# Applications

This page gives a list of works on the topic of Fokas' method, organised thematically. The list is growing and we hope to make it comprehensive soon. In the mean time, if you know of any omissions please send us an email with as much information as possible on the work.

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## Problems in two dimensions (either 1+1 or 2+0)

### Linear evolution PDEs formulated on the half-line

The novel integral representations obtained by the new method have both analytical and numerical advantages in comparison with the classical integral and series representations: (i) They are uniformly convergent at the boundaries; analytically, this makes it easier to prove rigorously the validity of such representations without the apriori assumption of existence, whereas numerically, using appropriate contour deformations, it makes it possible to obtain integrands which decay exponentially as |k| →∞ and this leads to efficient numerical computations [FF2008a], [KP2009a], [Vet2010a]. (ii) These representations retain their form even for more complicated boundary conditions, whereas the classical representations involve infinite series over a spectrum determined by a transcendental equation. For example, in the case of the heat equation with Robin boundary conditions, the classical representation involves an infinite series over where kn satisfy a transcendental equation, whereas the new method yields an explicit integral representation.

[BF2008a] Initial-boundary-value problems for linear and integrable nonlinear dispersive partial differential equations, Nonlinearity 21 (2008), 195-203,
[DV2011a] Well-posedness of boundary-value problems for the linear Benjamin-Bona-Mahony equation, (2011), submitted for review
[Fok2004b] Boundary-value problems for linear PDEs with variable coefficients, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2004), 1131-1151,
[Fok2008a] A Unified Approach to Boundary Value Problems, CBMS-SIAM, CBMS-SIAM (2008),
[FS1999a] Initial-Boundary Value Problems for Linear Dispersive Evolution Equations on the Half-Line, (1999), [ps] (unpublished)
[Pel2006a] Linear and nonlinear generalized Fourier transforms, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 364 1849 (2006), 3231-3249,
[Ros2014a] Education, SIAM Rev. 56 1 (2014), 157-158,
[Sun2000a] Initial-boundary value problems for linear dispersive evolution equations on the half-line, Fifth International Conference on Mathematical and Numerical Aspects of Wave Propagation, SIAM (2000), 374-378,

### Linear evolution PDEs formulated on a finite interval

The new transform method yields integral representations for evolution PDEs containing x-derivatives of arbitrary order formulated on the finite interval [Fok2008a], whereas it can be shown that even for the linearised KdV on the finite interval with generic boundary conditions, there does not exist an infinite series representation [Pap2011a], [Smi2011a]! Note that some of the works listed descibing linear evolution PDEs formulated on the half-line also refer to finite interval formulations.

[DS2014b] Heat conduction on the ring: interface problems with periodic boundary conditions, Appl. Math. Lett. 37 (2014), 107-111,
[DV2011a] Well-posedness of boundary-value problems for the linear Benjamin-Bona-Mahony equation, (2011), submitted for review
[Fok2008a] A Unified Approach to Boundary Value Problems, CBMS-SIAM, CBMS-SIAM (2008),
[Pap2011a] An example where separation of variable fails, J. Math. Anal. Appl. 373 2 (2011), 739-744,
[Pel2004a] Well-posed boundary value problems for linear evolution equations on a finite interval, Math. Proc. Cambridge Philos. Soc. 136 (2004), 361-382,
[Ros2014a] Education, SIAM Rev. 56 1 (2014), 157-158,
[Smi2014b] Birkhoff regularity and well-posedness of linear initial-boundary value problems, (2014), in preparation

### Linear evolution PDEs formulated in a time-dependent domain

The new method can be used to study linear evolution equations in domains of the form l(t)<x<∞, 0<t<T, with prescribed initial and boundary conditions and with l(t) a given differentiable function. In this case, the solution is characterised in general by a system of linear integral equations with explicit, weakly singular kernel. For example, for a second order problem such as the heat equation qt=qxx, given the Dirichlet boundary datum q(l(t),t), the unknown Neumann boundary value qx(l(t),t) can be computed as the solution of a single such linear Volterra integral equation.

[FS2003a] Long-time asymptotics of moving boundary problems using an Ehrenpreis-type representation and its Riemann-Hilbert nonlinearization, Comm. Pure Appl. Math. 56 4 (2003), 517-548,

### Linear elliptic PDEs on the interior of a polygon

The new method yields novel integral representations for the solution of several classical PDEs including the Laplace, the Helmholtz, and the modified Helmholtz equations formulated in the interior of a convex polygon. These representations involve certain transforms of the associated boundary values. There now exist two global relations. For a simple domain, it is possible, using the global relations and their invariant properties, to express all transforms in terms of the given boundary conditions using only algebraic manipulations. For example, the solution of the Dirichlet problem for the modified Helmholtz equation in the interior of an equilateral triangle is expressed in terms of an integral in the complex k-plane which involves an integrand which decays exponentially as |k| →∞. This solution has analytical and numerical advantages in comparison with the classical series solution of Lamé; furthermore, problems with more complicated boundary conditions (for which the classical approach apparently fails) can also be solved in a similar way. For more complicated domains, the global relation provides novel effective numerical techniques for the determination of the unknown boundary values [FFSS2009a], [FSS2010a], [FFSS2008a], [FF2011a], [FS2009a] and [FFX2004a]. Several boundary value problems for the biharmonic equation are analysed in [CF2004a] and [DF2011a].

