John Dold is a professor of applied mathematics, having trained in physics, mathematics and (briefly) history. Apart from some purely experimental studies, particularly of fire behaviour, his research primarily makes use of partial differential equations to model practical problems, including water waves and combustion phenomena. He founded the journal Combustion Theory and Modelling and he has organised or helped to organise major conferences on combustion research. In his teaching of mathematical methods and their application to various physical phenomena he has written a number of coursework texts. One of these has been expanded and improved to create the current volume.David Griffiths trained originally as an applied mathematician and has spent his academic career as a numerical analyst at the University of Dundee specialising in the numerical solution of partial differential equations. He taught the subject at both undergraduate and postgraduate levels for many years andpublished over 50 scientific articles on finite difference and finite element methods. He has previously published two textbooks on finite difference methods for ordinary and partial differential equations. David was joint organising secretary of the Dundee Biennial Conferences on Numerical Analysis from 1983 to 2005. He was awarded a “Certificate of Recognition” from NASA in both 1991 and 1993 for his work on spurious solutions that can arise from approximations of nonlinear differential equations.David Silvester is a Professor in the School of Mathematics at The University of Manchester. His research concerns numerical solution of partial differential equations, computational fluid dynamics, uncertainty quantification, and high performance computing. He received his doctorate in mathematics from the University of Manchester Institute of Science and Technology in 1984 and has had visiting positions at Stanford University, the University of Maryland at College Park, andthe Université du Littoral, France. He has served on the editorial boards of SIAM Journal on Scientific Computing and the International Journal for Numerical Methods in Fluids.

"The book is written in an engaging and lively style that will appeal to students. ... aim of the Springer SUMS series is to take a 'fresh and modern approach' to core foundational material through to final year topics. This book delivers on that promise with great success. ... As a first text that is set at the appropriate level ... which recognizes and incorporates numerical computation as an essential tool for learning and understanding, it looks hard to beat." (Mark Blyth, SIAM Review, Vol. 59 (1), March, 2017) "UK mathematicians Griffiths (Univ. of Dundee) and Dold and Silvester (both, Univ. of Manchester) introduce undergraduates to partial differential equations (PDEs) from both the analytical and numerical points of view. ... Summing Up: Recommended. Upper-division undergraduates through professionals/practitioners." (D. P. Turner, Choice, Vol. 53 (11), July, 2016) "This introduction to partial differential equations is designed for upper level undergraduates in mathematics. ... The writing is lively, the authors make appealing use of computational examples and visualization, and they are very successful at conveying and integrating physical intuition. ... This is probably the best introductory book on PDEs that I have seen in some time. It is well worth a look." (William J. Satzer, MAA Reviews, maa.org, April, 2016) "This textbook offers a nice introduction to analytical and numerical methods for partial differential equations. ... The book is self-contained and the prerequisites is a standard course in calculus and linear algebra. The textbook appeals to undergraduate students in both scientific and engineering programs in which PDEs are of practical importance." (Marius Ghergu, zbMATH 1330.35001, 2016)

• Setting the scene.- Boundary and initial data.- The origin of PDEs.- Classification of PDEs.- Boundary value problems in R1.- Finite difference methods in R1.- Maximum principles and energy methods.- Separation of variables.- The method of characteristics.- Finite difference methods for elliptic PDEs.- Finite difference methods for parabolic PDEs.- Finite difference methods for hyperbolic PDEs.- Projects.

John Dold is a professor of applied mathematics, having trained in physics, mathematics and (briefly) history. Apart from some purely experimental studies, particularly of fire behaviour, his research primarily makes use of partial differential equations to model practical problems, including water waves and combustion phenomena. He founded the journal Combustion Theory and Modelling and he has organised or helped to organise major conferences on combustion research. In his teaching of mathematical methods and their application to various physical phenomena he has written a number of coursework texts. One of these has been expanded and improved to create the current volume.David Griffiths trained originally as an applied mathematician and has spent his academic career as a numerical analyst at the University of Dundee specialising in the numerical solution of partial differential equations. He taught the subject at both undergraduate and postgraduate levels for many years andpublished over 50 scientific articles on finite difference and finite element methods. He has previously published two textbooks on finite difference methods for ordinary and partial differential equations. David was joint organising secretary of the Dundee Biennial Conferences on Numerical Analysis from 1983 to 2005. He was awarded a “Certificate of Recognition” from NASA in both 1991 and 1993 for his work on spurious solutions that can arise from approximations of nonlinear differential equations.David Silvester is a Professor in the School of Mathematics at The University of Manchester. His research concerns numerical solution of partial differential equations, computational fluid dynamics, uncertainty quantification, and high performance computing. He received his doctorate in mathematics from the University of Manchester Institute of Science and Technology in 1984 and has had visiting positions at Stanford University, the University of Maryland at College Park, andthe Université du Littoral, France. He has served on the editorial boards of SIAM Journal on Scientific Computing and the International Journal for Numerical Methods in Fluids.

