By Kenneth S. Miller
Commences with the historic improvement of fractional calculus, its mathematical theory—particularly the Riemann-Liouville model. quite a few examples and theoretical purposes of the speculation are awarded. beneficial properties issues linked to fractional differential equations. Discusses Weyl fractional calculus and a few of its makes use of. comprises chosen actual difficulties which result in fractional differential or essential equations.
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Extra info for An Introduction to the Fractional Calculus and Fractional Differential Equations
8 Ϫ 17͒ ϩ 3 8. Ϫ3͑5 Ϫ 2͒ 9. ͑4 Ϫ 7͒͑Ϫ2͒ 10. Ϫ5͑Ϫ2 Ϫ 6͒ In Exercises 11–16, evaluate the expression for each value of x. ) Expression Values 11. 4x Ϫ 6 (a) x ϭ Ϫ1 (b) x ϭ 0 12. 5 Ϫ 3x (a) x ϭ Ϫ3 (b) x ϭ 2 13. x 2 Ϫ 3x ϩ 4 (a) x ϭ Ϫ2 (b) x ϭ 2 (a) x ϭ 0 (b) x ϭ 2 14. Ϫx3 15. x xϪ2 (a) x ϭ Ϫ2 (b) x ϭ 2 16. xϩ3 xϪ3 (a) x ϭ 3 (b) x ϭ Ϫ3 ϩ 2x Ϫ 1 Խ 10. Ϫ 8 Ϫ 10 In Exercises 23–38, identify the rule(s) of algebra illustrated by the statement. 23. 3 ϩ 4 ϭ 4 ϩ 3 24. x ϩ 9 ϭ 9 ϩ x 25. Ϫ15 ϩ 15 ϭ 0 26. ͑x ϩ 2͒ Ϫ ͑x ϩ 2͒ ϭ 0 27.
X2 и xn x 3 и xn 42. xn и x2n x3n 45. ͑2x5͒0, x x Ϫ5 49. ͑4yϪ2͒͑8y4͒ 50. ͑Ϫ2x2͒3͑4x3͒Ϫ1 Ϫ1 Ϫ5 Ϫ1 x4 2x2 x ϭ Ϫ6 53. x 5y 24. 8x0 Ϫ ͑8x͒0 46. ͑x ϩ 5͒0, 0 48. ͑x ϩ y͒ ͑x ϩ y͒9 10x xϭ4 и 23m 47. ͑ y ϩ 2͒ ͑ y ϩ 2͒ 51. xϭ2 3 In Exercises 45–54, rewrite the expression with positive exponents and simplify. Value 7xϪ2 2 4 44. 2m Expression 22. 4xϪ3 23. ͑12 Ϫ 13 ͒ Ϭ 16 Ϫ2 In Exercises 21–24, evaluate the expression for the indicated value of x. 21. ϩ 12 Ϫ 56 43. 3n и 32n 20. ͑Ϫ2͒0 19. com for worked-out solutions to odd-numbered exercises.
8x4͒͑2x3͒ 28. 5x4͑x2͒ 58. 0000000000000003 gram 29. 10͑x2͒2 30. ͑4x 4͒3 59. 0000001 meter 31. ͑Ϫz͒3͑3z4͒ 32. ͑6y 2͒͑2y 3͒3 60. 00003937 inch 3 5 Fundamental Concepts of Algebra 67. 0052 68. 0061 69. 3 70. 34 ϫ 10Ϫ3͒22 In Exercises 71 and 72, write each number in scientific notation. Perform the operation and write your answer in scientific notation. 71. 0000064 72. 00000006͒ (b) 18,000,000,000 2,400,000 73. 75% for 12 years. Determine the balance in the account when the interest is compounded (a) daily ͑n ϭ 365͒, (b) weekly, (c) monthly, and (d) quarterly.