Seminars (WHT)

A multivariable, in particular two complex variables (2D), Wiener-Hopf (WH) method is one of the desired generalisations of the classical and celebrated WH technique that are easily conceived but very hard to implement (the second one, indeed, is the matrix WH). A 2D WH method, could potentially be used e.g. for finding a solution to the canonical problem of diffraction by a quarter-plane.  

Unfortunately, multidimensional complex analysis seems to be way more complicated than complex analysis of a single variable. There exists a number of powerful theorems in it, but they are organised into several disjoint theories, and, generally all of them are far from the needs of WH. In this mini-lecture course, we hope to introduce topics in complex analysis of several variables that we think are important for a generalisation of the WH technique. We will focus on the similarities and differences between functions of one complex variable and functions of two complex variables. Elements of differential forms and homotopy theory will be addressed.

We will start by reviewing some known attempts at building a 2D WH and explain why they were not successful. The framework of Fourier transforms and analytic functions in 2D will be introduced, leading us naturally to discuss multidimensional integration contours and their possible deformations. One of our main focus will be on polar and branch singularity sets and how to describe how a multidimensional contour bypasses these singularities. We will explain how multidimensional integral representation can be used in order to perform an analytical continuation of the unknowns of a 2D functional equation and why we believe it to be important. Finally, time permitting, we will discuss the branching structure of complex integrals depending on some parameters and introduce the so-called Picard-Lefschetz formulae.”

The Wiener-Hopf factorization is a fundamental result in the theory of Levy processes; it provides a wealth of information about the first exit of the underlying process from a half-line. The main goal of these lectures is to show how to use complex-analytic methods to obtain explicit formulas for Wiener-Hopf factors for several important classes of Levy processes. We will start with processes with jumps of rational transform, then we will discuss the class of stable processes, explaining how one could recover from the Wiener-Hopf factors the distribution of the supremum of the process at a fixed time. Finally, we will talk about the difficult problem of how a Levy process exits an interval, which turns out to be related to Wiener-Hopf factorization for certain 2x2 matrices. This latter problem is wide open for processes with double-sided jumps and we will discuss what is currently known for stable processes. 

The Sommerfeld Malyuzhinets (SM) method and the Wiener Hopf (WH) technique are different but closely related methods. In particular in the paper “Progress and Prospects in The Theory of Linear Waves Propagation” SIAM SIREV vol.21, No.2, April 1979, pp. 229-245, J.B. Keller posed the following question “What features of the methods account for this difference?”.  Furthermore  J.B. Keller notes “it might be helpful to understand this in order to predict the success of other methods”.

We agree with this opinion expressed by the giant of  Diffraction. Furthermore we think that SM and WH applied to the same problems (for instance the polygon diffraction)  can determine a helpful synergy. In the past the SM and WH methods were considered disconnected in particular because the SM method was traditionally defined with the angular complex representation while the WH method was traditionally defined in the Laplace domain.

In this course we show that the two methods have significant points of similarity when the representation of problems in both methods are expressed in terms of difference equations. The two methods show their diversity in the solution procedures that are completely different and effective. Both similarity and diversity properties are of advantage in  “Progress and Prospects in The Theory of Linear Waves Propagation”.

Moreover both methods have demonstrated their efficacy in studying particularly complex problems, beyond the traditional problem of scattering by a wedge: in particular the scattering by a three part polygon that we will present. Recent progress in both methods: One of the most relevant recent progress in SM is the derivation of functional difference equations without the use of Maliuzhinets inversion theorem.

One of the most relevant recent progress in WH is transformation of WH equations into integral equations for their effective solution

It is a little nearly 90 years since two of the most important mathematicians of the 20th century collaborated on finding the exact solution of a particular equation with semi-infinite convolution type integral operator. The elegance and analytical sophistication of the method, now called the Wiener-Hopf technique, impress all who use it. Its applicability to almost all branches of engineering, mathematical physics and applied mathematics is borne out by the many thousands of papers published on the subject since its conception. This series of three lectures will be informal in nature and directed at researchers who are either at an early stage of their career or else unfamiliar with particular aspects of the subject. Their aim is to demonstrate the beauty of the topic and its wide range of applications, and will be delivered in a traditional applied mathematical style. The lectures will not try to offer a comprehensive overview of the literature but will instead focus on specific topics that have been of interest over the years to the speaker.  

