Fokas Method Research Articles

The Riemann–Hilbert Approach to the Higher-Order Gerdjikov–Ivanov Equation on the Half Line

The Fokas method exhibits remarkable versatility in solving boundary value problems associated with both linear and nonlinear partial differential equations, particularly when conventional approaches encounter challenges in handling intricate boundary conditions. The existing literature often lacks the incorporation of unconventional boundary conditions, and this study addresses this issue by extending the application of the Fokas method to the higher-order Gerdjikov-Ivanov equation on the half line (−∞,0]. We have demonstrated the exclusive representation of the potential function u(z,t) in the higher-order Gerdjikov–Ivanov equation through the solution of a Riemann–Hilbert problem. The characteristic function is partitioned on the complex plane, and we obtain the jump matrix between each partition based on the positive and negative values of the partition as well as the spectral matrix determined by the initial data and boundary value data. The findings suggest that the spectral functions are not mutually independent; instead, they conform to a global relationship. The novel aspect of this study is the application of the Fokas method to a previously unexplored case, contributing to the theoretical and practical understanding of complex partial differential equations and filling a gap in the treatment of boundary conditions.

Variations of heat equation on the half‐line via the Fokas method

In this review paper, we discuss some of our recent results concerning the rigorous analysis of initial boundary value problems (IBVPs) and newly discovered effects for certain evolution partial differential equations (PDEs). These equations arise in the applied sciences as models of phenomena and processes pertaining, for example, to continuum mechanics, heat‐mass transfer, solid–fluid dynamics, electron physics and radiation, chemical and petroleum engineering, and nanotechnology. The mathematical problems we address include certain well‐known classical variations of the traditional heat (diffusion) equation, including (i) the Sobolev–Barenblatt pseudoparabolic PDE (or modified heat or second‐order fluid equation), (ii) a fourth‐order heat equation and the associated Cahn–Hilliard (or Kuramoto–Sivashinsky) model, and (iii) the Rubinshtein–Aifantis double‐diffusion system. Our work is based on the synergy of (i) the celebrated Fokas unified transform method (UTM) and (ii) a new approach to the rigorous analysis of this method recently introduced by one of the authors. In recent works, we considered forced versions of the aforementioned PDEs posed in a spatiotemporal quarter‐plane with arbitrary, fully non‐homogeneous initial and boundary data, and we derived formally effective solution representations, for the first time in the history of the models, justifying a posteriori their validity. This included the reconstruction of the prescribed initial and boundary conditions, which required careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. In each IBVP, the novel formula was utilized to rigorously deduce the solution's regularity properties near the boundaries of the spatiotemporal domain. Importantly, this analysis is indispensable for proving (non)uniqueness of solution. These works extend previous investigations. The usefulness of our closed‐form solutions will be demonstrated by studying their long‐time asymptotics. Specifically, we will briefly review some asymptotic results about Barenblatt's equation.

An analytical solution for vertical infiltration in homogeneous bounded profiles

AbstractIn this study, we derive an analytical solution to address the problem of vertical infiltration within 1D homogeneous bounded profiles. Initially, we consider the Richards equation together with Dirichlet boundary conditions. We assume constant diffusivity and linear dependence between the conductivity and the water content, resulting to a linear partial differential equation of diffusion type. To solve the simplified initial boundary value problem over a finite interval, we apply the unified transform, commonly known as the Fokas method. Through this methodology, we obtain an integral representation of the solution that can be efficiently and directly computed numerically, yielding a convergent scheme. We examine various cases, and we compare our solution with well‐known approximate solutions. This work can be seen as a first step to derive analytical solutions for the far more difficult and complex problem of modelling water flow in heterogeneous layered soils.

Scattering problems for the wave equation in 1D: D’Alembert-type representations and a reconstruction method

We derive the extension of the classical d’Alembert formula for the wave equation, which provides the analytical solution for the direct scattering problem for a medium with constant refractive index. Analogous formulae exist already in the literature, but in the current work this is derived in a natural way for general incident field, by employing results obtained via the Fokas method. This methodology is further extended to a medium with piecewise constant refractive index, providing the apparatus for the solution of the associated inverse scattering problem. Hence, we provide an exact reconstruction method which is also valid for phaseless data.

