9 research outputs found
Approximate Analytical Methods For Solving Fredholm Integral Equations
Persamaan kamiran memainkan peranan penting dalam banyak bidang sains seperti
matematik, biologi, kimia, fizik, mekanik dan kejuruteraan. Oleh yang
demikian,pelbagai teknik berbeza telah digunakan untuk menyelesaikan persamaan
jenis ini. Kajian ini, memfokus kepada analisis secara matematik dan berangka bagi
beberapa kes persamaan kamiran Fredholm yang linear dan bukan linear. Kes-kes ini
termasuklah persamaan kamiran Fredholm satu dimensi jenis pertama dan kedua,
persamaan kamiran Fredholm dua dimensi jenis pertama dan kedua dan sistem
persamaan kamiran Fredholm satu dimensi dan dua dimensi.
Integral equations play an important role in many branches of sciences such as
mathematics, biology, chemistry, physics, mechanics and engineering. Therefore, many
different techniques are used to solve these types of equations. This study focuses on the
mathematical and numerical analysis of some cases of linear and nonlinear Fredholm
integral equations. These cases are one-dimensional Fredholm integral equations of the
first kind and second kind, two-dimensional Fredholm integral equations of the first kind
and second kind and systems of one and two-dimensional Fredholm integral equations
Proximal Galerkin: A structure-preserving finite element method for pointwise bound constraints
The proximal Galerkin finite element method is a high-order, low iteration
complexity, nonlinear numerical method that preserves the geometric and
algebraic structure of bound constraints in infinite-dimensional function
spaces. This paper introduces the proximal Galerkin method and applies it to
solve free boundary problems, enforce discrete maximum principles, and develop
scalable, mesh-independent algorithms for optimal design. The paper leads to a
derivation of the latent variable proximal point (LVPP) algorithm: an
unconditionally stable alternative to the interior point method. LVPP is an
infinite-dimensional optimization algorithm that may be viewed as having an
adaptive barrier function that is updated with a new informative prior at each
(outer loop) optimization iteration. One of the main benefits of this algorithm
is witnessed when analyzing the classical obstacle problem. Therein, we find
that the original variational inequality can be replaced by a sequence of
semilinear partial differential equations (PDEs) that are readily discretized
and solved with, e.g., high-order finite elements. Throughout this work, we
arrive at several unexpected contributions that may be of independent interest.
These include (1) a semilinear PDE we refer to as the entropic Poisson
equation; (2) an algebraic/geometric connection between high-order
positivity-preserving discretizations and certain infinite-dimensional Lie
groups; and (3) a gradient-based, bound-preserving algorithm for two-field
density-based topology optimization. The complete latent variable proximal
Galerkin methodology combines ideas from nonlinear programming, functional
analysis, tropical algebra, and differential geometry and can potentially lead
to new synergies among these areas as well as within variational and numerical
analysis
Third International Conference on Inverse Design Concepts and Optimization in Engineering Sciences (ICIDES-3)
Papers from the Third International Conference on Inverse Design Concepts and Optimization in Engineering Sciences (ICIDES) are presented. The papers discuss current research in the general field of inverse, semi-inverse, and direct design and optimization in engineering sciences. The rapid growth of this relatively new field is due to the availability of faster and larger computing machines
Ahlfors circle maps and total reality: from Riemann to Rohlin
This is a prejudiced survey on the Ahlfors (extremal) function and the weaker
{\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e.
those (branched) maps effecting the conformal representation upon the disc of a
{\it compact bordered Riemann surface}. The theory in question has some
well-known intersection with real algebraic geometry, especially Klein's
ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a
gallery of pictures quite pleasant to visit of which we have attempted to trace
the simplest representatives. This drifted us toward some electrodynamic
motions along real circuits of dividing curves perhaps reminiscent of Kepler's
planetary motions along ellipses. The ultimate origin of circle maps is of
course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass.
Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found
in Klein (what we failed to assess on printed evidence), the pivotal
contribution belongs to Ahlfors 1950 supplying an existence-proof of circle
maps, as well as an analysis of an allied function-theoretic extremal problem.
Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree
controls than available in Ahlfors' era. Accordingly, our partisan belief is
that much remains to be clarified regarding the foundation and optimal control
of Ahlfors circle maps. The game of sharp estimation may look narrow-minded
"Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to
contemplate how conformal and algebraic geometry are fighting together for the
soul of Riemann surfaces. A second part explores the connection with Hilbert's
16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by
including now Rohlin's theory (v.2