Fractional Differential Equations: Theoretical Aspects and Applications presents the latest mathematical and conceptual developments in the field of Fractional Calculus and explores the scope of applications in research science and computational modelling. Fractional derivatives arise as a generalization of integer order derivatives and have a long history: their origin can be found in the work of G. W. Leibniz and L. Euler. Shortly after being introduced, the new theory turned out to be very attractive for many famous mathematicians and scientists, including P. S. Laplace, B. Riemann, J. Liouville, N. H. Abel, and J. B. J. Fourier, due to the numerous possibilities it offered for applications.Fractional Calculus, the field of mathematics dealing with operators of differentiation and integration of arbitrary real or even complex order, extends many of the modelling capabilities of conventional calculus and integer-order differential equations and finds its application in various scientific areas, such as physics, mechanics, engineering, economics, finance, biology, and chemistry, among others. However, many aspects from the theoretical and practical point of view have still to be developed in relation with models based on fractional operators. Efficient analytical and numerical methods have been developed but still need particular attention. Fractional Differential Equations: Theoretical Aspects and Applications delves into these methods and applied computational modelling techniques, including analysis of equations involving fractional derivatives, fractional derivatives and the wave equation, analysis of FDE on groups, direct and inverse problems, functional inequalities, and computational methods for FDEs in physics and engineering. Other modelling techniques and applications explored by the authors include general fractional derivatives involving the special functions in analysis, fractional derivatives with respect to another function in analysis, new fractional operators in real-world applications, fractional order dynamical systems, hidden attractors in complex systems, nonlinear dynamics and chaos in engineering applications, quantum chaos, and self-excited attractors.
1. 3.1 Introduction - p. 27 2. 3.2 Moment estimates - p. 28 3. 3.3 Rate of convergence - p. 31 4. 3.4 Rate of convergence in Lipschitz-type spaces - p. 33 5. 3.5 A-statistical convergence - p. 35 6. 3.6 A-statistical convergence - p. 37 7. Acknowledgements - p. 41 8. References - p. 41 9. 4. Differential operational techniques and applications - p. 43 10. 5. Some results on Wright function - p. 55 11. 6. Computational methods for the fractional differential equations in physics and engineering - p. 65 12. 6.1 Introduction - p. 65 13. 6.2 Spline function methods and solutions of linear and nonlinear fractional differential equations - p. 65 14. 6.3 Spline approximation method for a system of fractional differential equations - p. 71 15. 6.4 Modified homotopy perturbation method for the fractional Bagley.Torvik equation - p. 80 16. 6.5 Conclusion - p. 92 17. References - p. 92 18. 7. A study on the properties of new generalized special functions, their integral transformations, and applications to fractional differential equations - p. 95 19. 7.1 Introduction and preliminaries - p. 95 20. 7.2 New generalized special functions - p. 97 21. 7.3 Properties - p. 98 22. 7.4 Integral transforms - p. 103 23. 7.5 Solution of fractional differential equations - p. 109 24. 7.6 Conclusion - p. 111 25. References - p. 113 26. 8. Congruence between mild and classical solutions of generalized fractional impulsive evolution equation - p. 115 27. 9. Application of various methods to solve some fractional differential equations in different fields - p. 123 28. 9.1 Introduction - p. 123 29. 9.2 Preliminaries - p. 123 30. 9.3 Idea of the used methods - p. 125 31. 9.4 Results and discussion - p. 128 32. 9.5 Conclusion - p. 149 33. References - p. 150 34. 10. Modeling capillary absorption in building materials with emphasis on the fourth root time law: time-fractional models, solutions, and analysis - p. 153 35. 11. Fuzzy fractional Caputo-type numerical scheme for solving fuzzy nonlinear equations - p. 167 36. 12. Approximate solutions of epidemic model of Zika virus - p. 177 37. 13. On the nonlocal boundary value problem for the coupled system - p. 191 38. 13.1 Introduction - p. 191 39. 13.2 Existence of solution - p. 192 40. 13.3 Uniqueness of solution - p. 198 41. 13.4 Continuous dependence - p. 199 42. 13.5 Both analytical and numerical approaches - p. 203 43. 13.6 Numerical examples - p. 204 44. 13.7 Conclusion - p. 216 45. References - p. 216 46. 14. Solutions of nonlinear time fractional Klein–Gordon equations using composite fractional derivatives - p. 219 47. 15. Some generating functions for I -function - p. 229 48. 16. Approximate solution to the KdV equation in superthermal plasmas - p. 235 49. 17. Turátype inequalities for generalized fractional operators and special functions - p. 245 50. 17.1 Introduction - p. 245 51. 17.2 Preliminaries - p. 245 52. 17.3 Turátype inequality of k-fractional integral operators - p. 248 53. 17.4 Turátype inequality for generalized Caputo-type k-fractional operator - p. 250 54. 17.5 Turátype inequalities for extended hypergeometric functions - p. 252 55. Acknowledgements - p. 254 56. References - p. 254 57. Index - p. 257
