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On the Accurate Long-Time Solution of the Wave Equation in Exterior Domains: Asymptotic Expansions and Corrected Boundary Conditions

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On the Accurate Long-Time Solution of the Wave Equation in Exterior Domains: Asymptotic Expansions and Corrected Boundary Conditions

4.85 43,423 Ratings
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Editor's Note:
We consider the solution of scattering problems for the wave equation using approximate boundary conditions at artificial boundaries. These conditions are explicitly viewed as approximations to an exact boundary condition satisfied by the solution on the unbounded domain. We study the short and long term behavior of the error. It is provided that, in two space dimensions, no local in time, constant coefficient boundary operator can lead to accurate results uniformly in time for the class of problems we consider. A variable coefficient operator is developed which attains better accuracy (uniformly in time) than is possible with constant coefficient approximations. The theory is illustrated by numerical examples. We also analyze the proposed boundary conditions using energy methods, leading to asymptotically correct error bounds. Hagstrom, Thomas and Hariharan, S. I. and Maccamy, R. C. Glenn Research Center NCC3-233; RTOP 505-90-54...

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Length 48 Pages
Imprint Createspace Independent Publishing Platform
Released N/A
Identifier (ISBN) 41667220
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National Aeronautics and Space Administration

National Aeronautics and Space Administration is a writer and avid reader based in the Pacific Northwest. Before pursuing writing full-time, they spent over a decade working in education. When not working on their next book, National Aeronautics and Space Administration can usually be found hiking with their dog or trying out new coffee shops in their neighborhood.

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Gary Torres
Gary Torres June 19, 2025
"Absolutely brilliant. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. An effortless 5 stars."
David Rodriguez
David Rodriguez February 02, 2024
"Masterpiece from start to finish. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. Worth every moment spent reading."
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Sandra Miller
Sandra Miller April 22, 2025
"Excellent, though slightly flawed. It held my rapt attention, even when certain twists were telegraphed. Quite impressive overall."
Joseph Rodriguez
Joseph Rodriguez January 01, 2025
"A very strong entry. The protagonist was compelling, though some subplots felt slightly overshadowed. A very distinguished novel."
Elizabeth Cruz
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"Honestly one of the most premium reading experiences lately. I adored how the story unfolded. It kept me captivated without sacrificing elegance. An absolute must-read for connoisseurs of the genre."
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"I was completely blown away. I adored how the story unfolded. It kept me captivated without sacrificing elegance. Worth every moment spent reading."
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Nicholas Smith November 08, 2024
"A highly sophisticated read. The protagonist was compelling, though some subplots felt slightly overshadowed. Quite impressive overall."
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"I thoroughly enjoyed the elegance of it. The middle had a deliberate pace, but the climax was phenomenal. A very distinguished novel."
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Mary Miller
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"Such a rich narrative. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. I will undoubtedly revisit this masterpiece."
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"Absolutely brilliant. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. An effortless 5 stars."
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"Absolutely brilliant. I adored how the story unfolded. It kept me captivated without sacrificing elegance. An absolute must-read for connoisseurs of the genre."
Joseph Clark
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"Took a moment to build, but the payoff was spectacular. It held my rapt attention, even when certain twists were telegraphed. Quite impressive overall."
Sandra Robinson
Sandra Robinson July 28, 2024
"Exceeded all my expectations. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. A true literary gem."
Donna Baker
Donna Baker November 20, 2025
"Masterpiece from start to finish. I adored how the story unfolded. It kept me captivated without sacrificing elegance. A true literary gem."
George Lee
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Edward Adams
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"Took a moment to build, but the payoff was spectacular. Beautiful writing, though I craved slightly more depth in the lore. Still, undeniably engaging. A wonderful companion for a quiet weekend."
Karen Martinez
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"A highly sophisticated read. The middle had a deliberate pace, but the climax was phenomenal. A wonderful companion for a quiet weekend."
Ryan  Parker
Ryan Parker July 01, 2025
"Such a rich narrative. The prose is exquisite. It’s accessible yet undeniably sophisticated. I am now a lifelong fan of this author."
John Lopez
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"I was completely blown away. The author orchestrated a breathtaking finale. Truly a work of art. Worth every moment spent reading."
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"Absolutely brilliant. The prose is exquisite. It’s accessible yet undeniably sophisticated. An effortless 5 stars."
