# 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 Length: 2-3 days
AIAA CEU's available: Yes

Outline

Course Outline:

I. Introduction, Classification, and Advantages of Perturbation Methods
A. Landau gauge orders and series expansions
B. Systematic solution of transcendental equations

II. Regular Perturbation Methods
A. Regular expansion series and Bürmann’s inverse approach
B. Successive approximations and the method of ansatz
C. Dimensional analysis and scaling
D. The von Karman and Laplace equations

III. Rayleigh-Janzen Expansion Technique

IV. Singular Perturbation Methods
A. Identification of secular terms and singular behavior
B. Construction of uniformly valid approximations

V. Strained Coordinate Expansion (SCE) Techniques
A. Lindstedt’s method of stretched coordinates
B. PLK’s method of deformed coordinates
C. Pritulo’s method of renormalization

VI. Matched Asymptotic Expansion (MAE) Technique
A. Treatment of boundary layers and inner/outer approximations
B. Distinguished limits; Prandtl’s and Van Dyke’s matching principles
C. Uniformly valid composite solutions

VII. Multiple Scales Expansion (MSE) Technique
A. Slow and fast scales
B. Stretched and compressed scales

VIII. WKB Technique

IX. Van der Pol’s Method of Averaging

X. Latta’s Method of Composite Expansions (MCE)

XII. Asymptotic Expansion of Integrals:
A. Laplace’s Method
B. Watson’s Lemma

Materials
Instructors

Dr. Majdalani presently serves as Professor and Francis Chair of Aerospace Engineering at Auburn University. He previously served as the Auburn Alumni Engineering Council Endowed Professor and Department Chair of Aerospace Engineering at Auburn University (2013–2016) as well as the Jack D. Whitfield Professor and H. H. Arnold Chair of Excellence in Advanced Propulsion at the University of Tennessee (2003–2013).  Dr. Majdalani is known for his work on acoustic instability theory and vortex-driven rocket engine technology encompassing solid, liquid and hybrid rocket applications. He is presently a Fellow of ASME, Chair of the AIAA Hybrid Rockets Technical Committee (2015–2017), Chair of the Solid Rockets Technical Committee (2017–2019), Chair of the SRTC Awards Subcommittee, Director of Honors & Awards within the Greater Huntsville Section, Associate Editor of the International Journal of Energetic Materials and Chemical Propulsion, ISICP President Elect, and AIAA Short Course Instructor.

Dr. Majdalani’s research devotes itself to the computational modeling and optimization of solid, liquid and hybrid rocket engines. His interests span rocket engine design and optimization, rocket internal ballistics, vorticity dynamics, computational mathematics, finite volume methods, and singular perturbation theory.  His research activities since 1997 have materialized in over 280 publications in first-rate journals, book chapters, and conference proceedings, mostly in the field of rocket propulsion.  His work on helical flow modeling has led to the discovery of new Trkalian and Beltramian families of solutions to describe cyclonic motions in self-cooled, multi-phase liquid and hybrid rocket engines.  These have paved the way to understand and optimize a family of cyclonically-driven hybrid and liquid rocket engines. His work on wave propagation has resulted in the development of a generalized-scaling technique in perturbation theory, and of a consistently compressible framework for capturing both vorticoacoustic and biglobal stability waves in simulated combustors.  These have led to a new framework for modeling combustion instability in rocket systems.  Recently, his work on compressible gas motions has required the inception of a systematic procedure for modeling high speed flow problems.  In fact, a total of eighteen dimensionless parameters have been newly identified in the course of his research investigations.  These parameters have played a key role in guiding experimental procedures and shaping research investigations that explore the technical benefits of swirl driven rocket engines, thus leading to a ground-up optimization framework that is presently being used by Sierra Nevada Corporation/ORBITEC.  This framework starts with CAD drawings, and iterates through a vorticoacoustic solver until a high performance and stable engine is achieved.

Throughout his career, Dr. Majdalani has received several professional awards such as: