Basic Partial Differential Equations

Quantities Which Depend On Space And/or Time Variables Are Often Governed By Differential Equations Which Are Based On Underlying Physical Principles. Partial Differential Equations (pdes) Not Only Accurately Express These Principles, But Also Help To Predict The Behavior Of A System From An Initial State Of The System And From Given External Influences. Thus, It Is Hard To Overestimate The Relevance Of Pdes In All Forms Of Science And Engineering, Or Any Endeavor Which Involves Reasonably Smooth, Predictable Changes Of Measurable Quantities. Having Taught From The Material In This Book For Ten Years With Much Feedback From Students, We Have Found That The Book Serves As A Very Readable Introduction To The Subject For Undergraduates With A Year And A Half Of Calculus, But Not Necessarily Any More. In Particular, One Need Not Have Had A Linear Algebra Course Or Even A Course In Ordinary Differential Equations To Understand The Material. As The Title Suggests, We Have Concentrated Only On What We Feel Are The Absolutely Essential Aspects Of The Subject, And There Are Some Crucial Topics Such As Systems Of Pdes Which We Only Touch On. Yet The Book Certainly Contains More Material Than Can Be Covered In A Single Semester, Even With An Exceptional Class. Given The Broad Relevance Of The Subject, We Suspect That A Demand For A Second Semester Surely Exists, But Has Been Largely Unmet, Partly Due To The Lack Of Books Which Take The Time And Space To Be Readable By Sophomores. Content: Front Matter....Pages i-xvii Review and Introduction....Pages 1-56 First — Order PDEs....Pages 57-120 The Heat Equation....Pages 121-186 Fourier Series and Sturm-Liouville Theory....Pages 187-280 The Wave Equation....Pages 281-338 Laplace’s Equation....Pages 339-414 Fourier Transforms....Pages 415-502 Numerical Solutions of PDEs — An Introduction....Pages 503-558 PDEs in Higher Dimensions....Pages 559-676 Back Matter....Pages 677-747