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Optimization Techniques in Modern Engineering

From Classical Methods to Metaheuristic Approaches

A comprehensive reference on classical and modern optimization methods including linear programming, convex optimization, genetic algorithms, swarm intelligence, and gradient-free methods. Chapters address applications in logistics, structural design, energy systems, and machine learning.

Adaptive Finite Element Methods for Elliptic Partial Differential Equations

Sofia Petrova, Dmitri Volkov

In: Recent Advances in Numerical Methods and Computational Mathematics

We present a novel a posteriori error estimation framework for adaptive finite element methods applied to second-order elliptic PDEs. Our approach combines residual-based error indicators with hierarchical basis functions, achieving optimal convergence rates on both structured and unstructured meshes. Numerical experiments on benchmark problems confirm the efficiency and robustness of the proposed algorithm.

High-Order Spectral Methods for Incompressible Navier-Stokes Equations

Amara Patel, Ravi Sharma

In: Recent Advances in Numerical Methods and Computational Mathematics

This chapter presents high-order spectral element discretizations for the incompressible Navier-Stokes equations. We introduce a novel pressure-velocity coupling strategy that ensures inf-sup stability while maintaining spectral accuracy. The method is validated on Taylor-Green vortex and lid-driven cavity benchmark problems at moderate Reynolds numbers.

Krylov Subspace Methods for Large-Scale Sparse Linear Systems

Sofia Petrova

In: Recent Advances in Numerical Methods and Computational Mathematics

We survey and extend Krylov subspace iterative methods for solving large-scale sparse linear systems arising in scientific computing. Special attention is paid to preconditioned GMRES, BiCGSTAB, and algebraic multigrid preconditioning. Performance comparisons on real-world application matrices from structural engineering and computational electromagnetics are provided.

Recent Advances in Numerical Methods and Computational Mathematics

Theory, Algorithms, and Applications in Science and Engineering

Eds. Priya Nair, Hans Müller

This edited volume brings together contributions from world-leading researchers on the forefront of numerical analysis and computational mathematics. The book covers finite element methods, spectral methods, iterative solvers, and high-performance computing techniques, with applications spanning fluid dynamics, structural mechanics, and quantum systems.

Stochastic Differential Equations in Financial Mathematics

Amara Patel, Sunita Krishnan

In: Mathematical Modeling in Physical Sciences

This paper reviews stochastic differential equations (SDEs) and their applications in financial modeling. We examine the Black-Scholes framework, Heston stochastic volatility model, and jump-diffusion processes. Numerical methods for SDE simulation including Euler-Maruyama, Milstein, and Runge-Kutta schemes are analyzed with convergence guarantees.

Mathematical Models for Epidemic Spread: From SIR to Network-Based Approaches

Sofia Petrova, Igor Morozov

In: Mathematical Modeling in Physical Sciences

We develop and analyze mathematical models for the spread of infectious diseases. Starting from classical compartmental models (SIR, SEIR), we extend to network-based epidemic models that incorporate heterogeneous contact patterns. Stability analysis, threshold theorems, and parameter estimation from COVID-19 data are presented.

Mathematical Modeling in Physical Sciences

Proceedings of the International Conference on Applied Mathematics 2022

Eds. Priya Nair, Yuki Tanaka

This conference proceeding captures the most significant contributions presented at ICAM 2022. Topics include partial differential equations, stochastic modeling, fluid mechanics, and mathematical biology. The volume represents cutting-edge interdisciplinary research from over 30 countries.