The first lecture shall offer a subjective review of the subject, introducing the notation to be employed in later lectures, and indicating a sample of the enormous range of applications that have been found for the technique. The second lecture will focus on exact and approximate solution methods for scalar and vector Wiener-Hopf equations, and indicate the similarities and differences of the various approaches used. The final lecture shall continue discussion of approximate approaches, combining these with one or more specific applications of current interest to the speaker.

This lecture series is devoted to the interplay between diffraction and operator theory, particularly between the so-called canonical diffraction problems (exemplified by half-plane problems) on one hand and operator factorisation theory on the other hand. It is shown how operator factorisation concepts appear naturally from applications and how they can help to find solutions rigorously in case of well-posed problems as well as for ill-posed problems after an adequate normalisation.  

The operator theoretical approach has the advantage of a compact presentation of results simultaneously for wide classes of diffraction problems and space settings and gives a different and deeper understanding of the solution procedures.  

The main objective is to demonstrate how diffraction problems guide us to operator factorisation concepts and how useful those are to develop and to simplify the reasoning in the applications.  

In eight widely independent sections we shall address the following questions:  
How can we consider the classical Wiener-Hopf procedure as an operator factorisation (OF) and what is the profit of that interpretation?        
What are the characteristics of Wiener-Hopf operators occurring in Sommerfeld half-plane problems and their features in terms of functional analysis?
What are the most relevant methods of constructive matrix factorisation in Sommerfeld problems?
How does OF appear generally in linear boundary value and transmission problems and why is it useful to think about this question?     
What are adequate choices of function(al) spaces and symbol classes in order to analyse the well-posedness of problems and to use deeper results of factorisation theory?      
A sharp logical concept for equivalence and reduction of linear systems (in terms of OF) – why is it needed and why does it simplify and strengthen the reasoning?
Where do we need other kinds of operator relations beyond OF?
What are very practical examples for the use of the preceding ideas, e.g., in higher dimensional diffraction problems? 

Historical remarks and corresponding references are provided at the end of each section.

A multivariable, in particular two complex variables (2D), Wiener-Hopf (WH) method is one of the desired generalisations of the classical and celebrated WH technique that are easily conceived but very hard to implement (the second one, indeed, is the matrix WH). A 2D WH method, could potentially be used e.g. for finding a solution to the canonical problem of diffraction by a quarter-plane.  

Unfortunately, multidimensional complex analysis seems to be way more complicated than complex analysis of a single variable. There exists a number of powerful theorems in it, but they are organised into several disjoint theories, and, generally all of them are far from the needs of WH. In this mini-lecture course, we hope to introduce topics in complex analysis of several variables that we think are important for a generalisation of the WH technique. We will focus on the similarities and differences between functions of one complex variable and functions of two complex variables. Elements of differential forms and homotopy theory will be addressed.

We will start by reviewing some known attempts at building a 2D WH and explain why they were not successful. The framework of Fourier transforms and analytic functions in 2D will be introduced, leading us naturally to discuss multidimensional integration contours and their possible deformations. One of our main focus will be on polar and branch singularity sets and how to describe how a multidimensional contour bypasses these singularities. We will explain how multidimensional integral representation can be used in order to perform an analytical continuation of the unknowns of a 2D functional equation and why we believe it to be important. Finally, time permitting, we will discuss the branching structure of complex integrals depending on some parameters and introduce the so-called Picard-Lefschetz formulae.”