Laura Green
Laura Green March 05, 2024
"I find myself somewhat conflicted. It verged on repetitive in the second act. A more concise edit would have served it well. Perhaps more aligned with a different audience's tastes."
John Gomez
John Gomez May 02, 2024
"Exceeded all my expectations. I adored how the story unfolded. It kept me captivated without sacrificing elegance. I will undoubtedly revisit this masterpiece."
John Jones
John Jones January 11, 2024
"I was completely blown away. The author orchestrated a breathtaking finale. Truly a work of art. Worth every moment spent reading."
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"Masterpiece from start to finish. The author orchestrated a breathtaking finale. Truly a work of art. I will undoubtedly revisit this masterpiece."
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"I was completely blown away. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. I will undoubtedly revisit this masterpiece."
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"Exceeded all my expectations. The prose is exquisite. It’s accessible yet undeniably sophisticated. I am now a lifelong fan of this author."
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"Absolutely brilliant. The author orchestrated a breathtaking finale. Truly a work of art. I will undoubtedly revisit this masterpiece."
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"Honestly one of the most premium reading experiences lately. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. I am now a lifelong fan of this author."
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"Excellent, though slightly flawed. A brilliant concept executed with flair. Just a few pacing choices I would change. A wonderful companion for a quiet weekend."
George Flores
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Donald Turner
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Paul  Parker
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Daniel Diaz
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"Such a rich narrative. The prose is exquisite. It’s accessible yet undeniably sophisticated. An effortless 5 stars."
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"Such a rich narrative. I adored how the story unfolded. It kept me captivated without sacrificing elegance. Worth every moment spent reading."
Elizabeth Evans
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"I was completely blown away. The prose is exquisite. It’s accessible yet undeniably sophisticated. Worth every moment spent reading."
Edward Thompson
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"A profoundly satisfying read. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. I am now a lifelong fan of this author."
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"A decent, if standard, experience. It possessed moments of brilliance, but the narrative felt slightly disjointed ultimately. Suitable if your reading queue is empty."
Lisa Johnson
Lisa Johnson February 16, 2025
"Honestly one of the most premium reading experiences lately. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. I will undoubtedly revisit this masterpiece."
Sharon Lee
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"Honestly one of the most premium reading experiences lately. The author orchestrated a breathtaking finale. Truly a work of art. An absolute must-read for connoisseurs of the genre."
Steven Adams
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"Masterpiece from start to finish. The author orchestrated a breathtaking finale. Truly a work of art. An absolute must-read for connoisseurs of the genre."
Barbara Martinez
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"I thoroughly enjoyed the elegance of it. Beautiful writing, though I craved slightly more depth in the lore. Still, undeniably engaging. A solid 4 stars from me."
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"Such a rich narrative. I adored how the story unfolded. It kept me captivated without sacrificing elegance. A true literary gem."
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"Such a rich narrative. I became deeply invested in the protagonist's journey. A masterclass in storytelling. An effortless 5 stars."
Cynthia Anderson
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"Such a rich narrative. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. An effortless 5 stars."
Deborah Harris
Deborah Harris November 03, 2024
"A highly sophisticated read. A brilliant concept executed with flair. Just a few pacing choices I would change. Quite impressive overall."
Brian Lee
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"Masterpiece from start to finish. I adored how the story unfolded. It kept me captivated without sacrificing elegance. I will undoubtedly revisit this masterpiece."
Paul Baker
Paul Baker March 30, 2025
"Honestly one of the most premium reading experiences lately. I became deeply invested in the protagonist's journey. A masterclass in storytelling. An effortless 5 stars."
Joseph Clark
Joseph Clark November 17, 2024
"I thoroughly enjoyed the elegance of it. It held my rapt attention, even when certain twists were telegraphed. Quite impressive overall."
Deborah Harris
Deborah Harris July 19, 2024
"Absolutely brilliant. I became deeply invested in the protagonist's journey. A masterclass in storytelling. A true literary gem."
Cynthia Wright
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Linda Williams
Linda Williams January 02, 2025
"Absolutely brilliant. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. A true literary gem."
Charles Hall
Charles Hall February 27, 2025
"Exceeded all my expectations. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. Worth every moment spent reading."
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"Exceeded all my expectations. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. I will undoubtedly revisit this masterpiece."
