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The paper investigates the multiple rogue wave solutions associated with the generalized Hirota-Satsuma-Ito (HSI) equation and the newly proposed extended (3+1)-dimensional Jimbo-Miwa (JM) equation with the help of a symbolic computation technique. By incorporating a direct variable transformation and utilizing Hirota’s bilinear form, multiple rogue wave structures of different orders are obtained for both generalized HSI and JM equation. The obtained bilinear forms of the proposed equations successfully investigate the 1 s t , 2 n d and 3 r d -order rogue waves. The constructed solutions are verified by inserting them into original equations. The computations are assisted with 3D graphs to analyze the propagation dynamics of these rogue waves. Physical properties of these waves are governed by different parameters that are discussed in details.

Transformation is an important means to study problems in analytical mechanics. It is often difficult to solve dynamic equations, and the use of variable transformation can make the equations easier to solve. The theory of canonical transformations plays an important role in solving Hamilton’s canonical equations. Birkhoffian mechanics is a natural generalization of Hamiltonian mechanics. This paper attempts to extend the canonical transformation theory of Hamilton systems to Birkhoff systems and establish the generalized canonical transformation of Birkhoff systems. First, the definition and criterion of the generalized canonical transformation for the Birkhoff system are established. Secondly, based on the criterion equation and considering the generating functions of different forms, six generalized canonical transformation formulas are derived. As special cases, the canonical transformation formulas of classical Hamilton’s equations are given. At the end of the paper, two examples are given to illustrate the application of the results.

Accurate prediction of blood toxin concentration during and after dialysis will greatly contribute to the determination of dialysis treatment conditions. Conventional models, namely single-compartment model and two-compartment model, have advantages and disadvantages in terms of accuracy and practical application. In this study, we attempted to derive the mathematical model that predicts blood toxin concentrations during and after dialysis, which has both accuracy and practicality. To propose the accurate model, a new two-compartment model was mathematically derived by adapting volume-averaging theory to the mass transfer around peripheral tissues. Subsequently, to propose a practical model for predicting the blood toxin concentration during dialysis, an analytical solution expressed as algebraic expression was derived by adopting variable transformation. Furthermore, the other analytical solution that predicts rebound phenomena after dialysis was also derived through similar steps. The comparisons with the clinical data revealed that the proposed analytical solutions can reproduce the behavior of the measured blood urea concentration during and after dialysis. The analytical solutions proposed as algebraic expressions will allow a doctor to estimate the blood toxin concentration of a patient during and after dialysis. The proposed analytical solutions may be useful to consider the treatment conditions for dialysis, including the rebound phenomenon.

This paper discusses joint rectangular chance or probabilistic constrained geometric programs. We present a new reformulation of the joint rectangular chance constrained geometric programs where the random parameters are elliptically distributed and pairwise independent. As this reformulation is not convex, we propose new convex approximations based on the variable transformation together with piecewise linear approximation methods. For the latter, we provide a theoretical bound for the number of segments in the worst case. Our numerical results show that our approximations are asymptotically tight.

General bright and dark soliton solutions to the partial reverse space–time nonlocal Mel’nikov equation with parity–time symmetry are constructed by the Hirota bilinear method with KP hierarchy reduction method. These solutions of arbitrary order are given in forms of Gram-type determinants. The properties of propagation and collision of analytical solution including both bright and dark solitons are discussed in details. In the end, we provide a simple variable transformation to convert the nonlocal Mel’nikov equation to a local Mel’nikov equation.

