Rational approximation solution of the fractional Sharma–Tasso–Olever equation

In the paper, we implement relatively new analytical techniques, the variational iteration method, the Adomian decomposition method and the homotopy perturbation method, for obtaining a rational approximation solution of the fractional Sharma–Tasso–Olever equation. The three methods in applied mathematics can be used as alternative methods for obtaining an analytic and approximate solution for different types of differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. The numerical results demonstrate the significant features, efficiency and reliability of the three approaches.     

Analysis of an elliptic-parabolic free boundary problem modelling the growth of non-necrotic tumor cord

We study a free boundary problem modelling the growth of a tumor cord in which tumor cells live around and receive nutrient from a central blood vessel. The evolution of the tumor cord surface is governed by Darcy's law together with a surface tension equation. The concentration of nutrient in the tumor cord satisfies a reaction-diffusion equation. In this paper we first establish a well-posedness result for this free boundary problem in some Sobolev-Besov spaces with low regularity by using the analytic semigroup theory. We next study asymptotic stability of the unique radially symmetric stationary solution. By making delicate spectrum analysis for the linearized problem, we prove that this stationary solution is locally asymptotically stable provided that the constant c representing the ratio between the diffusion time of nutrient and the birth time of new cells is sufficiently small.     

Solving a final value fractional diffusion problem by boundary condition regularization

The evolution process of fractional order describes some phenomenon of anomalous diffusion and transport dynamics in complex system. The equation containing fractional derivatives provides a suitable mathematical model for describing such a process. The initial boundary value problem is hard to solve due to the nonlocal property of the fractional order derivative. We consider a final value problem in a bounded domain for fractional evolution process with respect to time, which means to recover the initial state for some slow diffusion process from its present status. For this ill-posed problem, we construct a regularizing solution using quasi-reversible method. The well-posedness of the regularizing solution as well as the convergence property is rigorously analyzed. The advantage of the proposed scheme is that the regularizing solution is of the explicit analytic solution and therefore is easy to be implemented. Numerical examples are presented to show the validity of the proposed scheme.     

On the Analytic Solutions of a Special Boundary Value Problem for a Nonlinear Heat Equation in Polar Coordinates

The paper addresses a nonlinear heat equation (the porous medium equation) in the case of a power-law dependence of the heat conductivity coefficient on temperature. The equation is used for describing high-temperature processes, filtration of gases and fluids, groundwater infiltration, migration of biological populations, etc. The heat waves (waves of filtration) with a finite velocity of propagation over a cold background form an important class of solutions to the equation under consideration. A special boundary value problem having solutions of such type is studied. The boundary condition of the problem is given on a sufficiently smooth closed curve with variable geometry. The new theorem of existence and uniqueness of the analytic solution is proved.     

The multiple-scale wave equation

An analytic and numerical study of the behavior of the linear nonhomogeneous wave equation of the form ε²u_tt = Δu + tf with high wave speed (ε 1) is carried out. This study was initially motivated by meteorological observations which have indicated the presence of large spatial scale gravity waves in the neighborhood of a number of summer and winter storms, mainly from visible images of ripples in clouds in satellite photos. There is a question as to whether the presence of these waves is caused by the nearby storms. Since the linear wave equation is an approximation to the full system describing pressure waves in the atmosphere, yet is considerably more tractable, we have chosen to analyze the behavior of the linear nonhomogeneous wave equation with high wave speed. The analysis is shown to be valid in one, two, and three space dimensions. Partly because of the high wave speed, the solution is known to consist of behavior which changes on two different time scales, one rapid and one slow. Additionally, because of the presence of the nonhomogeneous forcing term tf, we show that there is a component of the solution which will vary only on a very large spatial scale. Since even the linearized wave equation can give rise to persistent large spatial scale waves under the right conditions, the implication is that certain storms could be responsible for the generation of large-scale waves. Numerical simulations in one and two dimensions confirm analytic results.     

Kinetic Equation for Quantum Fermi Gases and the Analytic Solution of Boundary Value Problems

We construct a kinetic equation describing the behavior of quantum Fermi gases with the molecule collision frequency proportional to the molecule velocity. We obtain an analytic solution of the generalized Smoluchowski problem with the temperature gradient and the mass flow velocity specified away from the surface. We find exact formulas for jumps of the gas temperature, concentration, and chemical potential. Analysis of limit cases demonstrates a transition of the quantum Fermi gas to the classical or degenerate gas.     