The Wiener-Hopf factorization is a fundamental result in the theory of Levy processes; it provides a wealth of information about the first exit of the underlying process from a half-line. The main goal of these lectures is to show how to use complex-analytic methods to obtain explicit formulas for Wiener-Hopf factors for several important classes of Levy processes. We will start with processes with jumps of rational transform, then we will discuss the class of stable processes, explaining how one could recover from the Wiener-Hopf factors the distribution of the supremum of the process at a fixed time. Finally, we will talk about the difficult problem of how a Levy process exits an interval, which turns out to be related to Wiener-Hopf factorization for certain 2x2 matrices. This latter problem is wide open for processes with double-sided jumps and we will discuss what is currently known for stable processes. 

The Sommerfeld Malyuzhinets (SM) method and the Wiener Hopf (WH) technique are different but closely related methods. In particular in the paper “Progress and Prospects in The Theory of Linear Waves Propagation” SIAM SIREV vol.21, No.2, April 1979, pp. 229-245, J.B. Keller posed the following question “What features of the methods account for this difference?”.  Furthermore  J.B. Keller notes “it might be helpful to understand this in order to predict the success of other methods”.

We agree with this opinion expressed by the giant of  Diffraction. Furthermore we think that SM and WH applied to the same problems (for instance the polygon diffraction)  can determine a helpful synergy. In the past the SM and WH methods were considered disconnected in particular because the SM method was traditionally defined with the angular complex representation while the WH method was traditionally defined in the Laplace domain.

In this course we show that the two methods have significant points of similarity when the representation of problems in both methods are expressed in terms of difference equations. The two methods show their diversity in the solution procedures that are completely different and effective. Both similarity and diversity properties are of advantage in  “Progress and Prospects in The Theory of Linear Waves Propagation”.

Moreover both methods have demonstrated their efficacy in studying particularly complex problems, beyond the traditional problem of scattering by a wedge: in particular the scattering by a three part polygon that we will present. Recent progress in both methods: One of the most relevant recent progress in SM is the derivation of functional difference equations without the use of Maliuzhinets inversion theorem.

One of the most relevant recent progress in WH is transformation of WH equations into integral equations for their effective solution

Crack propagation is a process accompanied by multiple phenomena at different scales. In particular, when a crack grows, microstructural  vibrations are released, emanating from the crack tip. Continuous   models   of  dynamic   cracks   are  well  known   to  omit  information   concerning   these microstructural processes [1]. On the other hand, tracing these vibrations on the microscale is possible if one considers a crack propagating in a structured system, such as a lattice [2, 3].  These models have a particular relevance in the design of metamaterials,  whose microstructure  can be tailored to control dynamic effects for a variety of practical purposes [4]. Similar approaches have been recently paving new pathways to understanding failure processes in civil engineering systems [5, 6].  

In this lecture, we aim to demonstrate the importance of the Wiener-Hopf technique in the analysis and solution  of problems  concerning  waves and crack propagation  in discrete periodic  media. We begin with the model of a lattice system containing  a crack and show how this can be reduced to a scalar Wiener-Hopf  equation  through  the Fourier  transform.  From  this functional  equation  we identify  all possible  dynamic  processes  complementing   the  crack  growth.  We  determine  the  solution  to  the problem  and  how  this  is  used  to  predict  crack  growth  regimes  in  numerical  simulations.  Other applications of the adopted method, including the analysis of the progressive collapse of large-scale structures, are discussed.  

References [1] Marder, M. and Gross, S. (1995): Origin of crack tip instabilities, J. Mech. Phys. Solids 43, no. 1, 1- 48.   [2] Slepyan, L.I. (2001): Feeding and dissipative  waves in fracture and phase transition  I. Some 1D structures and a square-cell lattice, J. Mech. Phys. Solids 49, 469-511.   [3] Slepyan, L.I. (2002): Models and Phenomena  in Fracture Mechanics, Foundations  of Engineering Mechanics, Springer.   [4] Mishuris, G.S., Movchan, A.B. and Slepyan, L.I., (2007): Waves and fracture in an inhomogeneous lattice structure, Wave Random Complex 17, no. 4, 409-428.   [5] Brun, M., Giaccu, G.F., Movchan, A.B., and Slepyan, L. I., (2014): Transition wave in the collapse of the San Saba Bridge, Front. Mater. 1:12. doi: 10.3389/fmats.2014.00012.   [6] Nieves, M.J., Mishuris,  G.S., Slepyan,  L.I., (2016): Analysis  of dynamic  damage propagation  in discrete beam structures, Int. J. Solids Struct. 97-98, 699-713.