The Generalized Pareto Distribution (GPD) plays a central role in modelling heavy tail phenomena in many applications. Applying the GPD to actual datasets however is a non-trivial task. One common way suggested in the literature to investigate the tail behaviour is to take logarithm to the original dataset in order to reduce the sample variability. Inspired by this, we propose and study the Exponentiated Generalized Pareto Distribution (exGPD), which is created via log-transform of the GPD variable. After introducing the exGPD we derive various distributional quantities, including the moment generating function, tail risk measures. As an application we also develop a plot as an alternative to the Hill plot to identify the tail index of heavy tailed datasets, based on the moment matching for the exGPD. Various numerical analyses with both simulated and actual datasets show that the proposed plot works well., Comment: 24 pages, 10 figures, To appear in the proceedings of 2017 Joint Statistical Meetings, Baltimore, Maryland

In this letter, we study the periodic solutions of the equation of barotropic Friedmann–Robertson–Walker cosmologies. Using variable transformation, the original second order ordinary differential equation is converted to a planar dynamical system. We prove that the planar dynamical system has two isochronous centers under certain parameter conditions by using Picard–Fuchs equation. Consequently, we find that there exist two families of periodic solutions with equal period for the Friedmann–Robertson–Walker model.

In this paper, we propose a variable transformation way and obtain the optimal stop-loss reinsurance under value at risk (VaR) and conditional tail expectation (CTE) criteria, respectively. Let X be the initial loss of an insurer with cumulative distribution function $$F_X(x)=P(X\le x)$$ and survival function $$S_X(x)=1-F_X(x)$$. Denote a transformation variable $$Y=-\,\ln (S_X(X))$$. Firstly, we analyze properties of the variables X and Y. Then, under VaR- and CTE-optimization criteria, we provide the necessary and sufficient conditions for the optimal retention existence of Y, respectively. Further, the optimal retention of X is obtained. Some examples are given to illustrate these results.

A Sinc-collocation method was proposed by Stenger, who also gave a theoretical analysis of the method in the case of a “scalar” equation. This paper extends the theoretical results to the case of a “system” of equations. Furthermore, this paper proposes a more efficient method by replacing the variable transformation employed in Stenger’s method. The efficiency was confirmed by both a theoretical analysis and some numerical experiments. In addition to the existing and newly proposed Sinc-collocation methods, this paper also gives similar theoretical results for the Sinc-Nystrom methods proposed by Nurmuhammad et al. In terms of computational cost, the newly proposed Sinc-collocation method is the most efficient among these methods.

The following equation is considered in this paper: x ″ + ( − α + β cos ⁡ ( γ t ) ) x = 0 , where α, β and γ are real parameters and γ > 0 . This equation is referred to as Mathieu's equation when γ = 2 . The parameters determine whether all solutions of this equation are oscillatory or nonoscillatory. Our results provide parametric conditions for oscillation and nonoscillation; there is a feature in which it is very easy to check whether these conditions are satisfied or not. Parametric oscillation and nonoscillation regions are drawn to help understand the obtained results.

In this article, a Jacobi spectral Galerkin method is developed for weakly singular Volterra integral equations of the second kind. To obtain the superconvergence results, we transform the domain of integration of Volterra integral equation to the standard interval $$[-1, 1]$$ using variable transformation and function transformation. We obtain the convergence rates both in infinity and weighted $$L^2$$-norm. We prove that the Jacobi spectral iterated Galerkin method shows improvement over the Jacobi spectral Galerkin method. We improve these results further by considering the Jacobi spectral iterated multi-Galerkin method. Theoretical results are justified by the numerical results.

Mixture experiments are experiments performed using ingredients whose proportions are restricted. This restriction may result in extremely small range in terms of the mixtures, causing difficulties in model fitting arising from ill-conditioning. The choice of model form is a very important factor in the numerical stability of the information matrix. In this paper, the intercept model is compared against the slack-variable model for mixture experiments. We analyzed if it matter which component is replaced for the constant term in the intercept model, in the sense on numerical stability. We also show by numerical examples that the Correlation Criterion, presented in Kang et al. (2015), does not work for the intercept model. Next, as suggested in the literature, we use linear transformation to alleviate the numerical instability. In addition, we try four transformation methods and choose the best one for the intercept model and the slack-variable model. Finally, we compare the intercept model with the slack-variable model based mainly on the prediction accuracy and numerical stability.

