This book is focused on the qualitative theory of general quantum calculus, the modern name for the investigation of calculus without limits. It centers on designing, analysing and applying computational techniques for general quantum differential equations.
The quantum calculus or q-calculus began with F.H. Jackson in the early twentieth century, but this kind of calculus had already been worked out by Euler and Jacobi. Recently, it has aroused interest due to high demand of mathematics that models quantum computing and the connection between mathematics and physics.
Quantum calculus has many applications in different mathematical areas such as number theory, combinatorics, orthogonal polynomials, basic hyper-geometric functions and other sciences such as quantum theory, mechanics and the theory of relativity.
The authors summarize the most recent contributions in this area. General Quantum Numerical Analysis is intended for senior undergraduate students and beginning graduate students of engineering and science courses. The twelve chapters in this book are pedagogically organized, each concluding with a section of practical problems.
1 General Quantum Differentiation
1.1 The β -Operator
1.2 Definition for β -Derivative. Examples
1.3 Properties of the β -Derivative
1.4 Rules for β -Differentiation
1.5 Properties of β -Differentiable Functions
1.6 Chain Rules
1.7 A Mean Value Theorem
1.8 Higher Order β -Derivatives
1.9 The β -Rolle Theorem
1.10 Advanced Practical Problems
1.11 Notes and References
2 General Quantum Integration
2.1 β -Antiderivatives
2.2 Definition for β -Integral. Examples
2.3 Properties of β -Integrals
2.4 Inequalities and β -Integrals
2.5 General Quantum Monomials
2.6 The Taylor Formula
2.7 Improper Integrals of the First Kind
2.8 Improper Integrals of the Second Kind
2.9 Advanced Practical Problems
2.10 Notes and References
3 β -Elementary Functions
3.1 β - Regressive Functions
3.2 β -Exponential Functions
3.3 β -TrigonometricFunctions
3.4 β -HyperbolicFunctions
3.5 Advanced Practical Problems
3.6 Notes and References
4 General Quantum Polynomial Interpolation
4.1 General Quantum Lagrange Interpolation
4.2 β -Lagrange Interpolation
4.3 Hermite Interpolation
4.4 β -Hermite Interpolation
4.5 β - Differentiation
4.6 Advanced Practical Problems
4.7 Notes and References
5 Numerical β -Integration
5.1 Newton-Cotes Formula
5.2 β -Newton-Cotes Formula
5.3 Error Estimates
5.4 β -Error Estimates
5.5 CompositeQuadratureRules
5.6 β -Composite Quadrature Rules
5.7 TheEuler-MaclaurenExpansion
5.8 The β -Euler-Maclauren Expansion
5.9 Construction of Gauss Quadrature Rules
5.10 Error Estimation for Gauss Quadrature Rules
5.11 β -Gauss Quadrature Rules
5.12 Error Estimation for β -GaussQuadrature
5.13 Advanced Practical Problems
5.14 Notes and References
6 Piecewise Polynomial Approximation
6.1 LinearInterpolatingSplines
6.2 Linear Interpolating β -Splines
6.3 CubicSplines.
6.4 HermiteCubic
6.5 Advanced Practical Problems
6.6 Notes and References
7 The Euler Method
7.1 Analysing the Method
7.2 LocalTruncationError
7.3 Global Truncation Error
7.4 Advanced Practical Problems
7.5 Notes and References
8 The Order-Two Taylor Series Method-TS(2)
8.1 Analysing of the Method
8.2 Convergence of the Order-Two Taylor Series Method
8.3 Trapezoidal rule
8.4 Advanced Practical Problems
8.5 Notes and References
9. The Order-p Taylor Series Method-TS(p)
9.1 A Generalization of the Chain Rule
9.2 Analysing of the Order-p Taylor Series Method
9.3 Convergence and error analysis of the Taylor series method
9.4 The 2-step Adams-Bashforth method: AB(2)
9.5 Notes and References
10. Linear Multistep Methods-LLMs
10.1 Two-Step Methods
10.2 Consistency of Two-Step Methods
10.3 Construction of Two-Step Methods
10.4 k-Step Methods
10.5 Consistency of k-Set Methods
10.6 Advanced Practical Problems
10.7 Notes nad References
11. Runge-Kutta Methos-RMMs
11.1 One-Stage Methods
11.2 Two-Stage Methods
11.3 Three-Stage Methods
11.4 s-Stage Methods
11.5 Notes and References
12. The Adomain Polynomoials Method
12.1 Analysing of the Method
12.2 Examples of First Order Nonlinear β-Differential Equations
12.3 Numerical Examples
12.4 Noes and References
Bibliography
Index
Biography
Svetlin G. Georgiev is a mathematician who has worked in various areas of mathematics. He currently focuses on harmonic analysis, functional analysis, partial differential equations, ordinary differential equations, Clifford and quaternion analysis, integral equations and dynamic calculus on time scales.
Khaled Zennir earned his PhD in mathematics in 2013 from Sidi Bel Abbès University, Algeria. In 2015, he received his highest diploma in Habilitation in mathematics from Constantine University, Algeria. He is currently an assistant professor at Qassim University in the Kingdom of Saudi Arabia. His research interests lie in the subjects of nonlinear hyperbolic partial differential equations: global existence, blowup and longtime behavior.
