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This volume presents current trends in analysis and partial differential equations from researchers in developing countries. The fruit of the project 'Analysis in Developing Countries', whose aim was to bring together researchers from around the world, the volume also includes some contributions from researchers from developed countries.

Focusing on topics in analysis related to partial differential equations, this volume contains selected contributions from the activities of the project at Imperial College London, namely the conference on Analysis and Partial Differential Equations held in September 2016 and the subsequent Official Development Assistance Week held in November 2016. Topics represented include Fourier analysis, pseudo-differential operators, integral equations, as well as related topics from numerical analysis and bifurcation theory, and the countries represented range from Burkina Faso and Ghana to Armenia, Kyrgyzstan and Tajikistan, including contributions from Brazil, Colombia and Cuba, as well as India and China.

Suitable for postgraduate students and beyond, this volume offers the reader a broader, global perspective of contemporary research in analysis.

Inhaltsverzeichnis

Frontmatter

Analysis in Developing Countries

Frontmatter

Francis K. A. Allotey (Saltpond, Ghana 9 August 1932 – Accra, Ghana 2 November 2017)

Abstract
This is a brief Biography of Professor Francis K. Allotey.
Emmanuel K. Essel

State of Mathematics in Africa and the Way Forward

Abstract
The paper discusses some of the factors influencing students poor performance in mathematics in Africa and suggests ways to improve it. Examples of international centres and networks for capacity building in mathematical sciences are given.
Francis K. A. Allotey

The Optimal Vector Control for the Elastic Oscillations Described by Fredholm Integral-Differential Equations

Abstract
In this paper, we investigate the nonlinear problem of the optimal vector control for oscillation processes described by Fredholm integro-differential equations in partial derivatives when function of external sources nonlinearly depend on control parameters. It was found that the system of nonlinear integral equations,which obtained relatively to the components of the optimal vector control, have the property of equal relations. This fact lets us to simplify the procedure of the constructing the solution of the nonlinear optimization problem. We have developed algorithm for constructing the solution of the nonlinear optimization problem.
Elmira Abdyldaeva, Akylbek Kerimbekov

On Nuclear -Multipliers Associated to the Harmonic Oscillator

Abstract
In this paper we study multipliers associated to the harmonic oscillator (also called Hermite multipliers) belonging to the ideal of r-nuclear operators on Lebesgue spaces. We also study the nuclear trace and the spectral trace of these operators.
Edgardo Samuel Barraza, Duván Cardona

Hermite Multipliers on Modulation Spaces

Abstract
We study multipliers associated to the Hermite operator \(H=-\varDelta + |x|^2\) on modulation spaces \(M^{p,q}(\mathbb R^d)\). We prove that the operator m(H) is bounded on \(M^{p,q}(\mathbb R^d)\) under standard conditions on m,  for suitable choice of p and q. As an application, we point out that the solutions to the free wave and Schrödinger equations associated to H with initial data in a modulation space will remain in the same modulation space for all times. We also point out that Riesz transforms associated to H are bounded on some modulation spaces.
Divyang G. Bhimani, Rakesh Balhara, Sundaram Thangavelu

Potential Theory Results for a Class of PDOs Admitting a Global Fundamental Solution

Abstract
We outline several results of Potential Theory for a class of linear partial differential operators \(\mathcal {L}\) of the second order in divergence form. Under essentially the sole assumption of hypoellipticity, we present a non-invariant homogeneous Harnack inequality for \(\mathcal {L}\); under different geometrical assumptions on \(\mathcal {L}\) (mainly, under global doubling/Poincaré assumptions), it is described how to obtain an invariant, non-homogeneous Harnack inequality. When \(\mathcal {L}\) is equipped with a global fundamental solution \(\varGamma \), further Potential Theory results are available (such as the Strong Maximum Principle). We present some assumptions on \(\mathcal {L}\) ensuring that such a \(\varGamma \) exists.
Andrea Bonfiglioli

Semiclassical Analysis of Dispersion Phenomena

Abstract
Our aim in this work is to give some quantitative insight on the dispersive effects exhibited by solutions of a semiclassical Schrödinger-type equation in \(\mathbf{R}^d\). We describe quantitatively the localisation of the energy in a long-time semiclassical limit within this non compact geometry and exhibit conditions under which the energy remains localized on compact sets. We also explain how our results can be applied in a straightforward way to describe obstructions to the validity of smoothing type estimates.
Victor Chabu, Clotilde Fermanian-Kammerer, Fabricio Macià

Convergence of Fourier-Walsh Double Series in Weighted

Abstract
In this work we discuss the behavior of Fourier coefficients with respect to the Walsh double system, as well as \(L_{\mu }^{p}[0,1)^{2}\)-convergence of the spherical partial sums of the double Fourier-Walsh series after modification of functions.
Martin G. Grigoryan, Tigran M. Grigoryan, L. S. Simonyan