An operator factorisation conception is investigated for a general Wiener-Hopf operator $W = P_2 A |_{P_1 X}$ where $X,Y$ are Banach spaces, $P_1 \in \mathcal{L}(X), P_2 \in \mathcal{L}(Y)$ are projectors and $A \in \mathcal{L}(X,Y)$ is invertible. Namely we study a particular factorisation of $A = A_- C A_+$ where $A_+ : X \rightarrow Z$ and $A_- : Z \rightarrow Y$ have certain invariance properties and the cross factor $C : Z \rightarrow Z$ splits the "intermediate space" $Z$ into complemented subspaces closely related to the kernel and cokernel of $W$, such that $W$ is equivalent to a "simpler" operator, $W \sim P C|_{P Z}$.  

The main result shows equivalence between the generalised invertibility of the Wiener-Hopf operator and this kind of factorisation (provided $P_1 \sim P_2$) which implies a formula for a generalised inverse of $W$. It embraces I.B. Simonenko's generalised factorisation of matrix measurable functions in $L^p$ spaces and various other factorisation approaches.  

As applications we consider interface problems in weak formulation for the n-dimensional Helmholtz equation in $\Omega = \mathbb{R}^n_+ \cup \mathbb{R}^n_-$ (due to $x_n > 0$ or $x_n < 0$, respectively), where the interface $\Gamma = \partial \Omega$ is identified with $\mathbb{R}^{n-1}$ and divided into two parts, $\Sigma$ and $\Sigma'$, with different transmission conditions of first and second kind. These two parts are half-spaces of $\mathbb{R}^{n-1}$ (half-planes for $n = 3$). We construct explicitly resolvent operators acting from the interface data into the energy space $H^1(\Omega)$. The approach is based upon the present factorisation conception and avoids an interpretation of the factors as unbounded operators. In a natural way, we meet anisotropic Sobolev spaces which reflect the edge asymptotic of diffracted waves.

I will provide certain examples of scattering problems, motivated by lattice waves (phonons), electronic waves under certain assumptions, nanoscale effects, etc in crystals. The mathematical formulation is posed on lattices and involves difference equations that can be reduced to the problem of Wiener-Hopf on the unit circle (in an annulus in complex plane). In some of these examples, the Wiener-Hopf  problem is scalar, while in other cases it is a matrix Wiener-Hopf problem. For the latter, in a few cases it may be reduced to a scalar problem but it appears to be not the case in others. Some of these problems can be considered as discrete analogues of well-known Wiener-Hopf equations in continuum models on the real line (in an strip in complex plane), a few of which are still open problems.

For several classes  of  functions invertibility and factorability are equivalent; such is the case, e.g., for the Wiener class W or the algebra APW of almost periodic functions with absolutely convergent Bohr-Fourier series. The result for W extends to the matrix setting; not so for APW. Moreover, the factorability criterion even for 2-by-2 triangular matrix functions with APW entries and constant determinant remains a mystery. We will discuss some known results in this direction, and more specific open problems.

We give an introduction to the the theory of Wiener-Hopf factoirsations for Levy processes and discuss some very recent examples which are stimulated by some remarkable connections with self-similar Markov processes.

The field equations of gravitational theories in 4 dimensions are non-linear PDE's that are difficult to solve in general. By restricting to a subspace of solutions that only depend on two space-time coordinates, alternative approaches to solving those equations become available. We present here the Riemann-Hilbert approach, looking at the dimensionally reduced field equations as an integrable system associated to a certain Lax pair, whose solutions can be obtained by factorizing a so called monodromy matrix. This approach allows for the explicit construction of solutions to the non-linear gravitational field equations using simple complex analytic methods.