A Tridimensional Inverse Shaping Problem

We study the following tridimensional magnetostatic inverse shaping problem: can one find a distribution of currents around a levitating liquid metal bubble so that it takes a given shape? It leads to the resolution of an Eilonal equation on the surface of the bubble which has self-contained interest. We answer the question for closed smooth surface which are homeomorphic to a sphere. We give a necessary and sufficient condition on the data for existence and uniqueness of a C¹ solution. When the desired shape is axisymmetric and analytic, the solution is also analytic and the problem can be completely solved. A counterexample proves that not all analytic perturbations of such surface are shapable.     

On a Moving Boundary Problem Aristing in Fluidized Bed Combustion

We consider a model describing the combustion of a coal particlein a fluidized bed, in which attrition plays a dominant role.The model consists of (1) a quasi-linear elliptic equation forthe oxygen concentration, supplemented by boundary conditionson the moving surface representing the burning particle interface,(2) an evolution equation for the carbon consumption, and (3)an equation governing the motion of the interface in terms ofa specified function of the carbon consumption at the interface.We prove a global existence and uniqueness result, togetherwith a priori bounds for the solution; the existence of travellingwaves will also be established.     

Quasiexact Theory of Three-Dimensional Optical Self-Focusing

Theoretical and Mathematical Physics - We find a quasiexact three-dimensional analytic solution of the nonlinear Schrödinger equation describing the field of a stationary optical beam in an...     

Self-Similar Parabolic Optical Solitary Waves

We study solutions of the nonlinear Schrödinger equation (NLSE) with gain, describing optical pulse propagation in an amplifying medium. We construct a semiclassical self-similar solution with a parabolic temporal variation that corresponds to the energy-containing core of the asymptotically propagating pulse in the amplifying medium. We match the self-similar core through Painlevé functions to the solution of the linearized equation that corresponds to the low-amplitude tails of the pulse. The analytic solution accurately reproduces the numerically calculated solution of the NLSE.     

Model of <Emphasis Type="Italic">p</Emphasis>-Adic Random Walk in a Potential

We consider the p-adic random walk model in a potential which can be viewed as a generalization of p-adic random walk models used for describing protein conformational dynamics. This model is based on the Kolmogorov-Feller equations for the distribution function defined on the field of p-adic numbers in which the transition rate depends on ultrametric distance between the transition points as well as on function of potential violating the spatial homogeneity of a random process. This equation which will be called the equation of p-adic random walk in a potential, is equivalent to the equation of p-adic random walk with modified measure and reaction source. With a special choice of a power-law potential the last equation is shown to have an exact analytic solution. We find the analytic solution of the Cauchy problem for such equation with an initial condition, whose support lies in the ring of integer p-adic numbers.We also examine the asymptotic behaviour of the distribution function for large times. It is shown that in the limit t→∞ the distribution function tends to the equilibrium solution according to the law, which is bounded from above and below by power laws with the same exponent. Our principal conclusion is that the introduction of a potential in the model of p-adic random walk conserves the power-law behaviour of relaxation curves for large times.     

Integral equations of a two-dimensional lightguide

An integral equation describing oscillations of a two-dimensional periodic lightguide is studied. The integral operator is an analytic function of a spectral parameter. The homogenous integral equation has only the zero solution for all values of the spectral parameter expect for some isolated values. Bibliography: 9 titles. Translated fromProblemy Matematicheskogo Analiza, No. 18, 1998, pp. 78–84.     

A mixed finite-element method for fourth-order elliptic problems with variable coefficients

A new mixed finite-element method for the solution of the Dirichlet problem of fourth-order elliptic partial differential equations with variable coefficients on a convex polygon has been developed in this paper. Biharmonic and bending problems of elastic plates, for which this technique allows a simultaneous approximation to the deflection, components of the change in curvature tensor and the bending and twisting moments, are the particular cases of the problem considered in the paper. Error estimate for the mixed finite-element solution has been given.     

