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Maite Grau - One of the best experts on this subject based on the ideXlab platform.
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Generalized Hopf Bifurcation for Planar Vector Fields via the Inverse Integrating Factor
Journal of Dynamics and Differential Equations, 2011Co-Authors: Isaac A. Garcia, Hector Giacomini, Maite GrauAbstract:In this paper we study the maximum number of limit cycles that can bifurcate from a focus singular point p _0 of an analytic, autonomous differential system in the real plane under an analytic perturbation. We consider p _0 being a focus singular point of the following three types: non-degenerate, degenerate without characteristic directions and nilpotent. In a neighborhood of p _0 the differential system can always be brought, by means of a change to (generalized) polar coordinates ( r , θ ), to an equation over a cylinder in which the singular point p _0 corresponds to a limit cycle γ _0. This equation over the cylinder always has an inverse Integrating Factor which is smooth and non-flat in r in a neighborhood of γ _0. We define the notion of vanishing multiplicity of the inverse Integrating Factor over γ _0. This vanishing multiplicity determines the maximum number of limit cycles that bifurcate from the singular point p _0 in the non-degenerate case and a lower bound for the cyclicity otherwise. Moreover, we prove the existence of an inverse Integrating Factor in a neighborhood of many types of singular points, namely for the three types of focus considered in the previous paragraph and for any isolated singular point with at least one non-zero eigenvalue.
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Generalized Hopf Bifurcation for planar vector fields via the inverse Integrating Factor
Journal of Dynamics and Differential Equations, 2011Co-Authors: Isaac A. Garcia, Hector Giacomini, Maite GrauAbstract:In this paper we study the maximum number of limit cycles that can bifurcate from a focus singular point $p_0$ of an analytic, autonomous differential system in the real plane under an analytic perturbation. We consider $p_0$ being a focus singular point of the following three types: non-degenerate, degenerate without characteristic directions and nilpotent. In a neighborhood of $p_0$ the differential system can always be brought, by means of a change to (generalized) polar coordinates $(r, \theta)$, to an equation over a cylinder in which the singular point $p_0$ corresponds to a limit cycle $\gamma_0$. This equation over the cylinder always has an inverse Integrating Factor which is smooth and non--flat in $r$ in a neighborhood of $\gamma_0$. We define the notion of vanishing multiplicity of the inverse Integrating Factor over $\gamma_0$. This vanishing multiplicity determines the maximum number of limit cycles that bifurcate from the singular point $p_0$ in the non-degenerate case and a lower bound for the cyclicity otherwise. Moreover, we prove the existence of an inverse Integrating Factor in a neighborhood of many types of singular points, namely for the three types of focus considered in the previous paragraph and for any isolated singular point with at least one non-zero eigenvalue.
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a survey on the inverse Integrating Factor
Qualitative Theory of Dynamical Systems, 2010Co-Authors: Isaac A. Garcia, Maite GrauAbstract:The relation between limit cycles of planar differential systems and the inverse Integrating Factor was first shown in an article of Giacomini, Llibre and Viano appeared in 1996. From that moment on, many research articles are devoted to the study of the properties of the inverse Integrating Factor and its relation with limit cycles and their bifurcations. This paper is a summary of all the results about this topic. We include a list of references together with the corresponding related results aiming at being as much exhaustive as possible. The paper is, nonetheless, self-contained in such a way that all the main results on the inverse Integrating Factor are stated and a complete overview of the subject is given. Each section contains a different issue to which the inverse Integrating Factor plays a role: the integrability problem, relation with Lie symmetries, the center problem, vanishing set of an inverse Integrating Factor, bifurcation of limit cycles from either a period annulus or from a monodromic ω-limit set and some generalizations.