By an explicit solution of the factorization problem we mean the solution that can be found by finite number of some steps which we call "explicit". When we solve a specific factorization problem we must rigorously define these steps. In this talk we will do this for matrix polynomials, rational matrix functions, analytic matrix functions, meromorphic matrix functions, triangular matrix functions and others. For these classes we describe the data and procedures that are necessary for the explicit solution of the factorization problem. Since the factorization problem is unstable, the explicit solvability of the problem does not mean that we can get its numerical solution. This is the principal obstacle to use the Wiener-Hopf techniques in applied problems. For the above mentioned classes the main reason of the instability is the instability of the rank of a matrix. Numerical experiments show that the use of SVD for computation of the ranks often allows us to correctly find the partial indices for matrix polynomials. To create a test case set for numerical experiments we have to solve the problem exactly. By the exact solutions of the factorization problem we mean those solutions that can be found by symbolic computation. In the talk we obtain necessary and sufficient conditions for the existence of the exact solution to the problem for matrix polynomials and propose an algorithm for constructing of the exact solution. The solver modules in SymPy and in Maple that implement this algorithm are designed.

Following an example set by Slepyan, it proves possible to employ the Wiener-Hopf method to obtain exact solutions for fracture and friction problems at the atomic scale. I will describe a number of physical phenomena that have been analyzed in this way. These include the velocity gap and micro-branching instability for dynamic cracks, a connection of friction with self-healing pulses, and resolution of the energy transport paradox for supersonic cracks.

In this talk, I will discuss listed below problems with attendant circumstances and the results following straightforwardly from the formulation:

Mechanical wave momentum from the first principles. Wave Motion, 2016, 68, 283-290.
On the energy partition in oscillations and waves. Proc. R. Soc. A, 2015, 471: 20140838.
Betty Theorem and Orthogonality Relations for Eigenfunctions. Mechanics of Solids, 1979, 14, 74-77.
On a displacement of a deformable body in an acoustic medium. J. Appl. Math. Mech., 1963, 27, 1402-1411,
and possibly some others.

To show that steadily propagating nonlinear waves in active matter can be driven internally, we develop a prototypical model of a topological kink moving with a constant supersonic speed in a discrete bi-stable FPU chain capable of generating active stress. In contrast to subsonic kinks in such systems, that are necessarily dissipative, the obtained supersonic solutions are purely anti-dissipative. Joint work with N. Gorbushin.

This talk would focus on one of the main themes of this workshop: the diverse applications of the Wiener-Hopf technique for aerospace in general and turbofan noise problems in particular. First, I will give a theoretical model based on the Wiener-Hopf method (and matrix kernel factorisation) to unveil possible noise control mechanisms due to trailing-edge chevrons on the bypass duct of aircraft engine. Next, I will propose a new testing approach that relies on the forward propagation model based on the Wiener-Hopf method. The key contribution is the development of the inverse acoustic scattering approach for a sensor array by combining compressive sensing in a non-classical way. Last but not least, I will demonstrate some of the new aerospace applications of the Wiener-Hopf technique with recently popular deep neural networks.

In spite of the great efforts by many  studies, there have been little progresses towards a general method of constructive factorizations to get exact solution of vector WH equations.   The aim of  this talk  is the presentation of an alternative solution technique that is based to the reduction of the WH equations to  Fredholm equations of second kind (Fredholm factorization). The presentation will focus to the applications of the  Fredholm factorization  to  WH equations occurring in diffraction problem. In particular it   is based on five steps:1) Deduction of the WH equations of the problem,2) Reduction of the WH equations to Fredholm integral equations (FIE) ,3) Solution of the Fredholm integral equations , 4)Analytical continuation of the numerical solution of the FIE,5) Evaluation of the physical field components if present: reflected and refracted plane waves, diffracted fields, surface waves, lateral waves, leaky waves, mode excitations, near field. A characteristic  example of problem will be presented in the following talk.