This paper considers the exact solutions of the time-fractional classical Black-Scholes option pricing model through the application of a method known as He-Separation of Variable Transformation Method (HSVTM). The HSVTM combines the basic properties of the He’s polynomials, the Homo-separation variable, the modified DTM, which increases the efficiency and effectiveness of the proposed method. The proposed method is direct and straight forward. Hence, it is recommended for obtaining solutions to financial models resulting from either Ito or Stratonovich Stochastic Differential Equations (SDEs).

In this paper, the bilinear method is employed to investigate the multiple lump solutions of the generalized (3[Formula: see text]+[Formula: see text]1)-dimensional KP equation. With the aid of the variable transformation, this equation is reduced to a dimensionally reduced equation. The reduced equation can be transformed into a bilinear equation. On the basis of the bilinear forms and a special quadratic function, the 1-lump solution is constructed, which has a positive peak and two negative peaks. Three polynomial functions are introduced to derive the 3-lump solutions. The 3-lump wave has a “triangular” structure. As the parameters tend to zero, the 3-lump wave becomes the lump wave with two adjacent cusps. The 6-lump solutions are constructed. Four kinds of lump waves appear as the parameters increase. In addition, the high-order 8-lump solutions are also obtained. All the peaks of the 6-lump and the 8-lump wave tend to the same height when the parameters are sufficiently large. The dynamical behaviors of the multiple lump solutions are discussed. All the results are useful in explaining the dynamical phenomena of the generalized (3[Formula: see text]+[Formula: see text]1)-dimensional KP equation.

We discuss a few principles to guide the design of efficient Metropolis–Hastings proposals for well-behaved target distributions without deeply divided modes. We illustrate them by developing and evaluating novel proposal kernels using a variety of target distributions. Here, efficiency is measured by the variance ratio relative to the independent sampler. The first principle is to introduce negative correlation in the MCMC sample or to reduce positive correlation: to propose something new, propose something different. This explains why single-moded proposals such as the Gaussian random-walk is poorer than the uniform random walk, which is in turn poorer than the bimodal proposals that avoid values very close to the current value. We evaluate three new bimodal proposals called Box, Airplane and StrawHat, and find that they have similar performance to the earlier Bactrian kernels, suggesting that the general shape of the proposal matters, but not the specific distributional form. We propose the “Mirror” kernel, which generates new values around the mirror image of the current value on the other side of the target distribution (effectively the “opposite” of the current value). This introduces negative correlations, leading in many cases to efficiency of $\gt 100\%$ . The second principle, applicable to multidimensional targets, is that a sequence of well-designed one-dimensional proposals can be more efficient than a single $d$ -dimensional proposal. Thirdly, we suggest that variable transformation be explored as a general strategy for designing efficient MCMC kernels. We apply these principles to a high-dimensional Gaussian target with strong correlations, a logistic regression problem and a molecular clock dating problem to illustrate their practical utility.

We examine a 2-dimensional water-wave system, with gravitationally induced waves, consisting of a lower medium bound underneath by an impermeable flat bed and an upper medium bound above by an impermeable lid such that the 2 media have a free common interface. Both media have constant density and constant (non-zero) vorticity. By examining the governing equations of the system we calculate the Hamiltonian of the system in terms of its conjugate variables and perform a variable transformation to show that it has canonical Hamiltonian structure.

We provide a new approach to the analysis of the optimal joint inventory and transshipment control with uncertain capacities by employing the concept of L ? -convexity. In this approach, we use variable transformation techniques and apply two recent results to establish the L ? -concavity of the profit-to-go functions, which significantly simplifies the analysis in the existing literature. Some variations of the basic model can also be analyzed using our approach with minor modifications.

This paper focuses on delay-dependent passivity analysis for a class of memristive impulsive inertial neural networks with time-varying delays. By choosing proper variable transformation, the memristive inertial neural networks can be rewritten as first-order differential equations. The memristive model presented here is regarded as a switching system rather than employing the theory of differential inclusion and set-value map. Based on matrix inequality and Lyapunov-Krasovskii functional method, several delay-dependent passivity conditions are obtained to ascertain the passivity of the addressed networks. In addition, the results obtained here contain those on the passivity for the addressed networks without impulse effects as special cases and can also be generalized to other neural networks with more complex pulse interference. Finally, one numerical example is presented to show the validity of the obtained results.