Fractional Differential Equations: Theoretical Aspects and Applications presents the latest mathematical and conceptual developments in the field of Fractional Calculus and explores the scope of applications in research science and computational modelling. Fractional derivatives arise as a generalization of integer order derivatives and have a long history: their origin can be found in the work of G. W. Leibniz and L. Euler. Shortly after being introduced, the new theory turned out to be very attractive for many famous mathematicians and scientists, including P. S. Laplace, B. Riemann, J. Liouville, N. H. Abel, and J. B. J. Fourier, due to the numerous possibilities it offered for applications.Fractional Calculus, the field of mathematics dealing with operators of differentiation and integration of arbitrary real or even complex order, extends many of the modelling capabilities of conventional calculus and integer-order differential equations and finds its application in various scientific areas, such as physics, mechanics, engineering, economics, finance, biology, and chemistry, among others. However, many aspects from the theoretical and practical point of view have still to be developed in relation with models based on fractional operators. Efficient analytical and numerical methods have been developed but still need particular attention. Fractional Differential Equations: Theoretical Aspects and Applications delves into these methods and applied computational modelling techniques, including analysis of equations involving fractional derivatives, fractional derivatives and the wave equation, analysis of FDE on groups, direct and inverse problems, functional inequalities, and computational methods for FDEs in physics and engineering. Other modelling techniques and applications explored by the authors include general fractional derivatives involving the special functions in analysis, fractional derivatives with respect to another function in analysis, new fractional operators in real-world applications, fractional order dynamical systems, hidden attractors in complex systems, nonlinear dynamics and chaos in engineering applications, quantum chaos, and self-excited attractors.
1. 3.1 Introduction - p. 27 2. 3.2 Moment estimates - p. 28 3. 3.3 Rate of convergence - p. 31 4. 3.4 Rate of convergence in Lipschitz-type spaces - p. 33 5. 3.5 A-statistical convergence - p. 35 6. 3.6 A-statistical convergence - p. 37 7. Acknowledgements - p. 41 8. References - p. 41 9. 4. Differential operational techniques and applications - p. 43 10. 5. Some results on Wright function - p. 55 11. 6. Computational methods for the fractional differential equations in physics and engineering - p. 65 12. 6.1 Introduction - p. 65 13. 6.2 Spline function methods and solutions of linear and nonlinear fractional differential equations - p. 65 14. 6.3 Spline approximation method for a system of fractional differential equations - p. 71 15. 6.4 Modified homotopy perturbation method for the fractional Bagley.Torvik equation - p. 80 16. 6.5 Conclusion - p. 92 17. References - p. 92 18. 7. A study on the properties of new generalized special functions, their integral transformations, and applications to fractional differential equations - p. 95 19. 7.1 Introduction and preliminaries - p. 95 20. 7.2 New generalized special functions - p. 97 21. 7.3 Properties - p. 98 22. 7.4 Integral transforms - p. 103 23. 7.5 Solution of fractional differential equations - p. 109 24. 7.6 Conclusion - p. 111 25. References - p. 113 26. 8. Congruence between mild and classical solutions of generalized fractional impulsive evolution equation - p. 115 27. 9. Application of various methods to solve some fractional differential equations in different fields - p. 123 28. 9.1 Introduction - p. 123 29. 9.2 Preliminaries - p. 123 30. 9.3 Idea of the used methods - p. 125 31. 9.4 Results and discussion - p. 128 32. 9.5 Conclusion - p. 149 33. References - p. 150 34. 10. Modeling capillary absorption in building materials with emphasis on the fourth root time law: time-fractional models, solutions, and analysis - p. 153 35. 11. Fuzzy fractional Caputo-type numerical scheme for solving fuzzy nonlinear equations - p. 167 36. 12. Approximate solutions of epidemic model of Zika virus - p. 177 37. 13. On the nonlocal boundary value problem for the coupled system - p. 191 38. 13.1 Introduction - p. 191 39. 13.2 Existence of solution - p. 192 40. 13.3 Uniqueness of solution - p. 198 41. 13.4 Continuous dependence - p. 199 42. 13.5 Both analytical and numerical approaches - p. 203 43. 13.6 Numerical examples - p. 204 44. 13.7 Conclusion - p. 216 45. References - p. 216 46. 14. Solutions of nonlinear time fractional Klein–Gordon equations using composite fractional derivatives - p. 219 47. 15. Some generating functions for I -function - p. 229 48. 16. Approximate solution to the KdV equation in superthermal plasmas - p. 235 49. 17. Turátype inequalities for generalized fractional operators and special functions - p. 245 50. 17.1 Introduction - p. 245 51. 17.2 Preliminaries - p. 245 52. 17.3 Turátype inequality of k-fractional integral operators - p. 248 53. 17.4 Turátype inequality for generalized Caputo-type k-fractional operator - p. 250 54. 17.5 Turátype inequalities for extended hypergeometric functions - p. 252 55. Acknowledgements - p. 254 56. References - p. 254 57. Index - p. 257
