Mathematical Physics Vol 1

7.1 Definitionsandnotation...............................33.4 7.2 Formation of partial differential equations . . . . . . . . . . . . . . . . . . . . .33. 5 7.3 Linear and quasilinear first order PDE . . . . . . . . . . . . . . . . . . . . . . .3.41 7.3.1 OnsolutionsforPDE............................34.2 7.3.2 A general method for integrating linear first order PDE. First integral . .34. 3 7.3.3 Symmetrical form of a system of ordinary differential equations . . . . .34. 4 7.3.4 General solution of the linear homogeneous first order PDE . . . . . . .34. 5 7.3.5 General solution of linear non-homogeneous first order PDE . . . . . . .34. 6 7.3.6 Pfaffianequation ..............................34.7 7.3.7 Nonlinear first order PDE. Lagrange-Charpit method . . . . . . . . . . .35. 0 7.4 LinearsecondorderPDE..............................35.5 7.4.1 Some properties of homogeneous second order partial LDE . . . . . . .35. 6 7.4.2 Classification of second order LDE with two variables . . . . . . . . . .35. 7 7.4.3 Reductiontocanonicalform . . . . . . . . . . . . . . . . . . . . . . . .35.9 7.4.4 Examples of classification of some equations of mathematical physics . .36. 2 7.5 AformalprocedureforsolvingLDE . . . . . . . . . . . . . . . . . . . . . . . .36.3 7.6 Thevariableseparationmethod . . . . . . . . . . . . . . . . . . . . . . . . . . .36.4 7.7 Greenformulas ...................................37.0 7.8 Examples ......................................38.2 7.8.1 Appendix ..................................43.0 435 8.1 Brief History of Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . .43. 5 8.2 Basic Definitions of Fractional Order Differintegrals . . . . . . . . . . . . . . .4.41 8.3 Basic Properties of Fractional Order Differintegrals . . . . . . . . . . . . . . . .44. 4 8.4 Some other types of fractional derivatives . . . . . . . . . . . . . . . . . . . . .44. 6 8.4.1 Left and right Liouville-Weyl fractional derivatives on the real axis . . .44. 6 8.4.2 Hilfer fractional derivative . . . . . . . . . . . . . . . . . . . . . . . . .44. 8 8.4.3 Marchaud fractional derivative . . . . . . . . . . . . . . . . . . . . . . .44. 8 Appendices 453 Appendix Chapter A Fractional Calculus: A Survey of Useful Formulas 455 A.1 Introduction.....................................45.5 A.2 NotationandSpecialFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . .45.5 A.2.1 Notation ..................................45.5 A.2.2 Definitions of some Special Functions . . . . . . . . . . . . . . . . . . .45. 6 A.2.3 Properties of the Mittag-Leffler functions: special values . . . . . . . . .45. 8 A.2.4 Generalized exponential functions . . . . . . . . . . . . . . . . . . . . .45. 8 A.3 Fractional Derivatives and Integrals . . . . . . . . . . . . . . . . . . . . . . . .45. 9 A.3.1 Definitions of some unidimensional fractional operators . . . . . . . . .45. 9 A.3.2 Properties..................................4.61 A.3.3 FractionalTaylorFormulas . . . . . . . . . . . . . . . . . . . . . . . . .46.3 A.4 Analytical Expressions of Some Fractional Derivatives . . . . . . . . . . . . . .46. 3 A.5 LaplaceandFourierTransforms . . . . . . . . . . . . . . . . . . . . . . . . . .46.4 A.5.1 Someproperties ..............................46.4 VI Fractional Calculus 433 Chapter 8 Introduction to the Fractional Calculus