The existence of classical solutions to the one-dimensional stationary quantum drift–diffusion model for semiconductor devices is investigated. The proof is based on an exponential variable transformation and the Leray–Schauder fixed-point theorem. Furthermore, the uniqueness of solutions to the isothermal model is proved, if the current density and the Planck constant are sufficiently small.

In this paper, by introducing an appropriate action-angle variable transformation and adopting a new estimate method, we prove the existence of Aubry-Mather sets to a class of semilinear asymmetric Duffing equations $$ x''+\alpha x^{+}-\beta x^{-}+f(t,x)=0, $$ where $x^{\pm }=\max \{\pm x,0\}$ , α and β are positive constants satisfying $$ \frac{1}{\sqrt{\alpha }}+\frac{1}{\sqrt{\beta }}=\frac{2}{\omega } $$ with $\omega \in \mathbb{R}^{+} $ , and $f(t,x)\in C^{0,1}(\mathbf{S}^{1}\times \mathbb{R})$ is a continuous function, 2π-periodic in the first variable and continuously differentiable in the second one, by virtue of a generalized version of Aubry-Mather theorem on cylinder with monotone twist assumption given by Pei. It should be pointed out that the perturbation term $f(t,x)$ satisfying some suitable growth conditions, can be allowed to be either a bounded function or an unbounded function, which differs from many existing results in the literature. Moreover, some examples are provided to illustrate the validity of the proposed results.

In this paper, by introducing an appropriate action-angle variable transformation and combining with new estimates, the sufficient conditions are obtained on the existence of Aubry-Mather sets to a class of sublinear asymmetric Duffing equations via generalized Aubry-Mather theorem on Poincare map of Duffing equations.

By virtue of a variable transformation, Whitham–Broer–Kaup (WBK) model describing the propagation of the shallow water waves is transformed into the generalized Ablowitz–Kaup–Newell–Segur (AKNS) systems, the bilinear forms of which are derived with a rational transformation. Explicit multi-soliton solutions are obtained subsequently by means of the Wronskian technique and symbolic computation. Furthermore, interactions of the solitons are investigated graphically for the WBK model and a phenomenon is revealed that the elastic and inelastic interactions occur simultaneously without affecting each other, i.e., the elastic and fission interactions coexist at the same time, and they do not disturb mutually during the collision. Our results could be useful to explain certain physical phenomena in the shallow water models.

In this paper, we are concerned with the following 2 n th-order differential equations on time scales { ( − 1 ) n y Δ 2 n ( t ) = g ( t ) f ( t , y ( t ) ) , t ∈ [ a , b ] , y Δ 2 i ( a ) − β i + 1 y Δ 2 i + 1 ( a ) = α i + 1 y Δ 2 i ( ν ) , γ i + 1 y Δ 2 i ( ν ) = y Δ 2 i ( b ) , 0 ≤ i ≤ n − 1 , where ν ∈ ( a , b ) , n ≥ 1 , β i > 0 , 1 γ i b − a + β i ν − a + β i , 0 ≤ α i b − γ i ν + ( γ i − 1 ) ( a − β i ) b − ν , i = 1 , 2 , … , n . The functions g : [ a , b ] → [ 0 , + ∞ ) and f : [ a , b ] × [ 0 , + ∞ ) → ( − ∞ , + ∞ ) are continuous, or g is singular at t = a and/or t = b . We obtain some properties and sharp estimates of the corresponding Green’s function and investigate the existence of positive solutions of the semipositone problems for 2 n -order differential equations by the use of the property of Green’s function, variable transformation and the fixed point index theorem.