Ryan Taylor
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"I thoroughly enjoyed the elegance of it. The middle had a deliberate pace, but the climax was phenomenal. A solid 4 stars from me."
George Brown
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"Exceeded all my expectations. I became deeply invested in the protagonist's journey. A masterclass in storytelling. A true literary gem."
Laura Sanchez
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Elizabeth Scott
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"A profoundly satisfying read. I adored how the story unfolded. It kept me captivated without sacrificing elegance. I am now a lifelong fan of this author."
Sandra  Parker
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"Such a rich narrative. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. I will undoubtedly revisit this masterpiece."
Kimberly Harris
Kimberly Harris April 22, 2024
"A profoundly satisfying read. The prose is exquisite. It’s accessible yet undeniably sophisticated. An absolute must-read for connoisseurs of the genre."
Brian Lee
Brian Lee July 19, 2025
"Took a moment to build, but the payoff was spectacular. The protagonist was compelling, though some subplots felt slightly overshadowed. A wonderful companion for a quiet weekend."
Mark  Parker
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"Such a rich narrative. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. I am now a lifelong fan of this author."
Matthew Mitchell
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"I was completely blown away. The prose is exquisite. It’s accessible yet undeniably sophisticated. An effortless 5 stars."
Carol Carter
Carol Carter January 31, 2025
"Absolutely brilliant. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. An absolute must-read for connoisseurs of the genre."
Jennifer Jones
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"Masterpiece from start to finish. The prose is exquisite. It’s accessible yet undeniably sophisticated. An effortless 5 stars."
George Thomas
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"Absolutely brilliant. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. I will undoubtedly revisit this masterpiece."
Elizabeth Adams
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"Took a moment to build, but the payoff was spectacular. The middle had a deliberate pace, but the climax was phenomenal. Quite impressive overall."
Betty Johnson
Betty Johnson October 23, 2024
"I was completely blown away. The prose is exquisite. It’s accessible yet undeniably sophisticated. An effortless 5 stars."
Laura Green
Laura Green March 07, 2024
"A very strong entry. It held my rapt attention, even when certain twists were telegraphed. I would highly recommend this to discerning readers."
Andrew Turner
Andrew Turner December 27, 2024
"Absolutely brilliant. The author orchestrated a breathtaking finale. Truly a work of art. A true literary gem."
Ryan Phillips
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"An acceptable diversion. It verged on repetitive in the second act. A more concise edit would have served it well. Suitable if your reading queue is empty."
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"An acceptable diversion. It verged on repetitive in the second act. A more concise edit would have served it well. Perfectly average."
John  Parker
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"Such a rich narrative. I adored how the story unfolded. It kept me captivated without sacrificing elegance. An absolute must-read for connoisseurs of the genre."
Michael Thompson
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"Not entirely to my refined tastes. Certain acts were fascinating, while others felt like a slight endeavor to complete. Perfectly average."
Barbara Rodriguez
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"Took a moment to build, but the payoff was spectacular. The protagonist was compelling, though some subplots felt slightly overshadowed. I would highly recommend this to discerning readers."
Dorothy Mitchell
Dorothy Mitchell June 21, 2024
"Masterpiece from start to finish. I adored how the story unfolded. It kept me captivated without sacrificing elegance. I will undoubtedly revisit this masterpiece."
Mark Gomez
Mark Gomez September 24, 2025
"Honestly one of the most premium reading experiences lately. The author orchestrated a breathtaking finale. Truly a work of art. I am now a lifelong fan of this author."
Elizabeth Evans
Elizabeth Evans November 27, 2024
"Honestly one of the most premium reading experiences lately. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. I will undoubtedly revisit this masterpiece."
Mary Hernandez
Mary Hernandez February 19, 2025
"A profoundly satisfying read. The prose is exquisite. It’s accessible yet undeniably sophisticated. An absolute must-read for connoisseurs of the genre."
Betty Edwards
Betty Edwards May 18, 2024
"Exceeded all my expectations. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. A true literary gem."
Gary Campbell
Gary Campbell September 26, 2024
"Absolutely brilliant. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. A true literary gem."
Patricia Lopez
Patricia Lopez March 18, 2024
"I was completely blown away. I became deeply invested in the protagonist's journey. A masterclass in storytelling. Worth every moment spent reading."
Robert Thomas
Robert Thomas January 15, 2024
"Excellent, though slightly flawed. The protagonist was compelling, though some subplots felt slightly overshadowed. Quite impressive overall."
