We present the solution of the classical problem of the heat equation formulated in the interior of an equilateral triangle with Dirichlet boundary conditions. This solution is expressed as an integral in the complex Fourier space, i. By choosing Dirichlet data so that their integral transforms can be computed explicitly, we show that the solution is expressed in terms of an integral whose integrand decays exponentially as.
Hence, it is possible to evaluate this integral numerically in an efficient and straightforward manner. Other types of boundary value problems, including the Neumman and Robin problems, can be solved similarly. In particular, we investigate the so-called linearizable boundary conditions, which in this case are of Robin type. Furthermore, we use a particular solution to verify explicitly all the steps needed for the solution of a well-posed problem. There exists a distinctive class of physically significant evolution PDEs in one spatial dimension which can be treated analytically.
A prototypical example of this class which is referred to as integrable is the Korteweg-de Vries equation. A particularly powerful aspect of the IST is its ability to predict the large t behaviour of the solution. One of the most important developments in this area has been the generalization of the IST from initial to initial-boundary value IBV problems formulated on the half-line. A major complication of IBV problems is that the computation of the shapes and speeds of the solitons involves the solution of a nonlinear Volterra integral equation.
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However, for a certain class of boundary conditions, called linearizable, this complication can be bypassed and the relevant computation is as effective as in the case of the problem on the line. Here, after reviewing the general theory for KdV, we analyse three different types of linearizable boundary conditions.
For each of these cases, by computing explicitly the shapes and the speeds of the asymptotic solitons, we elucidate the influence of the boundary. The stationary, axisymmetric reduction of the vacuum Einstein equations, the so-called Ernst equation, is an integrable nonlinear PDE in two dimensions. This analysis involves, in general, the solution of a nonlinear problem; however, for certain boundary value problems called linearizable, it is possible to determine the unknown boundary values using only linear operations.
Here, we employ the above methodology for the investigation of certain boundary value problems for the elliptic version of the Ernst equation. For this problem, the main novelty is the occurence of the spectral parameter in the form of a square root and this necessitates the introduction of a two-sheeted Riemann surface for the formulation of the relevant RH problem. As a concrete application of the general formalism, it is shown that the particular boundary value problem corresponding to the physically significant case of a rotating disk is a linearizable boundary value problem.
In this way the remarkable results of Neugebauer and Meinel are recovered.
Boundary value problems for integrable nonlinear evolution PDEs formulated on the half-line can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. Two of these functions depend on the initial data, whereas the other two depend on all boundary values.
The most difficult step of the new method is the characterization of the latter two spectral functions in terms of the given initial and boundary data, i. For certain boundary conditions, called linearizable, this can be achieved simply using algebraic manipulations. Here, we present an effective characterization of the spectral functions in terms of the given initial and boundary data for the general case of non-linearizable boundary conditions.
Here, we first present an effective characterization of the spectral functions in terms of the given initial and boundary data for the general case of non-linearizable boundary conditions. This characterization is based on the analysis of the so-called global relation and on the introduction of the so-called Gelfand-Levitan-Marchenko representations of the eigenfunctions defining the spectral functions. Boundary value problems for integrable nonlinear evolution PDEs formulated on the finite interval can be analyzed by the unified method introduced by one of the authors and used extensively in the literature.
Two of these functions depend on the initial data, whereas the other four depend on all boundary values.
The most difficult step of the new method is the characterization of the latter four spectral functions in terms of the given initial and boundary data, i. Here, we present an effective characterization of the spectral functions in terms of the given initial and boundary data. We present two different characterizations of this problem. The other is based on the analysis of the global relation and on the introduction of the so-called Gelfand-Levitan-Marchenko representations of the eigenfunctions defining the relevant spectral functions.
We also show that these two different characterizations are equivalent and that in the limit when the length of the interval tends to infinity, the relevant formulas reduce to the analogous formulas obtained recently for the case of boundary value problems formulated on the half-line. The Unified Transform provides a novel method for analyzing boundary value problems for linear and for integrable nonlinear PDEs. One of the advantages of this new technique is that it actually yields directly the scattering amplitude.
Details are presented for the modified Helmholtz equation in the exterior of a square. We study boundary value problems posed in a semistrip for the elliptic sine-Gordon equation, which is the paradigm of an elliptic integrable PDE in two variables. We use the method introduced by one of the authors, which provides a substantial generalization of the inverse scattering transform and can be used for the analysis of boundary as opposed to initial-value problems. We first express the solution in terms of a 2 by 2 matrix Riemann-Hilbert problem whose "jump matrix" depends on both the Dirichlet and the Neumann boundary values.
For a well posed problem one of these boundary values is an unknown function. This unknown function is characterised in terms of the so-called global relation, but in general this characterisation is nonlinear. We then concentrate on the case that the prescribed boundary conditions are zero along the unbounded sides of a semistrip and constant along the bounded side. This corresponds to a case of the so-called linearisable boundary conditions, however a major difficulty for this problem is the existence of non-integrable singularities of the function q y at the two corners of the semistrip; these singularities are generated by the discontinuities of the boundary condition at these corners.
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Motivated by the recent solution of the analogous problem for the modified Helmholtz equation, we introduce an appropriate regularisation which overcomes this difficulty. Furthermore, by mapping the basic Riemann-Hilbert problem to an equivalent modified Riemann-Hilbert problem, we show that the solution can be expressed in terms of a 2 by 2 matrix Riemann-Hilbert problem whose jump matrix depends explicitly on the width of the semistrip L , on the constant value d of the solution along the bounded side, and on the residues at the given poles of a certain spectral function denoted by h.