On the Inverse Scattering Transform for the Kadomtsev-Petviashvili Equation

The initial value problem of the Kadomtsev-Petviashvili equation for one choice of sign in the equation has been recently investigated in the literature. Here we consider the other choice of sign. We introduce suitable eigenfunctions which though bounded are not analytic in the spectral parameter. This, in contrast to the known case, prevents us from formulating the inverse problem as a nonlocal Riemann-Hilbert boundary value problem. Nevertheless a suitable formulation is given and a formal solution is constructed via a linear integral equation.     

An optimum finite element technique to calculate oxygen concentration in tumoured tissue

The aim of this paper is to find a suitable finite element method for solving a tissue problem in a semi-infinite strip with a singularity at the origin. The problem is modelled using Laplace's equation in an infinite strip, simplified to a semi-infinite strip. An analytic form of the solution is derived and this is used for comparing with the finite element solution obtained in the semi-infinite strip using novel infinite elements. Numerical results for various values of the parameters present in the solution are reported and an optimum solution is presented.     

A Study on the Propagation of Plane Stress Waves across the Thickness of a Plate by the Method of Analytic Continuation in Time

The interaction of plane tension/compression waves propagating within a plate perpendicularly to its surface is considered. The analytic solution is obtained by a modified method of characteristics for the one-dimensional wave equation used in problems on an impact of a rigid body on the surface of a plate. The displacements, velocities, and stresses in the plate are determined by the edge disturbance caused by the initial velocity and the stationary force field of masses of the striker and the plate. The method of analytic continuation in time put forward allows a stress analysis for an arbitrary time interval by using finite expressions. Contrary to a stress analysis in the frequency domain, which is commonly used in harmonic expansion of disturbances, the approach advanced allows one to analyze the solution in the case of discontinuous first derivatives of displacements without calculating jumps in summing series. A generalized closed-form solution is obtained for stresses in an arbitrary cycle n(t), which is determined by the multiplicity of the time of wave travel across the double thickness of the plate. A method of recurrent solution based on calculating the convolution of repeated integrals of the initial form of disturbance at t = 0 is elaborated. The procedure can be used for evaluating the maximum stress and the contact time in a plane impact on the surface of a plate.     

Existence of analytic solutions to analytic nonlinear q-difference equations

In this paper, some analytic approaches are formulated for the existence of analytic solutions of analytic nonlinear difference equations. From the point of view of dynamical systems, analytic solutions of such kinds of equations can be generally expressed by formal power series of exponential variables, so we are interested in considering a difference equation as a q-difference equation via a suitable coordinate transformation. After stating analytic results for formal series solutions of nonlinear q-difference equations, we also derive some results for the existence of analytic solutions to autonomous rational difference equations.     

Representation of the Stieltjes Integral in Terms of the Riemann Integral Depending on a Parameter

In this paper, we obtain a new formula for the representation of the Riemann-Stieltjes integral of a continuous function in terms of the passage to the limit with respect to the parameter in a Riemann integral depending on this parameter. The derivation of this formula is based on the study of the functional properties of the solution of the auxiliary difference equation of first order representing the weighted first difference of a given function in the form of a simple first difference of an unknown function. The result obtained can be used for the analytic and approximate calculation of Stieltjes integrals.     

New Exact Solutions of One Nonlinear Equation in Mathematical Biology and Their Properties

The classical Lie approach and the method of additional generating conditions are applied to constructing multiparameter families of exact solutions of the generalized Fisher equation, which is a simplification of the known coupled reaction–diffusion system describing spatial segregation of interacting species. The exact solutions are applied to solving nonlinear boundary-value problems with zero Neumann conditions. A comparison of the analytic results and the corresponding numerical calculations shows the importance of the exact solutions obtained for the solution of the generalized Fisher equation.     

He's homotopy perturbation method for solving the shock wave equation

In this article, we discuss the analytic solution of the fully developed shock waves. The homotopy perturbation method is used to solve the shock wave equation, which describes the flow of gases. Unlike the various numerical techniques, which are usually valid for short period of time, the solution of the presented equation is analytic for 0 < t < ∞. The results presented converge very rapidly, indicating that the method is reliable and accurate.