[AF1999a] The Solution of the modified Helmholtz equation in a wedge and an application to diffusion-limited coalescence, Phys. Lett. A 263 (1999), 355-359, arXiv:cond-mat/9906351v1 [cond-mat.stat-mech]
[AF2001a] Solution of the modified Helmholtz equation in a triangular domain and an application to diffusion-limited coalescence, Phys. Rev. E 64 1 (2001), 16114, arXiv:cond-mat/0102175v1 [cond-mat.stat-mech]
[CD2013a] Stokes flow singularities in a two-dimensional channel: a novel transform approach with application to microswimming, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469 2157 (2013), 20130198,
[DF2011a] The Poisson and the Biharmonic Equations in the Interior of a Convex Polygon, (2011), (submitted)
[FF2011a] A numerical implementation of Fokas boundary integral approach: Laplace's equation on a polygonal domain, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467 2134 (2011), 2983-3003, http://rspa.royalsocietypublishing.org/content/early/2011/05/24/rspa.2011.0032.short?rss=1
[FK2014a] Eigenvalues for the Laplace operator in the interior of an equilateral triangle, (2014), submitted for review
[Fok2008a] A Unified Approach to Boundary Value Problems, CBMS-SIAM, CBMS-SIAM (2008),
[FPSS2009a] Direct and iterative solution of the generalized Dirichlet-Neumann map for elliptic PDEs on square domains, J. Comput. Appl. Math. 227 1 (2009), 171-184,
[FS2009a] Novel Analytical and Numerical Methods for Elliptic Boundary Value Problems, Highly Oscillatory Problems, Cambridge University Press (2009), 194-226,
[HZ2011a] High-frequency asymptotics for the modified Helmholtz equation in a quarter-plane, Appl. Anal. 90 12 (2011), 1927-1938,
[Pin2010b] Segre quaternions, spectral analysis and a four-dimensional Laplace equation, Progress in analysis and its applications, World Scientific (2010), 240-247,

### Linear elliptic PDEs in polar coordinates

The new method has also been implemented in the case of boundary value problems formulated in polar coordinates, and in particular has produced: (a) novel integral representations for the Poissonand Helmholtz equations, and (b) the solutions of several boundary value problems for these equations.

### Spectral theory for non-self-adjoint boundary value problems

The new method permits investigation of initial-boundary value problems on rectangular domains without a priori assumptions on the eigenfunctions of the spatial differential operator. Moreover, the global relation may be used to determine the eigenfunctions directly and the methods of spetral theory applied to test their completeness and basic properties.

[Smi2014b] Birkhoff regularity and well-posedness of linear initial-boundary value problems, (2014), in preparation

### Linear and integrable nonlinear hyperbolic PDEs

Boundary value problems for these equations are simpler than boundary value problems for evolution and elliptic PDEs because all boundary values needed for the solution representation are prescribed as boundary conditions.

### Integrable nonlinear evolution PDEs formulated on the half-line

The new method yields novel integral representations formulated in the complex k-plane (the Fourier plane). These integrals, in addition to the exponentials which appear in the integrals of the linearized version of these nonlinear PDEs, also contain the entries of a matrix-valued function M(x,t,k), which is the solution of a matrix Riemann-Hilbert (RH) problem. This RH problem is uniquely defined in terms of appropriate nonlinear transforms of boundary values, called spectral functions. These functions are defined in the Fourier space, i.e. they do not involve the independent variables of the PDE. This crucial feature of the new method makes it possible to obtain useful asymptotic information about the solution even before characterising the spectral functions in terms of the given boundary conditions. This can be achieved by using the Deift-Zhou (for the long-time asymptotics) [DZ1992a], [DZ1993a] and the Deift-Zhou-Venakides (for the zero-dispersion limit) [DVZ1994a], [DVZ1997a] techniques for the asymptotic analysis of these RH problems [Kam2003a], [FI1992a], [FI1996a], [FI1992b] and [FI1994a]. For certain nonlinear boundary value problems, called linearisable, it is possible, by analysing the global relation, to express the spectral functions in terms of the given initial and boundary conditions using only algebraic manipulations. Thus, for linearisable boundary conditions, the new method yields a representation which is as effective as the classical representation for the Cauchy problem obtained by the inverse scattering transform method. For general, non-linearisable boundary conditions the global relation yields an effective characterisation of the generalised Dirichlet to Neumann map. This important development was first achieved in [BFS2003a] by employing the so-called Gelfand-Levitan-Marchenko (GLM) representations [BK2000a], and was implemented numerically in [Zhe2006a] and [Zhe2007a]. It was later realised that this approach also provides a direct characterisation of the spectral functions in terms of the given initial and boundary conditions, bypassing the GLM representation, [Fok2005a] and [FT2008a]. Further progress in this problem was achieved in [FL2011b], where the relevant formulae were obtained without the need to introduce the GLM representations. However, it must be noted that for non-linearisable boundary value problems, the above characterisation involves a nonlinear equation.

