topological vector space setting using the customary vector space operations together with the metric given in 4 we will define differentiation and integration of fuzzy valued functions in ways that parallel closely the corresponding definitions for real differentiation and integration

pfaffianization of the generalized variable coefficient kadomtsev petviashvili equation abstract in this paper the pfaffianization procedure is applied to the generalized variable coefficient kadomtsev petviashvili vckp equation which can describe the realistic nonlinear phenomena in the fluid dynamics and plasmas using the pfaffianization procedure the coupled system for the generalized vckp equation is derived together with the wronski type pfaffian solution for this generalized coupled vckp system under certain coefficient constraint furthermore the gramm type pfaffian solution for such a coupled system is presented and verified by virtue of the pfaffian identities

solving systems of linear fuzzy equations abstract in this paper we wish to construct solutions for x to the fuzzy matrix equation ax b when the elements in a and b are triangular fuzzy numbers a is square and always non singular we argue that the previous method of solving for x based on the extension principle and regular fuzzy arithmetic should be abandoned since it too often fails to produce a solution we present six new solutions of which we show that five are identical we then adopt the common value x of these five new solutions as our solution to ax b we show that x is a fuzzy vector which is the generalization to rn of real fuzzy numbers we also show how these results pertain to and extend previous research on this problem within the area of interval analysis

mixed rational soliton solutions of two differential difference equations in casorati determinant form by reconsidering soliton solutions of the toda lattice and differential difference kdv equation in the casorati determinant form with new entries we obtain rational and mixed rational soliton solutions in the casorati determinant form all these solutions are verified by direct substitutions into bilinear equations the method used is general and can apply to other discrete systems

on the general variable coefficient kp equation with self consistent sources a new general variable coefficient kp equation with self consistent sources and its grammian solutions is obtained by using the source generation procedure moreover by applying the pfaffianization procedure a new coupled variable coefficient kp equation is constructed its wronski type pfaffian solution and gramm type pfaffian solution are given at last a new integrable system of the general variable coefficient kp equation with self consistent sources is established

solitonic interactions and double wronskian type solutions for a variable coefficient variant boussinesq model in the long gravity water waves abstract under investigation in this paper is a variable coefficient variant boussinesq vcvb model for the nonlinear and dispersive long gravity waves in shallow water traveling in two horizontal directions with varying depth connection between the vcvb model and a variable coefficient ablowitz kaup newell segur system is revealed under certain constraints with the help of the symbolic computation multi solitonic solutions in terms of the double wronskian determinant for the vcvb model are derived interactions among the vcvb solitons are discussed a novel dynamic property is observed i e the coexistence of elastic inelastic interactions

the formation of the dead core in parabolic reaction diffusion problems on donne des conditions pour l existence d une region a concentration de reactant nulle pour des problemes paraboliques non lineaires de reaction diffusion

fuzzy eigenvalues and input output analysis abstract we first generalize the perron frobenius results for nonnegative matrices to nonnegative fuzzy matrices which says that these fuzzy matrices have a dominant nonnegative fuzzy eigenvalue this result is then used to extend leontief s closed input output analysis to fuzzy economies and obtain a fuzzy model of the world s economy

solving fuzzy equations in economics and finance abstract in this paper we apply our new solution procedure for solving fuzzy equations linear non linear differential to three problems in economics and finance the three applications are 1 leontief s input output model 2 the internal rate of return and 3 a dynamic supply demand model where price and supply are governed by a system of differential equations

asymptotic behavior of the solutions of the dead core problem u u pup 0 1 1 this equation originates from the study of a class of models for the reaction diffusion processes of a gas inside a chemical reactor 2 81 from the mathematical point of view the major interest of equation 1 1 lies in the competition of the linear diffusion and the nonlinear fast p 0 t 0 with initial and boundary conditions