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the inverse Integrating Factor and the poincare map
Transactions of the American Mathematical Society, 2010Co-Authors: Isaac A. Garcia, Hector Giacomini, Maite GrauAbstract:This work is concerned with planar real analytic differential systems with an analytic inverse Integrating Factor defined in a neighborhood of a regular orbit. We show that the inverse Integrating Factor defines an ordinary differential equation for the transition map along the orbit. When the regular orbit is a limit cycle, we can determine its associated Poincare return map in terms of the inverse Integrating Factor. In particular, we show that the multiplicity of a limit cycle coincides with the vanishing multiplicity of an inverse Integrating Factor over it. We also apply this result to study the homoclinic loop bifurcation. We only consider homoclinic loops whose critical point is a hyperbolic saddle and whose Poincare return map is not the identity. A local analysis of the inverse Integrating Factor in a neighborhood of the saddle allows us to determine the cyclicity of this polycycle in terms of the vanishing multiplicity of an inverse Integrating Factor over it. Our result also applies in the particular case in which the saddle of the homoclinic loop is linearizable, that is, the case in which a bound for the cyclicity of this graphic cannot be determined through an algebraic method.
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The inverse Integrating Factor and the Poincaré map
Transactions of the American Mathematical Society, 2010Co-Authors: Isaac A. Garcia, Hector Giacomini, Maite GrauAbstract:This work is concerned with planar real analytic differential systems with an analytic inverse Integrating Factor defined in a neighborhood of a regular orbit. We show that the inverse Integrating Factor defines an ordinary differential equation for the transition map along the orbit. When the regular orbit is a limit cycle, we can determine its associated Poincaré return map in terms of the inverse Integrating Factor. In particular, we show that the multiplicity of a limit cycle coincides with the vanishing multiplicity of an inverse Integrating Factor over it. We also apply this result to study the homoclinic loop bifurcation. We only consider homoclinic loops whose critical point is a hyperbolic saddle and whose Poincaré return map is not the identity. A local analysis of the inverse Integrating Factor in a neighborhood of the saddle allows us to determine the cyclicity of this polycycle in terms of the vanishing multiplicity of an inverse Integrating Factor over it. Our result also applies in the particular case in which the saddle of the homoclinic loop is linearizable, that is, the case in which a bound for the cyclicity of this graphic cannot be determined through an algebraic method.
Jaume Llibre - One of the best experts on this subject based on the ideXlab platform.
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PHASE PORTRAITS OF THE QUADRATIC SYSTEMS WITH A POLYNOMIAL INVERSE Integrating Factor
International Journal of Bifurcation and Chaos, 2009Co-Authors: Bartomeu Coll, Antoni Ferragut, Jaume LlibreAbstract:We classify the phase portraits of all planar quadratic polynomial differential systems having a polynomial inverse Integrating Factor.
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limit cycles for singular perturbation problems via inverse Integrating Factor
Boletim da Sociedade Paranaense de Matemática, 2008Co-Authors: Jaume Llibre, Joao C Medrado, Paulo Ricardo Da SilvaAbstract:In this paper singularly perturbed vector fields X_{\varepsilon} defined in R^ 2 are discussed. The main results use the solutions of the linear partial diferential equation X_{\varepsilon} V = div( X_{\varepsilon} ) V to give conditions for the existence of limit cycles converging to a singular orbit with respect to the Hausdor distance.
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on the remarkable values of the rational first integrals of polynomial vector fields
Journal of Differential Equations, 2007Co-Authors: Antoni Ferragut, Jaume LlibreAbstract:Abstract The remarkable values for polynomial vector fields in the plane having a rational first integral were introduced by Poincare. He was mainly interested in their algebraic aspects. Here we are interested in their dynamic aspects; i.e. how they contribute to the phase portrait of the system, to its separatrices, to its singular points, etc. The relationship between remarkable values and dynamics mainly takes place through the inverse Integrating Factor.
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chapter 5 integrability of polynomial differential systems
Handbook of Differential Equations: Ordinary Differential Equations, 2004Co-Authors: Jaume LlibreAbstract:This chapter presents the planar polynomial differential systems. The notion of first integral is introduced; and the definition of Integrating Factor is discussed. The chapter introduces the notion of an exponential Factor due to Christopher. An exponential Factor appears when an invariant algebraic curve has in some sense multiplicity larger than 1.