“Strong” Turing-Hopf Instability for Reaction-Diffusion Systems

Abstract
Turing-Hopf instabilities for reaction-diffusion systems provide spatially inhomogeneous time-periodic patterns of chemical concentrations. In this presentation, it is shown the parameter space in which the reaction-diffusion system modelling glycolysis and the Lengyel-Epstein model could show twinkling patterns. To do so, we follow the Ricard-Mischler procedure in Ricard and Mischler (J Nonlinear Sci 19(5):467–496, 2009, [18]), i.e., considering this phenomenom as a consequence of the instability generated by diffusion on the limit cycle which appears due to a Hopf bifurcation about the spatially homogeneous steady state.
Giani Egaña Fernández, J Sarría González, Mariano Rodríguez Ricard

Correspondence Between Multiscale Frame Shrinkage and High-Order Nonlinear Diffusion

Abstract
Wavelet frame and nonlinear diffusion filters are two popular tools for signal denoising. The correspondence between Ron-Shen’s framelet and high-order nonlinear diffusion has been proved at multilevel setting. However, for the general framelet, the correspondence is established only at one level. In this paper we extend the relationship between framelet shrinkage and high-order nonlinear diffusion in Jiang (Appl Numerical Math 51–66, 2012 [19]) from one level framelet shrinkage to the multilevel framelet shrinkage setting. Subsequently, we complete the correspondence between framelet shrinkage and high-order nonlinear diffusion. Furthermore, we propose a framelet-diffused denoising method for processing the dynamic pressure signals which are generated by a transonic axial compressor. Numerical results show that our algorithm has superior noise removal ability than traditional algorithms and presents the ability in analyzing the pressure signals from an axial transonic compressor.
Haihui Wang, Qi Huang, Bo Meng

Pseudo-differential Operators Associated to General Type I Locally Compact Groups

Abstract
In a recent paper by M. Măntoiu and M. Ruzhansky, a global pseudo-differential calculus has been developed for unimodular groups of type I. In the present article we generalize the main results to arbitrary locally compact groups of type I. Our methods involve the use of Plancherel’s theorem for non-unimodular groups. We also make connections with a \(C^*\)-algebraic formalism, involving dynamical systems, and give explicit constructions for the group of affine transformations of the real line.
Marius Măntoiu, Maximiliano Sandoval

Existence and Numerical Computation of Standing Wave Solutions for a System of Two Coupled Schrödinger Equations

Abstract
In this paper, we consider the existence of a type of stationary wave of a system of two coupled Schrödinger equations with variable coefficients, which can be employed to describe the interaction among propagating modes in nonlinear optics and Bose-Einstein condensates (BECs), for instance. To prove existence of these solutions, we use some existing fixed point theorems for completely continuous operators defined in a cone in a Banach space. Furthermore, some numerical approximations of stationary waves are computed by using a spectral collocation technique combined with a Newton’s iteration.
Juan Carlos Muñoz Grajales, Luisa Fernanda Vargas

Shannon Sampling and Weak Weyl’s Law on Compact Riemannian Manifolds

Abstract
The well known Weyl’s asymptotic formula gives an approximation to the number \(\mathcal {N}_{\omega }\) of eigenvalues (counted with multiplicities) on an interval \([0,{\,}\omega ]\) of an elliptic second-order differential self-adjoint non-negative operator on a compact Riemannian manifold \(\mathbf{M}\). In this paper we approach this question from the point of view of Shannon-type sampling on compact Riemannian manifolds. Namely, we give a direct proof that \(\mathcal {N}_{\omega }\) is comparable to cardinality of certain sampling sets for the subspace of \(\omega \)-bandlimited functions on \(\mathbf{M}\).
Isaac Z. Pesenson

Well-posed Boundary Value Problems for New Classes of Singular Integral Equations in Cylindrical Domains

Abstract
In this work a class of three-dimensional complex integral equation in cylindrical domains is investigated in the case when the lateral surface may have singularity or super-singularity. For this type of integral equations condition for kernels are found under which the problem of finding solution is reduced to the problem of finding two splitting systems of integral equations which can be treated by existing methods. In this case the solution are obtained in an explicit form. In the case of more general kernels the, inversion formula is found in terms of the values on the surface of the cylinder. In model cases the solution of the integral equation is found in the form of absolutely and uniformly convergent generalised power series in powers of \((t-a)\) and the inversion formula is presented. It is used to investigate further Dirichlet-type boundary problems.
Nusrat Rajabov

Weighted Stepanov-Like Pseudo Almost Automorphic Solutions of Class r for Some Partial Differential Equations

Abstract
The aim of this work is to study weighted Stepanov-like pseudo almost automorphic functions using the measure theory. We present a new concept of weighted ergodic functions which is more general than the classical one. Then we establish many interesting results on the functional space of such functions. We also study the existence and uniqueness of \((\mu ,\nu )\) -Weighted Stepanov-like pseudo almost automorphic solutions of class r for some neutral partial functional differential equations in a Banach space when the delay is distributed using the spectral decomposition of the phase space developed by Adimy and co-authors. Here we assume that the undelayed part is not necessarily densely defined and satisfies the well-known Hille-Yosida condition, the delayed part are assumed to be pseudo almost automorphic with respect to the first argument and Lipschitz continuous with respect to the second argument.
Hamidou Toure, Issa Zabsonre
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