PurposeThe purpose of this paper is to present a modification of the Park variable transformation for a three‐phase wye‐connected winding without neutral wire. A new physical interpretation of the winding equivalent circuit is proposed.Design/methodology/approachAn equivalent circuit representing reluctance motor stator winding is rearranged to enable easier physical interpretation of obtained voltage equation. The Park transformation and constraints resulting from Kirchhof's laws are then applied to obtain a two‐axis mathematical model of the motor.FindingsA new physical two‐phase interpretation of the voltage equation for a three‐phase wye‐connected winding without neutral wire is proposed. A novel two‐axis transformation is formulated for all variables. Compared to the Park transformation, which is the same for all variables, in the proposed transformation its matrices for currents and voltages/flux linkages are different, yet strongly interconnected.Research limitations/implicationsThe proposed transformation is formulated for a specific type of winding connection scheme. Therefore, it is limited in its application.Practical implicationsFrom the practical point of view, the proposed transformation could be very useful as it applies to the most popular stator winding connection scheme. Its main advantages are fewer number of trigonometric parameters in the matrices and measurability of all currents and voltages present in its voltage equations. It could be of special importance for electric machines with non‐sinusoidal field distribution (e.g. Brushless DC).Originality/valueThe paper presents a new type of variable transformation for three‐phase electric machines with wye‐connected windings without neutral wire. Proposed transformation combines different transformations for currents and voltages/flux linkages.

Whitham–Broer–Kaup (WBK) equations describing the propagation of shallow-water waves, with a variable transformation, are transformed into a generalized Ablowitz–Kaup–Newell–Segur system, the bilinear forms of which are obtained via the rational transformations. Employing the matrix extension and symbolic computation, we derive types of solutions of the WBK equations through the selection of different canonical matrices, including solitons, rational solutions, and complexitons. Furthermore, dynamic properties of the solutions are discussed graphically and a novel phenomenon is observed, i.e., the coexistence of the elastic–inelastic interactions without disturbing each other.

Troesch’s problem is an inherently unstable two-point boundary value problem. A new and efficient algorithm based on the variational iteration method and variable transformation is proposed to solve Troesch’s problem. The underlying idea of the method is to convert the hyperbolic-type nonlinearity in the problem into polynomial-type nonlinearities by variable transformation, and the variational iteration method is then directly used to solve this transformed problem. Only the second-order iterative solution is required to provide a highly accurate analytical solution as compared with those obtained by other analytical and numerical methods.

This paper describes a simple and efficient approach to the Troesch’s problem. In this approach, the hyperbolic nonlinear term in the equation is first converted into polynomial nonlinear terms by variable transformation, and a simple shooting method is then used directly to solve this transformed problem. The calculated results are in excellent agreement with those obtained by other analytical and numerical methods.

The N-soliton solutions for an extended generalization of Vakhnenko equation are calculated by applying the improved Hirota method and the variable transformations. The N-soliton solutions can be expressed explicitly. Furthermore, the interaction processes are discussed for the N-soliton solutions.

By the variable transformation and generalized Hirota method, exact homoclinic and heteroclinic solutions for Davey-Stewartson II (DSII) equation are obtained. For perturbed DSII equation, the existence of a global attractor is proved. The persistence of homoclinic and heteroclinic flows is investigated, and the special homoclinic and heteroclinic structure in attractors is shown.

Asymptotic expansions of the distributions of typical estimators in canonical correlation analysis under nonnormality are obtained. The expansions include the Edgeworth expansions up to order O(1/n) for the parameter estimators standardized by the population standard errors, and the corresponding expansion by Hall's method with variable transformation. The expansions for the Studentized estimators are also given using the Cornish–Fisher expansion and Hall's method. The parameter estimators are dealt with in the context of estimation for the covariance structure in canonical correlation analysis. The distributions of the associated statistics (the structure of the canonical variables, the scaled log likelihood ratio and Rozeboom's between-set correlation) are also expanded. The robustness of the normal-theory asymptotic variances of the sample canonical correlations and associated statistics are shown when a latent variable model holds. Simulations are performed to see the accuracy of the asymptotic results in finite samples.