Many owl species rely on specialized plumage to mitigate their aerodynamic noise and achieve functionally-silent flight while hunting. One such plumage feature, a tattered arrangement of flexible trailing-edge feathers, is idealized as a semi-infinite poroelastic plate to model the effects that edge compliance and flow seepage have on the noise production. The interaction of the poroelastic edge with a turbulent eddy is examined analytically with respect to how efficiently the edge scatters the eddy as aerodynamic noise. The scattering event is formulated and solved as a scalar Wiener-Hopf problem to identify how the noise scales with the flight velocity, where special attention is paid to the limiting cases of rigid-porous and elastic-impermeable plate conditions. Results from this analysis identify new parameter spaces where the porous and/or elastic properties of a trailing edge may be tailored to diminish or effectively eliminate the edge scattering effect and may contribute to the owl hush-kit.

Wiener-Hopf equations involving $n\times n$ matrices can arise when solving mixed boundary value problems with $n$ junctions at which the boundary condition to be imposed changes form.  The offset Fourier transforms of the unknown boundary values lead to exponential factors which require careful consideration when applying the Wiener-Hopf technique. We consider the generalisation of an iterative method introduced recently (Kisil 2018) from $2\times 2$ to $n\times n$ problems. This may be effectively implemented numerically by employing a spectral method to compute Cauchy transforms. We illustrate the approach through various examples of scattering from collinear rigid plates and consider the merits of the iterative method relative to alternative approaches to similar problems.

Co-authors: Samar Chehade and Michel Darmon

Diffraction of the elastic plane wave by an infinite straight-edged 2D or 3D wedge made of an isotropic solid is a canonical problem that has no analytical solution. We review three major semi-analytical approaches to this problem and discuss their application in non-destructive evaluation as well as testing, cross-validation and experimental validation. We draw attention to high sensitivity of the backscatter diffraction coefficients to the Poisson ratio.

One-dimensional convolution integral operators play a crucial role in a variety of different contexts such as approximation and probability
theory, signal processing, physical problems in radiation transfer, neutron transport, diffraction problems, geological prospecting issues and quantum gases statistics,.
Motivated by this, we consider a generic eigenvalue problem for one-dimensional convolution integral operator on an interval where the kernel is
real-valued even $C^1$-smooth function which (in case of large interval) is absolutely integrable on the real line.
We show how this spectral problem can be solved by two different asymptotic techniques that take advantage of the
size of the interval.
In case of small interval, this is done by approximation with an integral operator for which there exists a commuting
differential operator thereby reducing the problem to a boundary-value problem for second-order ODE, and often
giving the solution in terms of explicitly available special functions such as prolate spheroidal harmonics.
In case of large interval, the solution hinges on solvability, by Riemann-Hilbert approach, of an approximate auxiliary
integro-differential half-line equation of Wiener-Hopf type, and culminates in simple characteristic equations for
eigenvalues, and, with such an approximation to eigenvalues, approximate eigenfunctions are given in an explicit form.
Besides the crude periodic approximation of Grenander-Szego, since 1960s, large-interval spectral results were available
only for integral operators with kernels of a rapid (typically exponential) decay at infinity or those whose symbols
are rational functions. We assume the symbol of the kernel, on the real line, to be continuous and, for the sake of
simplicity, strictly monotonically decreasing with distance from the origin. Contrary to other approaches, the proposed
method thus relies solely on the behavior of the kernel's symbol on the real line rather than the entire complex plane
which makes it a powerful tool to constructively deal with a wide range of integral operators.
We note that, unlike finite-rank approximation of a compact operator, the auxiliary problems arising in both small-
and large-interval cases admit infinitely many solutions (eigenfunctions) and hence structurally better represent
the original integral operator.
The present talk covers an extension and significant simplification of the previous author's result on
Love/Lieb-Liniger/Gaudin equation.