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darboux integrability and the inverse Integrating Factor
Journal of Differential Equations, 2003Co-Authors: Javier Chavarriga, Hector Giacomini, Jaume Gine, Jaume LlibreAbstract:Abstract We mainly study polynomial differential systems of the form dx / dt = P ( x , y ), dy / dt = Q ( x , y ), where P and Q are complex polynomials in the dependent complex variables x and y , and the independent variable t is either real or complex. We assume that the polynomials P and Q are relatively prime and that the differential system has a Darboux first integral of the form H=f 1 λ 1 ⋯f p λ p exp h 1 g 1 n 1 μ 1 ⋯ exp h q g q n q μ q , where the polynomials f i and g j are irreducible, the polynomials g j and h j are coprime, and the λ i and μ j are complex numbers, when i =1,…, p and j =1,…, q . Prelle and Singer proved that these systems have a rational Integrating Factor. We improve this result as follows. Assume that H is a rational function which is not polynomial. Following to Poincare we define the critical remarkable values of H . Then, we prove that the system has a polynomial inverse Integrating Factor if and only if H has at most two critical remarkable values. Under some assumptions over the Darboux first integral H we show, first that the system has a polynomial inverse Integrating Factor; and secondly that if the degree of the system is m , the homogeneous part of highest degree of H is a multi-valued function, and the functions exp( h j / g j ) are exponential Factors for j =1,…, q , then the system has a polynomial inverse Integrating Factor of degree m +1. We also present versions of these results for real polynomial differential systems. Finally, we apply these results to real polynomial differential systems having a Darboux first integral and limit cycles or foci.
Jaume Gine - One of the best experts on this subject based on the ideXlab platform.
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integrability of planar nilpotent differential systems through the existence of an inverse Integrating Factor
Communications in Nonlinear Science and Numerical Simulation, 2019Co-Authors: Antonio Algaba, Cristobal Garcia, Jaume GineAbstract:Abstract In this work is characterized the analytic integrability problem around a nilpotent singularity for differential systems in the plane under generic conditions. The analytic integrability problem is characterized via the existence of a formal inverse Integrating Factor. The relation between the analytic integrability and the existence of an algebraic inverse Integrating Factor is also given.
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The reversibility and the center problem
Nonlinear Analysis-theory Methods & Applications, 2011Co-Authors: Jaume Gine, Susanna MazaAbstract:Abstract In this work we study the narrow relation between reversibility and the center problem and also between reversibility and the integrability problem. It is well known that an analytic system having either a non-degenerate or nilpotent center at the origin is analytically reversible or orbitally analytically reversible, respectively. In this paper we prove the existence of a smooth map that transforms an analytic system having a degenerate center at the origin with either an analytic first integral or a C ∞ inverse Integrating Factor into a reversible linear system (after rescaling the time). Moreover, if the degenerate center has an analytic or a C ∞ reversing symmetry, then the transformed system by the map also has a reversing symmetry. From the knowledge of a first integral near the center we give a procedure to detect reversing symmetries.
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Linearizable planar differential systems via the inverse Integrating Factor
Journal of Physics A: Mathematical and Theoretical, 2008Co-Authors: Hector Giacomini, Jaume Gine, Maite GrauAbstract:Our purpose in this paper is to study when a planar differential system polynomial in one variable linearizes in the sense that it has an inverse Integrating Factor which can be constructed by means of the solutions of linear differential equations. We give several families of differential systems which illustrate how the integrability of the system passes through the solutions of a linear differential equation. At the end of the work, we describe some families of differential systems which are Darboux integrable and whose inverse Integrating Factor is constructed using the solutions of a second–order linear differential equation defining a family of orthogonal polynomials.
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the nondegenerate center problem and the inverse Integrating Factor
Bulletin Des Sciences Mathematiques, 2006Co-Authors: Jaume GineAbstract:In this paper we study some aspects of the nondegenerate center problem for analytic and, in particular, for polynomial vector fields. The relation between the existence of an inverse Integrating Factor and the center problem is studied. The relationship between the conditions for a center using the Poincare formal series and the inverse Integrating Factor formal series for systems with a linear center perturbed by homogeneous polynomials is proved.