The determination of the function h remains open. The phenomenon of stimulated Raman scattering SRS can be described by three coupled PDEs which define the pump electric field, the Stokes electric field, and the material excitation as functions of distance and time. In the transient limit these equations are integrable, i. Here we study this transient limit. The relevant physical problem can be formulated as an initial-boundary value IBV problem where both independent variables are on a finite domain. A general method for solving IBV problems for integrable equations has been introduced recently.
Using this method we show that the solution of the equations describing the transient SRS can be obtained by solving a certain linear integral equation. It is interesting that this equation is identical to the linear integral equation characterizing the solution of an IBV problem of the sine-Gordon equation in light-cone coordinates. This integral equation can be solved uniquely in terms of the values of the pump and Stokes fields at the entry of the Raman cell.
This result is consistent with both numerical simulations and experimental observations. We also discuss briefly the effect of frequency mismatch between the pump and the Stokes electric fields.
Bothe : Periodic solutions of a nonlinear evolution problem from heterogeneous catalysis
It provides a multidimensional generalisation of the renowned KdV equation. The analysis makes crucial use of the so-called d -bar formalism, as well as of the so-called global relation. In this work, we study the analogous problem for the Kadomtsev-Petviashvili I equation; in particular, through the simultaneous spectral analysis of the associated Lax pair via a d-bar formalism, we are able to obtain an integral representation for the solution, which involves certain transforms of all the initial and the boundary values, as well as an identity, the so-called global relation, which relates these transforms in appropriate regions of the complex spectral plane.
We use the heat equation as an illustrative example to show that the unified method introduced by one of the authors can be employed for constructing analytical solutions for linear evolution partial differential equations in one spatial dimension involving non-separable boundary conditions as well as non-local constraints. Furthermore, we show that for the particular case in which the boundary conditions become separable, the unified method provides an easier way for constructing the relevant classical spectral representations avoiding the classical spectral analysis approach.
We note that the unified method always yields integral expressions which, in contrast to the series or integral expressions obtained by the standard transform methods, are uniformly convergent at the boundary. Thus, even for the cases that the standard transform methods can be implemented, the unified method provides alternative solution expressions which have advantages for both numerical and asymptotic considerations. The former advantage is illustrated by providing the numerical evaluation of typical boundary value problems.
In the flat-bottom case, this formulation consists of the Bernoulli equation, as well as of a non-local equation. The variable-bottom case, which now involves two non-local equations, was outlined but not explored in the above paper. Here, the variable-bottom formulation is addressed in more detail. First, it is shown that in the weakly nonlinear, weakly dispersive regime, the above system of three equations can be reduced to a system of two equations.
Second, by combining the novel non-local formulation of the above authors with conformal mappings, it is shown that in the two-dimensional case, it is possible to obtain a system of two equations without any asymptotic approximations. Furthermore, for the weakly nonlinear, weakly dispersive regime, the nonlinear equations are simpler than the equations obtained without conformal mappings, since they contain lower order derivatives for the terms involving the bottom variable.
A new transform method for solving initial boundary value problems for linear and for integrable nonlinear PDEs in two independent variables is introduced.
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This unified method is based on the fact that linear and integrable nonlinear equations have the distinguished property that they possess a Lax pair formulation. The implementation of this method involves performing a simultaneous spectral analysis of both parts of the Lax pair and solving a Riemann-Hilbert problem. In addition to a unification in the method of solution, there also exists a unification in the representation of the solution.
It is also shown that appropriate deformations of the Lax pairs of linear equations can be used to construct Lax pairs for integrable nonlinear equations.
Unified Transform Method
A new method for studying boundary value problems for linear and for integrable nonlinear partial differential equations PDE's in two dimensions is reviewed. This method provides a unification as well as a significant extension of the following three seemingly different topics: a The classical integral transform method for solving linear PDE's and several of its variations such as the Wiener-Hopf technique. The detailed implementation of the method is presented for: a An arbitrary linear dispersive evolution equation on the half line.
In addition, several other applications are briefly considered. The possible extension of this method to multidimensions is also discussed. It consists of three algorithmic steps. These boundary values are in general related, thus only some of them can be prescribed. As an illustration, the following integral representations are obtained: a q x, t for a general dispersive evolution equation of order n in a domain bounded by a linearly moving boundary; b q x,y for the Laplace, modified Helmholtz and Helmholtz equations in a convex polygon.
This method is simple to implement and yet it yields integral representations which are particularly convenient for computing the long time asymptotics of the solution. Under the assumption that there exist functions such that the spectral functions satisfy a certain global algebraic relation, prove that the function q x,t is defined for all , it satisfies the given nonlinear PDE, and furthermore that.
In general this involves the solution of a nonlinear Volterra integral equation which is shown to have a global solution. We review a new method for studying boundary value problems for integrable PDEs in two dimensions. The starting point of the method is formulating the given PDE as the condition that an appropriate differential 1-form W x 1 ,x 2 ,k , k complex, is exact. The funcamental properties of an exact form W are the existence of an 0-form, and the vanishing of the integral of W around a closed contour.
The spectral analysis of the associated o-form gives rise to a Riemann-Hilbert RH problem with explicit exponential x 1 ,x 2 dependence, while the vanishing of the integral of W around the boundary of the domain gives rise gives rise to a global relation.
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