[BK2007a] The Focusing nonlinear Schrödinger equation on the quarter plane with time-periodic boundary condition: a Riemann-Hilbert approach, J. Inst. Math. Jussieu 6 4 (2007), 579-611,
[BKS2009a] Decaying long-time asymptotics for the focusing NLS equation with periodic boundary condition, Int. Math. Res. Not. 2009 3 (2009), 547-577,
[BKSZ2010a] Initial boundary value problems for integrable systems: towards the long time asymptotics, Nonlinearity 23 10 (2010), 2483,
[DVZ1994a] The collisionless shock region for the long-time behavior of solutions of the KdV equation, Comm. Pure Appl. Math. 47 2 (1994), 199-206,
[DVZ1997a] New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems, Int. Math. Res. Not. 1997 6 (1997), 285-299,
[DZ1993a] A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. Of Math. 137 2 (1993), 295-368,
[FG1994a] Integrability of Linear and Nonlinear Evolution Equations and the Associated Nonlinear Fourier Transforms, Lett. Math. Phys. 32 (1994), 189-210,
[FI1992a] Soliton generation for initial-boundary-value problems, Phys. Rev. Lett. 68 21 (1992), 3117-3120,
[FI1992b] An Initial-Boundary Value Problem for the Sine-Gordon Equation in Laboratory Coordinates, Theoret. and Math. Phys. 92 3 (1992), 964-978,
[FI1994a] An initial-boundary value problem for the Korteweg-de Vries equation, Math. Comput. Simulation 37 4 (1994), 293-321,
[FK2004a] Zero-dispersion limit for integrable equations on the half-line with linearisable data, Abstr. Appl. Anal. 2004 5 (2004), 361-370,
[FL2009a] An integrable generalization of the nonlinear Schrödinger equation on the half-line and solitons, Inverse Problems 25 11 (2009), 115006, arXiv:0812.1335v2 [nlin.SI]
[Fok2007a] From Green to Lax via Fourier, Recent Advances in Nonlinear Partial Differential Equations and Applications, American Mathematical Society (2007), 87-118,
[Fok2008a] A Unified Approach to Boundary Value Problems, CBMS-SIAM, CBMS-SIAM (2008),
[FX2013a] The unified method for the three-wave equation on the half-line, (2013), arXiv:1304.4339 [nlin.SI]
[FX2013b] The Fokas method to the Sasa-Satsuma equation on the half-line, (2013), arXiv:1304.4586 [nlin.SI]
[IS2012a] Initial boundary value problem for the focusing NLS equation with Robin boundary condition: half-line approach, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469 2149 (2012), 20120199, arXiv:1201.5948v2 [nlin.SI]
[Its2003a] The Riemann-Hilbert problem and integrable systems, Notices Amer. Math. Soc. 50 11 (2003), 1389-1400,
[Vek2011a] Lattice representation and dark solitons of the Fokas-Lenells equation, Nonlinearity 24 4 (2011), 1165, arXiv:1103.4701v1 [nlin.SI]
[Zhe2006a] Exact nonreflecting boundary conditions for one-dimensional cubic nonlinear Schrödinger equations, J. Comput. Phys. 215 2 (2006), 552-565,

### Integrable nonlinear evolution PDEs on the finite interval

The most interesting aspect of this class of problems is the realization that the x-periodic problem is linearizable.