Periodic finite-band solutions to the focusing nonlinear Schrödinger equation by the Fokas method: inverse and direct problems

We consider the Riemann–Hilbert (RH) approach to the construction of periodic finite-band solutions to the focusing nonlinear Schrödinger (NLS) equation. An RH problem for the solution of the finite-band problem has been recently derived via the Fokas method (Deconinck et al. 2021 Lett. Math. Phys. 111 , 1–18. ( doi:10.1007/s11005-021-01356-7 ); Fokas & Lenells. 2021 Proc. R. Soc. A 477 , 20200605. ( doi:10.1007/s11005-021-01356-7 )) Building on this method, a finite-band solution to the NLS equation can be given in terms of the solution of an associated RH problem, the jump conditions for which are characterized by specifying the endpoints of the arcs defining the contour of the RH problem and the constants (so-called phases) involved in the jump matrices. In our work, we solve the problem of retrieving the phases given the solution of the NLS equation evaluated at a fixed time. Our findings are corroborated by numerical examples of phases computation, demonstrating the viability of the method proposed.

On Barenblatt's pseudoparabolic equation with forcing on the half‐line via the Fokas method

AbstractA novel technique is presented for explicitly solving inhomogeneous initial‐boundary‐value problems (IBVPs) (Dirichlet, Neumann and Robin) on the half‐line, for a well‐known pseudo‐parabolic partial differential equation. This so‐called Barenblatt's equation arises in a plethora of important applications, ranging from heat‐mass transfer, solid‐fluid‐gas dynamics and materials science, to mechanical, chemical and petroleum engineering, as well as electron physics, radiation and diffusive processes. Our approach is based on the extension of the Fokas method, so that it can be applied to problems with mixed derivatives. First, we derive formally effective solution representations and then justify a posteriori their validity rigorously. This includes the reconstruction of the prescribed initial and boundary conditions, which requires careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. In each type of IBVP, the novel formulae are utilized to rigorously deduce the solution's regularity properties near the boundaries of the spatiotemporal domain and the problem's well‐posedness. Furthermore, importantly, our solutions’ numerical advantages are demonstrated and highlighted by way of a concrete and illustrative example. Our rigorous approach can be extended to IBVPs for other significant models.

Numerical Computation of Neumann Controls for the Heat Equation on a Finite Interval

This paper presents a new numerical method which approximates Neumann type null controls for the heat equation and is based on the Fokas method. This is a direct method for solving problems originating from the control theory, which allows the realisation of an efficient numerical algorithm that requires small computational effort for determining the null control with exponentially small error. Furthermore, the unified character of the Fokas method makes the extension of the numerical algorithm to a wide range of other linear PDEs and different type of boundary conditions straightforward.

Long-range instability of linear evolution PDE on semi-bounded domains via the Fokas method

Long-range instability of linear evolution PDE on semi-bounded domains via the Fokas method

Fokas Method for the Heat Equation on Metric Graphs

Fokas Method for the Heat Equation on Metric Graphs

Null controllability of the advection-dispersion equation with respect to the dispersion parameter using the Fokas method

We investigate the null controllability of the linear advection-dispersion (LAD) equation posed on a finite interval. To this end, we employ the Fokas method to determine boundary controls and the controlled solutions concerning varying dispersion parameters and final times. By employing the Fokas method for the LAD equation, which provides an explicit integral representation of the solution, we enable a comprehensive analysis of null controllability. Consequently, we derive a boundary control input that effectively steers the solution from its initial state to the zero state within a given final time. The proposed approach offers a straightforward means to achieve the null controllability for the LAD equation, even as the dispersion parameter and the final time vary. Importantly, the effectiveness of this approach can be extended to various types of boundary control problems, thereby enhancing its broader applicability and potential impact.