The paper is concerned with conservative solutions to the nonlinear wave equation $u_{tt} - c(u)\big(c(u) u_x\big)_x = 0$. For an open dense set of $C^3$ initial data, we prove that the solution is piecewise smooth in the $t$-$x$ plane, while the gradient $u_x$ can blow up along finitely many characteristic curves. The analysis is based on a variable transformation introduced in [7], which reduces the equation to a semilinear system with smooth coefficients, followed by an application of Thom's transversality theorem., Comment: 25 pages

The classical thermodynamic treatment of chemical equilibria in closed systems is founded on theoretical formulations by J.W. Gibbs and Th. De Donder. These two theories represent mathematically equivalent energy variation problems, related to each other through a mass variable transformation analogous to coordinate transformations in mechanics. Associated with the change in mass variables, chemical potentials of reactive substances are defined differently in the two theories, and are subject to different sets of constraints. Traditionally, the two sets of constraints have been merged into one, by assuming that the chemical potential represents the same variable in both theories, an assumption that is formally inconsistent with the difference in chemical potential definitions. Merging of constraints is still possible if chemical potentials remain invariant in value as a manifestation of symmetry under a Gibbs – De Donder mass transformation, but such symmetry has not been investigated. Here we demonstrate chemical potential invariance by reformulating De Donder’s theory using Lagrange multipliers, and combining the resulting constraints on chemical potentials with the classical Gibbs constraints, in a phase equilibration thought experiment. A new form of the Gibbs–Duhem (GD) relation is derived in the De Donder mass system. Comparing this relation to the classical GD equation in Gibbs’ component framework, subject to chemical potential invariance, directly yields the linear operator mapping the De Donder mass variables onto the set of Gibbs component masses. Results are illustrated for a simple ion exchange reaction, in the process solving the longstanding problem of determining solid exchanger activities in ion exchanging mixtures. It is pointed out that, even though the Gibbs and De Donder formulations are mathematically equivalent, the formal structure of De Donder’s theory allows a more explicit and systematic treatment of constraints, and also defines chemical masses in a form more naturally adapted to typical experimental measurements.

Starting from a special variable transformation and with the help of an extended mapping approach, the high-order Schrodinger equation (n = 3,4) is solved. A new family of variable separation solutions with arbitrary functions is derived.

Class S m {\mathcal S}_m variable transformations with integer m m , for numerical computation of finite-range integrals, were introduced and studied by the author in the paper [A. Sidi, A new variable transformation for numerical integration, Numerical Integration IV, 1993 (H. Brass and G. Hämmerlin, eds.), pp. 359–373.] A representative of this class is the sin m \sin ^m -transformation that has been used with lattice rules for multidimensional integration. These transformations “periodize” the integrand functions in a way that enables the trapezoidal rule to achieve very high accuracy, especially with even m m . In the present work, we extend these transformations to arbitrary values of m m , and give a detailed analysis of the resulting transformed trapezoidal rule approximations. We show that, with suitable m m , they can be very useful in different situations. We prove, for example, that if the integrand function is smooth on the interval of integration and vanishes at the endpoints, then results of especially high accuracy are obtained by taking 2 m 2m to be an odd integer. Such a situation can be realized in general by subtracting from the integrand the linear interpolant at the endpoints of the interval of integration. We also illustrate some of the results with numerical examples via the extended sin m \sin ^m -transformation.

Recently, a variable transformation for integrals over smooth surfaces in ℝ3 was introduced in a paper by Atkinson. This interesting transformation, which includes a ‘‘grading’’ parameter that can be fixed by the user, makes it possible to compute these integrals numerically via the product trapezoidal rule in an efficient manner. Some analysis of the approximations thus produced was provided by Atkinson, who also stated some conjectures concerning the unusually fast convergence of his quadrature formulas observed for certain values of the grading parameter. In a recent report by Atkinson and Sommariva, this analysis is continued for the case in which the integral is over the surface of a sphere and the integrand is smooth over this surface, and optimal results are given for special values of the grading parameter. In the present work, we give a complete analysis of Atkinson’s method over arbitrary smooth surfaces that are homeomorphic to the surface of the unit sphere. We obtain optimal results that explain the actual rates of convergence, and we achieve this for all values of the grading parameter.