[BS2004a] Initial boundary value problem for the mKdV equation on a finite interval, Ann. Inst. Fourier (Grenoble) 54 (2004), 1477-1495,
[Fok2008a] A Unified Approach to Boundary Value Problems, CBMS-SIAM, CBMS-SIAM (2008),

### Integrable nonlinear elliptic PDEs formulated in the interior of a polygon

The new method again yields novel integral representations in the complex k-plane. These integrals, in adddition to the exponentials which appear in the integrals of the linearised versions of these nonlinear PDEs, also contain the entries of a matrix-valued function M(x,y,k) which is the solution of a matrix RH problem. The spectral functions, i.e. the entries of the relevant jump matrix which depend only on k, can be computed in terms of the associate boundary values. For elliptic PDEs involving only second order derivatives, by employing the two relevant global relations, it is possible to express the unknown boundary values via a system of nonlinear Fredholm integral equations. However, for linearisable boundary conditions, it is again possible to avoid this nonlinear step and to express the spectral functions directly in terms of the given boundary conditions. The new method has already been implemented to the elliptic sine-Gordon formulated in the quarter plane and in a semistrip, as well as to the celebrated Ernst equation formulated in several complicated domains. Furthermore, particular linearisable boundary value problems have been solved for these two physically significant PDEs.

[Fok2008a] A Unified Approach to Boundary Value Problems, CBMS-SIAM, CBMS-SIAM (2008),
[Len2011b] Boundary Value Problems for the Stationary Axisymmetric Einstein Equations: A Disk Rotating Around a Black Hole, Comm. Math. Phys. 304 3 (2011), 585-635, arXiv:1003.1453v1 [nlin.SI]

## Problems in three dimensions

### Linear evolution PDEs in two space variables

The Fokas method has been applied to simple domains such as the quarter plane, as well as to more complicated domains such as the interior of the isosceles triangle.

[FM2012b] The unified method for the heat equation: II. Non-separable boundary conditions in two dimensions, (2012), (submitted)

### Linear elliptic PDEs in spherical coordinates

The new method has also been implemented in the case of boundary value problems formulated in spherical coordinates and, in particular, has produced: (a) the classical Poisson integral formula for the solutions of the Dirichlet problem for the Poisson equation in the interior of a sphere, the analogous formula for the Neumann problem, and the generalisations of these formulae to n dimensions, (b) the solutions of various boundary value problems for the inhomogeneous Helmholtz equation in the interior of a sphere, and (c) the solution of the Dirichlet problem for the Laplace equation in the interior of a spherical sector.

### Linear elliptic PDEs with nonlocal boundary conditions

In particular, the Fokas method has been applied to the problem of water waves, which involves the Laplace equation in three dimensions supplemented with certain nonlocal boundary conditions. This has led to a novel nonlocal formulation of this classical problem.

### Quaternion methods

[Pin2010b] Segre quaternions, spectral analysis and a four-dimensional Laplace equation, Progress in analysis and its applications, World Scientific (2010), 240-247,

### Elasticity and fluid loaded plate

The Fokas method yields novel results for the classical boundary value problems arising in elasticity and in fluid loaded plate in 2 and in 3 dimensions.

[Pin2011a] Integral representations of displacements in linear elasticity, Appl. Math. Lett. 24 10 (2011), 1670-1675,

### Nonlinear evolution PDEs in two spatial dimensions

The new method uses either a d-bar formalism (as in the case of the Davey-Stewartson (DS) equation), or a combination of the a d-bar formalism with a nonlocal Riemann-Hilbert formalism (as in the case of the Kadomtsev-Petviashvili (KP) equation), to produce novel integral representations together with the associated global relations. These representations depend on certain spectral functions which themselves are characterized via certain linear integral equations of Fredholm type in terms of the initial conditions and all the boundary values. The characterization of the spectral functions in terms of the given initial and boundary conditions remains open. Moreover, the question of identifying linearisable boundary conditions for the DS or the KP also remains open.

[AF2004a] A Generalization of both the Method of Images and of the Classical Integral Transforms, Advances in scattering and biomedical engineering: proceedings of the Sixth International Workshop, Tsepelovo, Greece, 18-21 September 2003 World Scientific, (2004), 260-276,

## Differential-difference equations

### Linear and integrable nonlinear differential-difference equations

Following the development of the Fokas method for PDEs, it has been shown that the method can also be used for solving differential-difference equations (DDEs) on the natural numbers. These equations can be considered as discrete analogues of IBVPs for PDEs, since lattice spacing are taken from the natural numbers instead of the integers. The new method can deal with IBVPs for discrete linear and integrable discrete nonlinear Schrödinger equations (also known as the Ablowitz-Ladik system [AL1975a], [AL1976a]) on the natural numbers [BH2008a]. Linearizable BCs for the integrable discrete nonlinear Schrödinger equation were analysed, which are the discrete analogue of the homogenous Robin-type BCs. When such BCs are given, the unknown boundary values can be eliminated using only algebraic manipulation of the global relation as shown in continuum problems. It is worth emphasizing that the Fokas method can be generalized to solve IBVPs for any discrete linear evolution equations with more general boundary conditions just as effectively [BW2010a]; this is to be contrasted with the classical Fourier series approach which is very limited.