In the first part we discuss a steady-state problem for an interface crack between two dissimilar elastic materials. We consider a model of the process zone described by imperfect transmission conditions that reflect the bridging effect along a finite part of the interface in front of the crack. By means of Fourier transform, the problem is reduced into a Wiener-Hopf equation with a 2x2 matrix, containing oscillatory terms. We factorize the kernel following an existing numerical method and analyse its performance for various parameters of the problem. We show that the model under consideration leads to the classic stress singularity at the crack tip. Finally, we derive conditions for the existence of an equilibrium state and compute admissible length of the process zone.  
For the second part of the talk, we consider propagation of a dynamic crack in a periodic structure with internal energy. The structural interface is formed by a discrete set of uniformly distributed alternating compressed and stretched bonds. In such a structure, the fracture of the initially stretched bonds is followed by that of the compressed ones with an unspecified time-lag. That, in turn, reflects the impact of both the internal energy accumulated inside the pre-stressed interface and the energy brought into the system by external loading. The application to the original problem of continuous (with respect to time) and  selective discrete (with respect to spatial coordinate) Fourier transforms yields another vector Wiener-Hopf equation with a kernel containing oscillating terms. We use a perturbation technique to factorise the matrix.   Finally, we show similarities and differences of the matrix-valued kernels mentioned above and discuss the chosen approaches for their factorisation.

The unified transform is typically associated with the solution of integrable nonlinear PDEs. However, after an appropriate linearisation, one can also apply the method to linear PDEs and develop a spectral boundary-based method. I will discuss recent advances of this method, in particular, the application of the method to problems in unbounded domains with solutions having corner singularities. Consequently, a wide variety of mixed boundary condition problems can be solved without the need for the Wiener-Hopf technique. Such problems arise frequently in acoustic scattering or in the calculation of electric fields in geometries involving finite and/or multiple plates. The new approach constructs a global relation that relates known boundary data, such as the scattered normal velocity on a rigid plate, to unknown boundary values, such as the jump in pressure upstream of the plate. This can be viewed formally as a domain dependent Fourier transform of the boundary integral equations. By approximating the unknown boundary functions in a suitable basis expansion and evaluating the global relation at collocation points, one can accurately obtain the expansion coefficients of the unknown boundary values. The local choice of basis functions is flexible, allowing the user to deal with singularities and complicated boundary conditions such as those occurring in elasticity models or spatially variant Robin boundary conditions modelling porosity.

In this work we study diffraction of a plane incident wave in a complex 2D domain composed by two shifted angular domains having a part of their common boundary. The perfect (Dirichlet or Neumann) boundary conditions are postulated on the polygonal boundary of such compound domain. By means of the Sommerfeld-Malyuzhinets technique the boundary-value problem at hand is reduced to a non-standard systems of Malyuzhinets-type functional-integral equations and then to a Fredholm integral equation of the second kind. Existence and uniqueness of the solution for the diffraction problem is studied and is based on the Fredholm alternative for the integral equation. The far field asymptotics of the wave field is also addressed.

The important open canonical problem of wave diffraction by a penetrable wedge is considered in the high-contrast limit. Mathematically, this means that the contrast parameter, the ratio of a specific material property of the host and the wedge scatterer, is assumed small. The relevant material property depends on the physical context and is different for acoustic and electromagnetic waves for example. Based on this assumption, a new asymptotic iterative scheme is constructed. The solution to the penetrable wedge is written in terms of an infinite sequence of (possibly inhomogeneous) impenetrable wedge problems. Each impenetrable problem is solved using a combination of the Sommerfeld-Malyuzhinets and Wiener-Hopf techniques. The resulting approximate solution to the penetrable wedge involves a large number of nested complex integrals and is hence difficult to evaluate numerically. In order to address this issue, a subtle method (combining asymptotics, interpolation and complex analysis) is developed and implemented, leading to a fast and efficient numerical evaluation. This asymptotic scheme is shown to have excellent convergent properties and leads to a clear improvement on extant approaches.