negatons positons rational like solutions and conservation laws of the korteweg de vries equation with loss and non uniformity terms solitons negatons positons rational like solutions and mixed solutions of a non isospectral equation the korteweg de vries equation with loss and non uniformity terms are obtained through the wronskian technique the non isospectral characteristics of the motion behaviours of some solutions are described with some figures made by using mathematica we also derive an infinite number of conservation laws of the equation

the evolution of travelling waves in fractional order autocatalysis with decay i permanent form travelling waves this paper considers a model isothermal autocatalytic reaction scheme with termination when the reaction orders m n are fractional i e 0 n and k in 0 k c where kc depends upon m and n it is further established that these wavefronts are of excitable rather than fisher kolmogorov type and have only semi infinite support that is the reaction is totally dormant ahead of the wavefront these features are in contrast to the case when m n geq 1 as studied by merkin needham and scott proc roy soc london ser a 424 1989 pp 187 209 merkin and needham j engrg math 23 1989 pp 343 356 and needham and merkin phil trans roy soc london ser a 337 1991 pp 261 274 it should be noted that the corresponding reaction diffusion equations are singular in the sense that the reaction and decay terms are not lipsch

on the non isospectral kadomtsev petviashvili equation with self consistent sources abstract a new procedure called source generation is applied to construct non isospectral soliton equations with self consistent sources as results the non isospectral kadomtsev petviashvili equation with self consistent sources kpescs and its gramm type determinant solutions are obtained furthermore the non isospectral pfaffianized kp equation with self consistent sources is constructed this coupled system can not only be reduced to the non isospectral pfaffianized kp equation but also reduced to the non isospectral kpescs

regularity for degenerate evolution equations with strong absorption abstract in this manuscript we study geometric regularity estimates for degenerate parabolic equations of p laplacian type 2 p under a strong absorption condition p u u t 0 u q in t 0 t where 0 q 1 this model is interesting because it yields the formation of dead core sets i e regions where non negative solutions vanish identically we shall prove sharp and improved parabolic c regularity estimates along the set f 0 u t u 0 t the free boundary where p p 1 q 1 1 p 1 some weak geometric and measure theoretical properties as non degeneracy positive density porosity and finite speed of propagation are proved as an application we prove a liouville type result for entire solutions a specific analysis for blow up type solutions will be done as well the results are new even for dead core problems driven by the heat operator

a modified kdv equation with self consistent sources in non uniform media and soliton dynamics two non isospectral modified kdv equations with self consistent sources are derived which correspond to the time dependent spectral parameter satisfying t and t 3 respectively gauge transformation between the first non isospectral equation corresponding to t and its isospectral counterpart is given from which exact solutions and conservation laws for the non isospectral one are easily listed besides solutions to the two nonisospectral modified kdv equations with self consistent sources are derived by means of the hirota method and the wronskian technique respectively nonisospectral dynamics and source effects including one soliton characteristics in non uniform media two solitons scattering and special behaviours related to sources for example the ghost solitons in the degenerate two soliton case are investigated analytically

perturbation analysis of fuzzy linear systems this paper deals with the perturbation analysis of fuzzy linear systems three cases of perturbation are considered a the right hand side is perturbed while the coefficient matrix remains unchanged b the coefficient matrix is perturbed while the right hand side remains unchanged and c both the coefficient matrix and the right hand side are perturbed for all of the three cases the respective relative error bounds for solutions of fuzzy linear system are derived the results are illustrated by numerical examples

solving fuzzy equations a new solution concept abstract we have previously shown 2 that many fuzzy equations do not have solutions when the solution concept is based on the extension principle we therefore introduce two new solution procedures one based on the unified extension 6 and the other based on possibility theory after we solve the non fuzzy equation for the unknown variable we show for many types of equations that 1 the two new solutions are identical 2 the solution is either a real or generalized complex fuzzy number all uncertain parameters are modeled as real fuzzy numbers and 3 the previous solution based on the extension principle when it exists is a subset of the new solution in particular we show that the fuzzy quadratic equation with real fuzzy number coefficients always has a new solution

