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American Institute of Aeronautics and Astronautics

# Perturbation Methods in Science and Engineering

#### Synopsis:

“Perturbation Methods in Science and Engineering” is a must for all engineers and scientists aspiring to develop theoretical solutions to accompany their numerical and/or experimental work, irrespective of their research discipline. The majority of problems confronting engineers, physicists, and applied mathematicians encompass nonlinear differential/integral equations, transcendental relations, equations with singularities/variable coefficients, and complex boundary conditions that cannot be solved exactly. For such problems, only approximate solutions may be obtained using either numerical and/or analytical techniques. Foremost amongst analytical approximation techniques are the systematic methods of asymptotic perturbation theory. Unlike numerical solutions that can be acquired using canned packages and/or commercial solvers, the ability to derive closed-form analytical approximations to complex problems is becoming a lost art. Numerical solvers are routinely relied on to the extent that mastery of approximation methods is becoming not only a desirable tool, but rather a must amongst engineers and scientists, especially those aspiring to establish new theories and/or achieve deeper physical insight than may be gained on the basis of numerical modeling alone.

#### Key Topics:

• Regular Perturbation Methods
• Singular Perturbation Methods: Strained Coordinates, Matched Asymptotic Expansions, and Multiple Scales
• Asymptotic Principles: Prandtl, Van Dyke, and Least Singularity
• Special Topics: WKB, van der Pol’s Method of Averaging, Latta’s Method of Composite Expansions, and the Generalized Scaling Technique
• Parameter-Fee Methods: Adomian Decomposition and Homotopy Analysis Methods
• Asymptotic Expansion of Integrals: Laplace's Method and Watson's Lemma

#### Who Should Attend:

This course is aimed at bringing together professionals with interest in both conventional and modern analytical modeling approaches. Examples include the Generalized Scaling Technique (GST) and the Homotopy Analysis Method (HAM) that have been developed for the treatment of problems exhibiting multiple scales and nonlinearities. These and other powerful tools can be used to obtain explicit approximations to a whole gambit of transcendental equations and nonlinear problems that arise in aeronautical, astronautical, and aerospace applications. The course is a must-have for all scientists and engineers who wish to augment their investigative capabilities by adding a theoretical component to their research arsenal. In practical applications, analytical solutions are complementary to numerical simulations as they enable us to gain newer physical insight into complex phenomena than is permitted through computations alone. They not only offer an avenue for limiting process verifications but also provide, in most situations, a compelling justification for dissemination in prestigious journals that either favor the advancement of mathematical formulations or require the existence of a theoretical framework to accompany laboratory and/or computer-based measurements.

#### Course Information:

Type of Course: Instructor-Led Short Course
Course Level: Fundamentals/Intermediate

Course scheduling available in the following formats:

• Course at Conference
• On-site Course
• Stand-alone/Public Course

Course Length: 2-3 days

AIAA CEU's available: yes