"The book is written in an engaging and lively style that will appeal to students. ... aim of the Springer SUMS series is to take a 'fresh and modern approach' to core foundational material through to final year topics. This book delivers on that promise with great success. ... As a first text that is set at the appropriate level ... which recognizes and incorporates numerical computation as an essential tool for learning and understanding, it looks hard to beat." (Mark Blyth, SIAM Review, Vol. 59 (1), March, 2017) "UK mathematicians Griffiths (Univ. of Dundee) and Dold and Silvester (both, Univ. of Manchester) introduce undergraduates to partial differential equations (PDEs) from both the analytical and numerical points of view. ... Summing Up: Recommended. Upper-division undergraduates through professionals/practitioners." (D. P. Turner, Choice, Vol. 53 (11), July, 2016) "This introduction to partial differential equations is designed for upper level undergraduates in mathematics. ... The writing is lively, the authors make appealing use of computational examples and visualization, and they are very successful at conveying and integrating physical intuition. ... This is probably the best introductory book on PDEs that I have seen in some time. It is well worth a look." (William J. Satzer, MAA Reviews, maa.org, April, 2016) "This textbook offers a nice introduction to analytical and numerical methods for partial differential equations. ... The book is self-contained and the prerequisites is a standard course in calculus and linear algebra. The textbook appeals to undergraduate students in both scientific and engineering programs in which PDEs are of practical importance." (Marius Ghergu, zbMATH 1330.35001, 2016)

• Setting the scene.- Boundary and initial data.- The origin of PDEs.- Classification of PDEs.- Boundary value problems in R1.- Finite difference methods in R1.- Maximum principles and energy methods.- Separation of variables.- The method of characteristics.- Finite difference methods for elliptic PDEs.- Finite difference methods for parabolic PDEs.- Finite difference methods for hyperbolic PDEs.- Projects.

John Dold is a professor of applied mathematics, having trained in physics, mathematics and (briefly) history. Apart from some purely experimental studies, particularly of fire behaviour, his research primarily makes use of partial differential equations to model practical problems, including water waves and combustion phenomena. He founded the journal Combustion Theory and Modelling and he has organised or helped to organise major conferences on combustion research. In his teaching of mathematical methods and their application to various physical phenomena he has written a number of coursework texts. One of these has been expanded and improved to create the current volume.David Griffiths trained originally as an applied mathematician and has spent his academic career as a numerical analyst at the University of Dundee specialising in the numerical solution of partial differential equations. He taught the subject at both undergraduate and postgraduate levels for many years andpublished over 50 scientific articles on finite difference and finite element methods. He has previously published two textbooks on finite difference methods for ordinary and partial differential equations. David was joint organising secretary of the Dundee Biennial Conferences on Numerical Analysis from 1983 to 2005. He was awarded a “Certificate of Recognition” from NASA in both 1991 and 1993 for his work on spurious solutions that can arise from approximations of nonlinear differential equations.David Silvester is a Professor in the School of Mathematics at The University of Manchester. His research concerns numerical solution of partial differential equations, computational fluid dynamics, uncertainty quantification, and high performance computing. He received his doctorate in mathematics from the University of Manchester Institute of Science and Technology in 1984 and has had visiting positions at Stanford University, the University of Maryland at College Park, andthe Université du Littoral, France. He has served on the editorial boards of SIAM Journal on Scientific Computing and the International Journal for Numerical Methods in Fluids.

"The book is written in an engaging and lively style that will appeal to students. ... aim of the Springer SUMS series is to take a 'fresh and modern approach' to core foundational material through to final year topics. This book delivers on that promise with great success. ... As a first text that is set at the appropriate level ... which recognizes and incorporates numerical computation as an essential tool for learning and understanding, it looks hard to beat." (Mark Blyth, SIAM Review, Vol. 59 (1), March, 2017) "UK mathematicians Griffiths (Univ. of Dundee) and Dold and Silvester (both, Univ. of Manchester) introduce undergraduates to partial differential equations (PDEs) from both the analytical and numerical points of view. ... Summing Up: Recommended. Upper-division undergraduates through professionals/practitioners." (D. P. Turner, Choice, Vol. 53 (11), July, 2016) "This introduction to partial differential equations is designed for upper level undergraduates in mathematics. ... The writing is lively, the authors make appealing use of computational examples and visualization, and they are very successful at conveying and integrating physical intuition. ... This is probably the best introductory book on PDEs that I have seen in some time. It is well worth a look." (William J. Satzer, MAA Reviews, maa.org, April, 2016) "This textbook offers a nice introduction to analytical and numerical methods for partial differential equations. ... The book is self-contained and the prerequisites is a standard course in calculus and linear algebra. The textbook appeals to undergraduate students in both scientific and engineering programs in which PDEs are of practical importance." (Marius Ghergu, zbMATH 1330.35001, 2016)

• Setting the scene.- Boundary and initial data.- The origin of PDEs.- Classification of PDEs.- Boundary value problems in R1.- Finite difference methods in R1.- Maximum principles and energy methods.- Separation of variables.- The method of characteristics.- Finite difference methods for elliptic PDEs.- Finite difference methods for parabolic PDEs.- Finite difference methods for hyperbolic PDEs.- Projects.