positons for the toda lattice and related spectral problems the concept of positons is introduced for the toda lattice equation it is shown that these multiparametric oscillating and slowly decaying solutions when inserted as potentials in the finite difference schrodinger equation of the corresponding lax pair lead to a trivial s matrix the resulting eigenvalues are embedded in the continuous spectrum of this infinite jacobi matrix the singularities connected with the one positon solution are discussed and compared with those of the positons of continuous integrable models the special features of the soliton positon interaction are analysed

perturbation analysis for a class of fuzzy linear systems in this paper the perturbation analysis is presented for a class of fuzzy linear systems which can be solved by an embedding method we deduce the nonlinear upper perturbation bounds for the solutions and discuss the normwise mixed and componentwise condition numbers the results show how the perturbations of the coefficient matrix and the right hand vector impact the solutions to fuzzy linear systems

optimal fuzzy profit for price in fuzzy sense the fuzzy price function can be derived from a linear or a quadratic demand function and a linear cost function in this article we find the optimal fuzzy profit using the fuzzy price function and linear cost function the results are identical when the fuzzy case was reduced to the crisp case

solving fuzzy equations using neural nets abstract in this paper we show how a neural net can be used to solve ax c for x even though for some values of a and c there is no fuzzy arithmetic solution for x the neural net solution is identified with our new solution 6 to fuzzy equations

solving fuzzy equations we start in the next section 10 2 with looking at possible solutions to the simple fuzzy linear equation overline a cdot overline x overline b overline c we discuss three different types of solution which we have studied before in solving fuzzy equations then we present a fourth type of solution based on our fuzzy monte carlo method in section 10 2 2 this new solution is based on random fuzzy numbers in section 10 3 we look at only classical solutions to the fuzzy quadratic equation and apply our fuzzy monte carlo method to obtain new solutions then in section 10 4 we consider the fuzzy matrix equation overline a cdot overline x overline b and a number of solution types for overline x and then another solution based on fuzzy monte carlo techniques the last section contains a brief summary and our conclusions

asymptotics of the multipositon soliton τ function of the korteweg de vries equation and the supertransparency the asymptotics of the function generating the n positon m soliton solution of the korteweg de vries equation is calculated this allows to prove that solitons do not experience any phase shift in a collision with positons the positons themselves survive mutual collisions unchanged this phenomenon is called the supertransparency or super reflectionless property of the multipositon solutions the linear aspects of this phenomenon are also discussed it is demonstrated that positons acquire two additional but always finite phase shifts in collision with solitons this result admits a natural extension to any number of solitons and positons in interaction

short time behaviour of a singular solution to the heat equation with absorption on considere l evolution aux temps petits de l interface qui apparait dans le probleme aux limites u t u xx u p u o t 1 t 0 u x o 0 x 0 pour p 1 on construit une solution interieure d ordre zero consistante avec une interface mobile ou u est zero

joint solution to fuzzy programming problems we propose a new solution concept for fuzzy programming problems it is based on our new method of solving fuzzy equations 101 for simplicity we discuss in detail only fuzzy linear programming in this paper we define and obtain the basic properties of the joint solution a fuzzy vector in rn and the optimal value of the objective function a fuzzy number three examples are presented illustrating these concepts

regularity for the fully nonlinear dead core problem we establish new geometric regularity estimates for reaction diffusion equations with strong absorption terms the model is given by a fully nonlinear elliptic equation with measurable coefficients and a mu holder continuous convection term f x d 2u f u the lack of lipschitz regularity of the map u mapsto f u allows the existence of plateaus i e nonnegative solutions may vanish identically within an a priori unknown region the dead core of the solution we prove that at any touching ground point z in partial u 0 solutions are varkappa mu differentiable for a sharp value varkappa mu ge 2 and in fact varkappa 1 infty the proof is based on a new flatness improvement method we apply this new regularity estimate to establish a liouville type theorem for entire solutions to dead core problems and also to obtain measure estimates on the touching ground boundary the results obtained in this article are new even for dead core problems ruled by linear equations