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darboux integrability and the inverse Integrating Factor
Journal of Differential Equations, 2003Co-Authors: Javier Chavarriga, Hector Giacomini, Jaume Gine, Jaume LlibreAbstract:Abstract We mainly study polynomial differential systems of the form dx / dt = P ( x , y ), dy / dt = Q ( x , y ), where P and Q are complex polynomials in the dependent complex variables x and y , and the independent variable t is either real or complex. We assume that the polynomials P and Q are relatively prime and that the differential system has a Darboux first integral of the form H=f 1 λ 1 ⋯f p λ p exp h 1 g 1 n 1 μ 1 ⋯ exp h q g q n q μ q , where the polynomials f i and g j are irreducible, the polynomials g j and h j are coprime, and the λ i and μ j are complex numbers, when i =1,…, p and j =1,…, q . Prelle and Singer proved that these systems have a rational Integrating Factor. We improve this result as follows. Assume that H is a rational function which is not polynomial. Following to Poincare we define the critical remarkable values of H . Then, we prove that the system has a polynomial inverse Integrating Factor if and only if H has at most two critical remarkable values. Under some assumptions over the Darboux first integral H we show, first that the system has a polynomial inverse Integrating Factor; and secondly that if the degree of the system is m , the homogeneous part of highest degree of H is a multi-valued function, and the functions exp( h j / g j ) are exponential Factors for j =1,…, q , then the system has a polynomial inverse Integrating Factor of degree m +1. We also present versions of these results for real polynomial differential systems. Finally, we apply these results to real polynomial differential systems having a Darboux first integral and limit cycles or foci.
Sigal Gottlieb - One of the best experts on this subject based on the ideXlab platform.
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strong stability preserving Integrating Factor two step runge kutta methods
Journal of Scientific Computing, 2019Co-Authors: Leah Isherwood, Zachary J Grant, Sigal GottliebAbstract:Problems with components that feature significantly different time scales, where the stiff time-step restriction comes from a linear component, implicit-explicit (IMEX) methods alleviate this restriction if the concern is linear stability. However, when nonlinear non-inner-product stability properties are of interest, such as in the evolution of hyperbolic partial differential equations with shocks or sharp gradients, linear inner-product stability is no longer sufficient for convergence, and so strong stability preserving (SSP) methods are often needed. Where the SSP property is needed, IMEX SSP Runge–Kutta (SSP-IMEX) methods have very restrictive time-steps. An alternative to SSP-IMEX schemes is to adopt an Integrating Factor approach to handle the linear component exactly and step the transformed problem forward using some time-evolution method. The strong stability properties of Integrating Factor Runge–Kutta methods were established in Isherwood et al. (SIAM J Numer Anal 56(6):3276–3307, 2018), where it was shown that it is possible to define explicit Integrating Factor Runge–Kutta methods that preserve strong stability properties satisfied by each of the two components when coupled with forward Euler time-stepping. It was proved that the solution will be SSP if the transformed problem is stepped forward with an explicit SSP Runge–Kutta method that has non-decreasing abscissas. However, explicit SSP Runge–Kutta methods have an order barrier of $$p=4$$, and sometimes higher order is desired. In this work we consider explicit SSP two-step Runge–Kutta Integrating Factor methods to raise the order. We show that strong stability is ensured if the two-step Runge–Kutta method used to evolve the transformed problem is SSP and has non-decreasing abscissas. We find such methods up to eighth order and present their SSP coefficients. Adding a step allows us to break the fourth order barrier on explicit SSP Runge–Kutta methods; furthermore, our explicit SSP two-step Runge–Kutta methods with non-decreasing abscissas typically have larger SSP coefficients than the corresponding one-step methods. A selection of our methods are tested for convergence and demonstrate the design order. We also show, for selected methods, that the SSP time-step predicted by the theory is a lower bound of the allowable time-step for linear and nonlinear problems that satisfy the total variation diminishing (TVD) condition. We compare some of the non-decreasing abscissa SSP two-step Runge–Kutta methods to previously found methods that do not satisfy this criterion on linear and nonlinear TVD test cases to show that this non-decreasing abscissa condition is indeed necessary in practice as well as theory. We also compare these results to our SSP Integrating Factor Runge–Kutta methods designed in Isherwood et al. (2018).