Rebecca Diaz
Rebecca Diaz January 21, 2025
"A profoundly satisfying read. The author orchestrated a breathtaking finale. Truly a work of art. A true literary gem."
Michelle Sanchez
Michelle Sanchez July 14, 2024
"I find myself somewhat conflicted. The prose was competent, yet the characters lacked the depth I usually seek. Perfectly average."
Paul Lewis
Paul Lewis February 22, 2025
"Masterpiece from start to finish. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. An absolute must-read for connoisseurs of the genre."
Michelle Gomez
Michelle Gomez October 08, 2024
"I was completely blown away. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. A true literary gem."
John Jones
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"It was adequate. It verged on repetitive in the second act. A more concise edit would have served it well. A fair 3 stars."
Michelle Gomez
Michelle Gomez April 15, 2024
"Absolutely brilliant. The author orchestrated a breathtaking finale. Truly a work of art. Worth every moment spent reading."
Robert Rodriguez
Robert Rodriguez May 22, 2025
"Masterpiece from start to finish. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. A true literary gem."
Charles Evans
Charles Evans October 02, 2024
"A highly sophisticated read. Beautiful writing, though I craved slightly more depth in the lore. Still, undeniably engaging. Quite impressive overall."
Steven Smith
Steven Smith November 25, 2025
"I thoroughly enjoyed the elegance of it. The middle had a deliberate pace, but the climax was phenomenal. A very distinguished novel."
Charles King
Charles King December 02, 2025
"Honestly one of the most premium reading experiences lately. I adored how the story unfolded. It kept me captivated without sacrificing elegance. I will undoubtedly revisit this masterpiece."
James Nelson
James Nelson October 30, 2025
"Exceeded all my expectations. I adored how the story unfolded. It kept me captivated without sacrificing elegance. I am now a lifelong fan of this author."
Jessica Hill
Jessica Hill July 04, 2025
"I was completely blown away. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. An effortless 5 stars."
Thomas White
Thomas White August 01, 2024
"A profoundly satisfying read. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. An effortless 5 stars."
John Martin
John Martin July 30, 2025
"A profoundly satisfying read. The author orchestrated a breathtaking finale. Truly a work of art. A true literary gem."
Jennifer Thompson
Jennifer Thompson June 06, 2025
"Honestly one of the most premium reading experiences lately. I became deeply invested in the protagonist's journey. A masterclass in storytelling. Worth every moment spent reading."
Elizabeth Adams
Elizabeth Adams December 31, 2024
"Excellent, though slightly flawed. A brilliant concept executed with flair. Just a few pacing choices I would change. A wonderful companion for a quiet weekend."
Kevin Nguyen
Kevin Nguyen March 15, 2025
"I was completely blown away. I adored how the story unfolded. It kept me captivated without sacrificing elegance. An absolute must-read for connoisseurs of the genre."
Nancy Baker
Nancy Baker March 20, 2025
"Exceeded all my expectations. I adored how the story unfolded. It kept me captivated without sacrificing elegance. An absolute must-read for connoisseurs of the genre."
Matthew Turner
Matthew Turner April 10, 2025
"A highly sophisticated read. It held my rapt attention, even when certain twists were telegraphed. Quite impressive overall."
Susan Sanchez
Susan Sanchez December 03, 2025
"Exceeded all my expectations. I adored how the story unfolded. It kept me captivated without sacrificing elegance. An absolute must-read for connoisseurs of the genre."
Rebecca Clark
Rebecca Clark August 27, 2025
"Masterpiece from start to finish. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. Worth every moment spent reading."
Stephanie Ramirez
Stephanie Ramirez January 18, 2025
"Exceeded all my expectations. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. An absolute must-read for connoisseurs of the genre."
Andrew Williams
Andrew Williams December 01, 2024
"A highly sophisticated read. The protagonist was compelling, though some subplots felt slightly overshadowed. Quite impressive overall."
Edward Clark
Edward Clark December 19, 2024
"I find myself somewhat conflicted. The prose was competent, yet the characters lacked the depth I usually seek. Perfectly average."
Andrew Perez
Andrew Perez September 15, 2025
"Not entirely to my refined tastes. The tempo was somewhat languid, and I found my mind wandering. Not poor, simply average. Perfectly average."
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