Article

# On The Solution Of Wiener-Hopf Problems Involving Noncommutative Matrix Kernel Decompositions

SIAM Journal on Applied Mathematics (Impact Factor: 1.41). 11/2000; DOI: 10.1137/S0036139995287673

Source: CiteSeer

- [Show abstract] [Hide abstract]

**ABSTRACT:**In contrast to scalar Riemann-Hilbert problems, a general matrix Riemann-Hilbert problem cannot be solved in terms of Sokhotskyi-Plemelj integrals. As far as the authors know, the exact solutions are known for a class of homogeneous matrix Riemann-Hilbert problems with commutative and factorable kernels. This article considered matrix Riemann-Hilbert problems in which all the partial indices are zero and the logarithms of the components of the kernels and their non-homogeneous vectors are exponential-type (equivalently, band-limited) functions. Then, it develops exact solutions for such matrix Riemann-Hilbert problems. Applications in a class of spectral factorizations and a class of the Wiener-Hopf system of integrations are given.IMA Journal of Applied Mathematics. 01/2014; 79(1). - [Show abstract] [Hide abstract]

**ABSTRACT:**We present a solution for the interaction of normally incident linear waves with a submerged elastic plate of semi-infinite extent, where the water has finite depth. While the problem has been solved previously by the eigenfunction-matching method, the present study shows that this problem is also amenable to the more analytical, and extremely efficient, Wiener–Hopf (WH) and residue calculus (RC) methods. We also show that the WH and RC solutions are actually equivalent for problems of this type, a result which applies to many other problems in linear wave theory. (e.g., the much-studied floating elastic plate scattering problem, or acoustic wave propagation in a duct where one wall has an abrupt change in properties.) We present numerical results and a detailed convergence study, and discuss as well the scattering by a submerged rigid dock, particularly the radiation condition beneath the dock.Journal of Engineering Mathematics 01/2012; 75(1). · 1.07 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**A matrix factorization problem is considered. The matrix is algebraic and belongs to the Jones-Moiseev class. A new method of factorization is proposed. The matrix factorization problem is reduced to a Riemann-Hilbert problem using Hurd’s method. The Riemann-Hilbert problem is embedded into a family of Riemann-Hilbert problems indexed by a variable b taking values on a half-line. A linear ordinary differential equation (ODE1) with respect to b is derived. The coefficient of this equation remains unknown at this step. Finally, the coefficient of the ODE1 is computed. For this, it is proven that this coefficient obeys a non-linear ordinary differential equation (ODE2) on a half-line. Thus, the numerical procedure of matrix factorization becomes reduced to two runs of solving ordinary differential equations on a half-line: first ODE2 for the coefficient of ODE1, and then ODE1 for the unknown function. The efficiency of the new method is demonstrated on some examples.The Quarterly Journal of Mechanics and Applied Mathematics 11/2013; 66(4). · 0.57 Impact Factor

Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed. The impact factor represents a rough estimation of the journal's impact factor and does not reflect the actual current impact factor. Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence agreement may be applicable.