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strong stability preserving Integrating Factor two step runge kutta methods
arXiv: Numerical Analysis, 2019Co-Authors: Leah Isherwood, Zachary J Grant, Sigal GottliebAbstract:Problems that feature significantly different time scales, where the stiff time-step restriction comes from a linear component, implicit-explicit (IMEX) methods alleviate this restriction if the concern is linear stability. However, where the SSP property is needed, IMEX SSP Runge-Kutta (SSP-IMEX) methods have very restrictive time-steps. An alternative to SSP-IMEX schemes is to adopt an Integrating Factor approach to handle the linear component exactly and step the transformed problem forward using some time-evolution method. The strong stability properties of Integrating Factor Runge--Kutta methods were previously established, where it was shown that it is possible to define explicit Integrating Factor Runge-Kutta methods that preserve strong stability properties satisfied by each of the two components when coupled with forward Euler time-stepping. It was proved that the solution will be SSP if the transformed problem is stepped forward with an explicit SSP Runge-Kutta method that has non-decreasing abscissas. However, explicit SSP Runge-Kutta methods have an order barrier of p=4, and sometimes higher order is desired. In this work we consider explicit SSP two-step Runge--Kutta Integrating Factor methods to raise the order. We show that strong stability is ensured if the two-step Runge-Kutta method used to evolve the transformed problem is SSP and has non-decreasing abscissas. We find such methods up to eighth order and present their SSP coefficients. Adding a step allows us to break the fourth order barrier on explicit SSP Runge-Kutta methods; furthermore, our explicit SSP two-step Runge--Kutta methods with non-decreasing abscissas typically have larger SSP coefficients than the corresponding one-step methods.
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strong stability preserving Integrating Factor runge kutta methods
SIAM Journal on Numerical Analysis, 2018Co-Authors: Leah Isherwood, Zachary J Grant, Sigal GottliebAbstract:Strong stability preserving (SSP) Runge--Kutta methods are often desired when evolving in time problems that have two components that have very different time scales. Where the SSP property is need...
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downwinding for preserving strong stability in explicit Integrating Factor runge kutta methods
Pure and Applied Mathematics Quarterly, 2018Co-Authors: Leah Isherwood, Zachary J Grant, Sigal GottliebAbstract:Strong stability preserving (SSP) Runge-Kutta methods are desirable when evolving in time problems that have discontinuities or sharp gradients and require nonlinear non-inner-product stability properties to be satisfied. Unlike the case for L2 linear stability, implicit methods do not significantly alleviate the time-step restriction when the SSP property is needed. For this reason, when handling problems with a linear component that is stiff and a nonlinear component that is not, SSP Integrating Factor Runge--Kutta methods may offer an attractive alternative to traditional time-stepping methods. The strong stability properties of Integrating Factor Runge--Kutta methods where the transformed problem is evolved with an explicit SSP Runge--Kutta method with non-decreasing abscissas was recently established. In this work, we consider the use of downwinded spatial operators to preserve the strong stability properties of Integrating Factor Runge--Kutta methods where the Runge--Kutta method has some decreasing abscissas. We present the SSP theory for this approach and present numerical evidence to show that such an approach is feasible and performs as expected. However, we also show that in some cases the Integrating Factor approach with explicit SSP Runge--Kutta methods with non-decreasing abscissas performs nearly as well, if not better, than with explicit SSP Runge--Kutta methods with downwinding. In conclusion, while the downwinding approach can be rigorously shown to guarantee the SSP property for a larger time-step, in practice using the Integrating Factor approach by including downwinding as needed with optimal explicit SSP Runge--Kutta methods does not necessarily provide significant benefit over using explicit SSP Runge--Kutta methods with non-decreasing abscissas.
