http://timescalewiki.org/index.php?title=Book:Svetlin_G._Georgiev/Integral_Equations_on_Time_Scales&feed=atom&action=history Book:Svetlin G. Georgiev/Integral Equations on Time Scales - Revision history 2021-01-20T01:13:46Z Revision history for this page on the wiki MediaWiki 1.28.0 http://timescalewiki.org/index.php?title=Book:Svetlin_G._Georgiev/Integral_Equations_on_Time_Scales&diff=1745&oldid=prev Tom: Created page with "{{Book|Integral Equations on Time Scales||||Svetlin G. Georgiev}} ===Contents=== :1 Elements of the Time Scale Calculus ::1.1 Forward and Backward Jump Operators, Graininess..." 2017-12-20T02:57:48Z <p>Created page with &quot;{{Book|Integral Equations on Time Scales||||Svetlin G. Georgiev}} ===Contents=== :1 Elements of the Time Scale Calculus ::1.1 Forward and Backward Jump Operators, Graininess...&quot;</p> <p><b>New page</b></p><div>{{Book|Integral Equations on Time Scales||||Svetlin G. Georgiev}}<br /> <br /> ===Contents===<br /> :1 Elements of the Time Scale Calculus<br /> ::1.1 Forward and Backward Jump Operators, Graininess Function<br /> ::1.2 Differentiation<br /> ::1.3 Mean Value Theorems<br /> ::1.4 Integration<br /> ::1.5 The Exponential Function<br /> :::1.5.1 Hilger's Complex Plane<br /> :::1.5.2 Definition and Properties of the Exponential Function<br /> :::1.5.3 Examples for Exponential Functions<br /> ::1.6 Hyperbolic and Trigonometric Functions<br /> ::1.7 Dynamic Equations<br /> ::1.8 Advanced Practical Exercises<br /> :2 Introductory Concepts of Integral Equations on Time Scales<br /> ::2.1 Reducing Double Integrals to Single Integrals<br /> ::2.2 Converting IVP to Generalized Volterra Integral Equations<br /> ::2.3 Converting Generalized Volterra Integral Equations to IVP<br /> ::2.4 Converting BVP to Generalized Fredholm Integral Equation<br /> ::2.5 Converting Generalized Fredholm Integral Equation to BVP<br /> ::2.6 Solutions of Generalized Integral Equations and Generalized Integro-Differential Equations<br /> ::2.7 Advanced Practical Exercises<br /> :3 Generalized Volterra Integral Equations<br /> ::3.1 Generalized Volterra Integral Equations of the Second Kind<br /> :::3.1.1 The Adomian Decomposition Method<br /> :::3.1.2 The Modified Decomposition Method<br /> :::3.1.3 The Noise Terms Phenomenon<br /> :::3.1.4 Differential Equations Method<br /> :::3.1.5 The Successive Approximations Method<br /> ::3.2 Conversion of a Generalized Volterra Integral Equation of the First Kind to a Generalized Volterra Integral Equation of the Second Kind<br /> ::3.3 Existence and Uniqueness of Solutions<br /> :::3.3.1 Preliminary Results<br /> :::3.3.2 Existence of Solutions of Generalized Volterra Integral Equations of the Second Kind<br /> :::3.3.3 Uniqueness of Solutions of Generalized Volterra Integral Equations of the Second Kind<br /> :::3.3.4 Existence and Uniqueness of Solutions of Generalized Volterra Integral Equations of the First Kind<br /> ::3.4 Resolvent Kernels<br /> ::3.5 Application to Linear Dynamic Equations<br /> ::3.6 Advanced Practical Exercises<br /> :4 Generalized Volterra Integro-Differential Equations<br /> ::4.1 Generalized Volterra Integro-Differential Equations of the Second Kind<br /> :::4.1.1 The Adomian Decomposition Method<br /> :::4.1.2 Converting Generalized Volterra Integro-Differential Equations of the Second Kind to Initial Value Problems<br /> :::4.1.3 Converting Generalized Volterra Integro-Differential Equations of the Second Kind to Generalized Volterra Integral Equations<br /> ::4.2 Generalized Volterra Integro-Differential Equations of the First Kind<br /> ::4.3 Advanced Practical Exercises<br /> :5 Generalized Fredholm Integral Equations<br /> ::5.1 Generalized Fredholm Integral Equations of the Second Kind<br /> :::5.1.1 The Adomian Decomposition Method<br /> :::5.1.2 The Modified Decomposition Method<br /> :::5.1.3 The Noise Terms Phenomenon<br /> :::5.1.4 The Direct Computation Method<br /> :::5.1.5 The Successive Approximations Method<br /> ::5.2 homogeneous Generalized Fredholm Integral Equations of the Second Kind<br /> ::5.3 Fredholm Alternative Theorem<br /> :::5.3.1 The Case When $\displaystyle\int_a^b \displaystyle\int_a^b |K(X,Y)|^2 \Delta X \Delta Y &lt; 1$<br /> :::5.3.2 The General Case<br /> :::5.3.3 Fredholm's Alternative Theorem<br /> ::5.4 The Schmidt Expansion Theorem and the Mercer Expansion Theorem<br /> :::5.4.1 Operator-Theoretical Notations<br /> :::5.4.2 The Schmidt Expansion Theorem<br /> :::5.4.3 Application to Generalized Fredholm Integral EQuation of the First Kind<br /> :::5.4.4 Positive Definite Kernels. Mercer's Expansion Theorem<br /> ::5.5 Advanced Practical Exercises<br /> :6 Hilbert-Schmidt Theory of Generalized Integral Equations with Symmetric Kernels<br /> ::6.1 Schmidt's Orthogonalization Process<br /> ::6.2 Approximations of Eigenvalues<br /> ::6.3 Inhomogeneous Generalized Integral Equations<br /> :7 The Laplace Transform Method<br /> ::7.1 The Laplace Transform<br /> :::7.1.1 Definition and Examples<br /> :::7.1.2 Properties of the Laplace Transform<br /> :::7.1.3 Convolution and Shifting Properties of Special Functions<br /> ::7.2 Applications to Dynamic Equations<br /> ::7.3 Generalized Voltera Integral Equations of the Second Kind<br /> ::7.4 Generalized Voltera Integral Equations of the First Kind<br /> ::7.5 Generalized Volterra Integro-Differential Equations of the Second Kind<br /> ::7.6 Generalized Volterra Integro-Differential Equations of the First Kind<br /> ::7.7 Advanced Practical Exercises<br /> :8 The Series Solution Method<br /> ::8.1 Generalized Volterra Integral Equations of the Second Kind<br /> ::8.2 Generalized Volterra Integral Equations of the First Kind<br /> ::8.3 Generalized Volterra Integro-Differential Equations of the Second Kind<br /> :9 Non-linear Generalized Integral Equations<br /> ::9.1 Non-linear Generalized Volterra Integral Equations<br /> ::9.2 Non-linear Generalized Fredholm Integral Equations<br /> :References<br /> :Index<br /> <br /> [[Category:Book]]</div> Tom