On the linearizable initial–boundary value problems for the Sasa–Satsuma equation on the half-line

The solutions u(x,t) of the Sasa–Satsuma equation was constructed in terms of the solution of a 3 × 3 Riemann–Hilbert problem by Fokas method. To formulate the associated Riemann–Hilbert problem, we need know all of the boundary value data, i.e. u(0,t),ux(0,t),uxx(0,t). However, for a well-posed problem, not all of these boundary data were prescribed, the remaining boundary data cannot be independently specified, but are determined by the so-called global relation. In this paper, we analyze particular linearizable boundary conditions, which means all the spectral functions can be constructed by an algebraic system in terms of the initial value data and the predicted boundary value data.

Initial-boundary value problem for a fractional heat equation on an interval

Abstract In this paper, we study a Dirichlet problem for a fractional heat equation, with spacial fractional derivative in the sense of Riemann–Liouville on a finite interval. The main ideas of Fokas method is employed, where the Lax pairs are used to obtain an integral representation of solutions.

Local well‐posedness of the higher‐order nonlinear Schrödinger equation on the half‐line: Single‐boundary condition case

AbstractWe establish local well‐posedness in the sense of Hadamard for a certain third‐order nonlinear Schrödinger equation with a multiterm linear part and a general power nonlinearity, known as higher‐order nonlinear Schrödinger equation, formulated on the half‐line . We consider the scenario of associated coefficients such that only one boundary condition is required and hence assume a general nonhomogeneous boundary datum of Dirichlet type at . Our functional framework centers around fractional Sobolev spaces with respect to the spatial variable. We treat both high regularity () and low regularity () solutions: in the former setting, the relevant nonlinearity can be handled via the Banach algebra property; in the latter setting, however, this is no longer the case and, instead, delicate Strichartz estimates must be established. This task is especially challenging in the framework of nonhomogeneous initial‐boundary value problems, as it involves proving boundary‐type Strichartz estimates that are not common in the study of Cauchy (initial value) problems. The linear analysis, which forms the core of this work, crucially relies on a weak solution formulation defined through the novel solution formulae obtained via the Fokas method (also known as the unified transform) for the associated forced linear problem. In this connection, we note that the higher‐order Schrödinger equation comes with an increased level of difficulty due to the presence of more than one spatial derivatives in the linear part of the equation. This feature manifests itself via several complications throughout the analysis, including (i) analyticity issues related to complex square roots, which require careful treatment of branch cuts and deformations of integration contours; (ii) singularities that emerge upon changes of variables in the Fourier analysis arguments; and (iii) complicated oscillatory kernels in the weak solution formula for the linear initial‐boundary value problem, which require a subtle analysis of the dispersion in terms of the regularity of the boundary data. The present work provides a first, complete treatment via the Fokas method of a nonhomogeneous initial‐boundary value problem for a partial differential equation associated with a multiterm linear differential operator.

A Python toolbox for the numerical solution of the Maxey‐Riley equation

AbstractThe Maxey‐Riley equation (MRE) models the motion of a finite‐sized, spherical particle in a fluid. It is a second‐order integro‐differential equation with a kernel with a singularity at initial time. Because solving the integral term is numerically challenging, it is often neglected despite its often non‐negligible impact. Recently, Prasath et al. showed that the MRE can be rewritten as a time‐dependent heat equation on a semi‐infinite domain with a nonlinear, Robin‐type boundary condition. This approach avoids the need to deal with the integral term. They also describe a numerical approach for solving the transformed MRE based on Fokas method. We provide a Python toolbox implementing their approach, verify it against some of their numerical examples and demonstrate its flexibility by computing the trajectory of a particle in a velocity field given by experimental data.

The analytic extension of solutions to initial-boundary value problems outside their domain of definition

AbstractWe examine the analytic extension of solutions of linear, constant-coefficient initial-boundary value problems outside their spatial domain of definition. We use the Unified Transform Method or Method of Fokas, which gives a representation for solutions to half-line and finite-interval initial-boundary value problems as integrals of kernels with explicit spatial and temporal dependence. These solution representations are defined within the spatial domain of the problem. We obtain the extension of these representation formulae via Taylor series outside these spatial domains and find the extension of the initial condition that gives rise to a whole-line initial-value problem solved by the extended solution. In general, the extended initial condition is not differentiable or continuous unless the boundary and initial conditions satisfy compatibility conditions. We analyse dissipative and dispersive problems, and problems with continuous and discrete spatial variables.