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strong stability preserving Integrating Factor runge kutta methods
arXiv: Numerical Analysis, 2017Co-Authors: Sigal Gottlieb, Zachary J Grant, Leah IsherwoodAbstract:Strong stability preserving (SSP) Runge-Kutta methods are often desired when evolving in time problems that have two components that have very different time scales. Where the SSP property is needed, it has been shown that implicit and implicit-explicit methods have very restrictive time-steps and are therefore not efficient. For this reason, SSP Integrating Factor methods may offer an attractive alternative to traditional time-stepping methods for problems with a linear component that is stiff and a nonlinear component that is not. However, the strong stability properties of Integrating Factor Runge-Kutta methods have not been established. In this work we show that it is possible to define explicit Integrating Factor Runge-Kutta methods that preserve the desired strong stability properties satisfied by each of the two components when coupled with forward Euler time-stepping, or even given weaker conditions. We define sufficient conditions for an explicit Integrating Factor Runge--Kutta method to be SSP, namely that they are based on explicit SSP Runge--Kutta methods with non-decreasing abscissas. We find such methods of up to fourth order and up to ten stages, analyze their SSP coefficients, and prove their optimality in a few cases. We test these methods to demonstrate their convergence and to show that the SSP time-step predicted by the theory is generally sharp, and that the non-decreasing abscissa condition is needed in our test cases. Finally, we show that on typical total variation diminishing linear and nonlinear test-cases our new explicit SSP Integrating Factor Runge-Kutta methods out-perform the corresponding explicit SSP Runge-Kutta methods, implicit-explicit SSP Runge--Kutta methods, and some well-known exponential time differencing methods.
Todd H Skaggs - One of the best experts on this subject based on the ideXlab platform.
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analytical solution for the advection dispersion transport equation in layered media
International Journal of Heat and Mass Transfer, 2013Co-Authors: J Perez S Guerrero, Luiz Claudio Gomes Pimentel, Todd H SkaggsAbstract:Abstract The advection–dispersion transport equation with first-order decay was solved analytically for multi-layered media using the classic integral transform technique (CITT). The solution procedure used an associated non-self-adjoint advection–diffusion eigenvalue problem that had the same form and coefficients as the original problem. The generalized solution of the eigenvalue problem for any numbers of layers was developed using mathematical induction, establishing recurrence formulas and a transcendental equation for determining the eigenvalues. The orthogonality property of the eigenfunctions was found using an Integrating Factor that transformed the non-self-adjoint advection–diffusion eigenvalue problem into a purely diffusive, self-adjoint problem. The performance of the closed-form analytical solution was evaluated by solving the advection–dispersion transport equation for two- and five-layer media test cases which have been previously reported in the literature. Additionally, a solution featuring first-order decay was developed. The analytical solution reproduced results from the literature, and it was found that the rate of convergence for the current solution was superior to that of previously published solutions.
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analytical solution for one dimensional advection dispersion transport equation with distance dependent coefficients
Journal of Hydrology, 2010Co-Authors: J Perez S Guerrero, Todd H SkaggsAbstract:Summary Mathematical models describing contaminant transport in heterogeneous porous media are often formulated as an advection–dispersion transport equation with distance-dependent transport coefficients. In this work, a general analytical solution is presented for the linear, one-dimensional advection–dispersion equation with distance-dependent coefficients. An Integrating Factor is employed to obtain a transport equation that has a self-adjoint differential operator, and a solution is found using the generalized integral transform technique (GITT). It is demonstrated that an analytical expression for the Integrating Factor exists for several transport equation formulations of practical importance in groundwater transport modeling. Unlike nearly all solutions available in the literature, the current solution is developed for a finite spatial domain. As an illustration, solutions for the particular case of a linearly increasing dispersivity are developed in detail and results are compared with solutions from the literature. Among other applications, the current analytical solution will be particularly useful for testing or benchmarking numerical transport codes because of the incorporation of a finite spatial domain.