The paper demonstrates how a more profound theoretical understanding concerning the errors governing the accuracy of the flux in digital simulations of electrochemical experiments can be obtained by investigating how a particular discretized scheme performs if it is considered a box method, i.e. a discretized form of Flick's first law of diffusion. The behaviour of the flux is then directly reflected in the discretized equations and the accuracy of the flux can be estimated in a simple and straightforward way by investigating the underlying discretization errors. When discretising Fick's second law of diffusion the discretization errors reflect only the properties of the second-order space derivative and say only little about the accuracy of the simulated flux.

In this paper we present a second order PVT (parallel variable transformation) algorithm converging to second order stationary points for minimizing smooth functions, based on the first order PVT algorithm due to Fukushima (1998). The corresponding stopping criterion, descent condition and descent step for the second order PVT algorithm are given.

A variable transformation is utilized in the computation of Sommerfeld integrals, which arise in the analysis of antennas radiating over a dielectric half-space. The method involves a change of variable, which transforms the integrand into a rapidly converging one. Several numerical examples, which illustrate the effectiveness of the method, are presented and the results are compared with those reported in the literature. © 2003 Wiley Periodicals, Inc. Microwave Opt Technol Lett 37: 177–180, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10860

Presents an application of the boundary element method to the analysis of magnetic fields in materials for which permeability depends non‐linearly on spatial co‐ordinates. A new approach is proposed, which relates the Green’s function for non‐linearly dependent permeability to Green’s function of the Laplace equation in free space by adequate variable transformation. This can be done for very broad class magnetic permeabilities. These permeable functions, which cannot be directly used in such transformations, can be approximated by series of admissible functions.

Developing robust and efficient numerical solution methods for Richards' equation (RE) continues to be a challenge for certain problems. We consider such a problem here: infiltration into unsaturated porous media initially at static conditions for uniform and non-uniform pore size media. For ponded boundary conditions, a sharp infiltration front results, which propagates through the media. We evaluate the resultant solution method for robustness and efficiency using combinations of variable transformation and adaptive time-stepping methods. Transformation methods introduce a change of variable that results in a smoother solution, which is more amenable to efficient numerical solution. We use adaptive time-stepping methods to adjust the time-step size, and in some cases the order of the solution method, to meet a constraint on nonlinear solution convergence properties or a solution error criterion. Results for three test problems showed that adaptive time-stepping methods provided robust solutions; in most cases transforming the dependent variable led to more efficient solutions than untransformed approaches, especially as the pore-size uniformity increased; and the higher-order adaptive time integration method was robust and the most efficient method evaluated.

The Lotka–Volterra dynamical system \(\left( {\dot x_1 = ax_1 - bx_1 x_2 ,\;\dot x_2 = - cx_2 + bx_1 x_2 } \right)\) is reduced to a single second‐order autonomous ordinary differential equation by means of a new variable transformation. Formal analytic solutions are presented for this latter differential equation.

A Gegenbauer approximation is discussed. Several imbedding inequalities and inverse inequalities are obtained. Some approximation results are given. By variable transformation, differential equations on the whole line are changed to certain equations on a finite interval. Gegenbauer polynomials are used for their numerical solutions. The stabilities and convergences of proposed schemes are proved. The main idea and techniques used in this paper are also applicable to other multiple-dimensional problems in unbounded domains.

Bi-Hamiltonian structure of a periodic modified Volterra model is studied. A new Hamiltonian structure of the model is derived by use of the variable transformation between the model and the original Volterra model. This result shows the bi-Hamiltonian structure of the model.

This paper considers seasonality in Australian macroeconomic time series, emphasizing the roles of unit roots and the selection of differencing filters. The consequences of seasonal unit roots and the importance of correct variable transformation are analyzed. For certain variables, in addition to unit roots at the usual zero frequency, it is found that the hypothesis of seasonal unit roots cannot be rejected. In many macroeconomic time series, the commonly used first-differencing filter is insufficient for the removal of seasonal unit roots, and the resultant bias in the critical values of various tests remains if seasonal integration is not considered.