The mixed nonlinear Schrödinger equation on the half-line

We analyze the initial-boundary value problem for the mixed nonlinear Schrödinger equation posed on the half-line by using the Fokas method. Assuming that a smooth solution exists, we show that the solution can be represented in term of the solution of a matrix Riemann–Hilbert problem formulated in the complex plane. The jump matrix is defined in terms of the spectral functions obtained from the initial and boundary values. We derive a certain relation, the so-called global relation that involves all initial and boundary values. Furthermore, it can be shown that the solution for the mixed nonlinear Schrödinger equation posed on the half-line exists, which can be determined by the unique solution of the associated Riemann–Hilbert problem, as long as the spectral functions satisfy the above global relation.

Construction of the differential surface admittance operator with an extended Fokas method for electromagnetic scattering at polygonal objects with arbitrary material parameters

This article presents a novel method to accurately simulate electromagnetic scattering at homogeneous polygonal cylinders with arbitrary material properties. A single source equivalence approach is invoked, allowing to substitute the background medium for the inner material of the scatterer, provided an equivalent surface current density is introduced. We construct the pertinent differential surface admittance operator by means of the Fokas method, establishing a map between the known Dirichlet boundary values and their unknown Neumann counterparts. However, to allow for lossy materials, we extend the Fokas method to complex wavenumbers. The novel formalism, employing pulse-shaped local basis functions, natively supports combined magnetic and dielectric contrast, accurately captures the skin effect, and is conveniently integrated in traditional boundary integral equation formulations. The correctness and versatility of our technique are verified for various examples by means of analytical validation and through comparison with a Poggio-Miller-Chan-Harrington-Wu-Tsai approach, a volume integral equation method and a commercial solver.

Boundary behavior of the solution to the linear Korteweg‐De Vries equation on the half line

AbstractIn this paper, we consider the solution to the linear Korteweg‐De Vries (KdV) equation, both homogeneous and forced, on the quadrant via the unified transform method of Fokas and we provide a complete rigorous study of the integrals of the formula provided by the method, especially focusing on the explicit verification of the considered initial‐boundary‐value problems (IBVPs), with generic data, as well as on the uniform convergence of all its derivatives, as approaches the boundary of the quadrant, and their rapid decay as .

Solving the heat equation with variable thermal conductivity

We consider the heat equation with spatially variable thermal conductivity and homogeneous Dirichlet boundary conditions. Using the Method of Fokas or Unified Transform Method, we derive solution representations as the limit of solutions of constant-coefficient interface problems where the number of subdomains and interfaces becomes unbounded. This produces an explicit representation of the solution, from which we can compute the solution and determine its properties. Using this solution expression, we can find the eigenvalues of the corresponding variable-coefficient eigenvalue problem as roots of a transcendental function. We can write the eigenfunctions explicitly in terms of the eigenvalues. The heat equation is the first example of more general variable-coefficient second-order initial–boundary value problems that can be solved using this approach.

A higher dispersion KdV equation on the half-line

The initial-boundary value problem (ibvp) for the m-th order dispersion Korteweg-de Vries (KdV) equation on the half-line with rough data and solution in restricted Bourgain spaces is studied using the Fokas Unified Transform Method (UTM). Thus, this work advances the implementation of the Fokas method, used earlier for the KdV on the half-line with smooth data and solution in the classical Hadamard space, consisting of function that are continuous in time and Sobolev in the spatial variable, to the more general Bourgain spaces framework of dispersive equations with rough data on the half-line. The spaces needed and the estimates required arise at the linear level and in particular in the estimation of the linear pure ibvp, which has forcing and initial data zero but non-zero boundary data. Using the iteration map defined by the Fokas solution formula of the forced linear ibvp in combination with the bilinear estimates in modified Bourgain spaces introduced by this map, well-posedness of the nonlinear ibvp is established for rough initial and boundary data belonging in Sobolev spaces.