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Linz P. — Analytical and numerical methods for Volterra equations
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Название: Analytical and numerical methods for Volterra equationsАвтор: Linz P. Аннотация: Presents an aspect of activity in integral equations methods for the solution of Volterra equations for those who need to solve real-world problems. Since there are few known analytical methods leading to closed-form solutions, the emphasis is on numerical techniques. The major points of the analytical methods used to study the properties of the solution are presented in the first part of the book. These techniques are important for gaining insight into the qualitative behavior of the solutions and for designing effective numerical methods. The second part of the book is devoted entirely to numerical methods.
The author has chosen the simplest possible setting for the discussion, the space of real functions of real variables. The text is supplemented by examples and exercises.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 1985Количество страниц: 242Добавлена в каталог: 18.11.2013Операции: Положить на полку |
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111 209 Abel’s equation, generalized 5 Abel’s equation, inversion formulas 74 Abel’s equation, numerical solution 165 Abel’s equation, simple 4 Accumulated error, consistency 104 Accumulated error, starting 104 Adams — Moulton methods 182 Aitken’s method 134 Asymptotic behavior of solutions 42 61 89 Beltyukov methods 122 Block-by-block methods, for Abel’s equation 169 Block-by-block methods, for equations of the first kind 154 Block-by-block methods, for equations of the second kind 114 Block-by-block methods, for integrodifferential equations 185 Block-by-block methods, using product integration 136 Causality 17 Characteristic polynomial of stability 112 Comparison theorems 40 47 58 Consistency condition, for integral equations 101 Consistency condition, for integrodifferential equations 183 Consistency error, accumulated 104 Consistency error, local 101 Consistency error, order 101 Contraction mapping argument 32 Convergence, and stability 185 Convergence, of an approximation method 100 Convergence, order of 101 Conversion, differential equation to integral equation 13 Conversion, first kind equation to second kind 67 68 Conversion, integral equation into differential equation 7 78 Conversion, integrodifferential equation to integral Equation 5 Convolution equation, linear 18 77 Convolution equation, nonlinear 77 Convolution theorem 84 Dahlquist theory 177 Degenerate kernel approximation methods 126 Delay differential equation 189 Difference kernel 37 Differential resolvent 83 Differentiation methods for first kind equations 158 Direct methods for equations of the first kind 143 Discretization error 100 Effect of data errors, on Abel’s equation 204 Effect of data errors, on equations of the first kind 204 Elongation of a filament 204 Error estimates 103 Error expansion 105 121 Euler’s method 133 144 Existence of a solution, Abel’s equation 73 75 Existence of a solution, equations with unbounded kernels 48 Existence of a solution, integrodifferential equations 49 Existence of a solution, linear first kind equations 67 Existence of a solution, linear second kind equations 29 Existence of a solution, nonlinear first kind equations 69 Existence of a solution, nonlinear second kind equations 52 55 Existence of a solution, systems of the second kind 48 Finite rank kernel 8 First kind equations, nonlinear 69 160 First kind equations, numerical solution 143 First kind equations, with smooth kernels 67 Fourier transform 19 Functional equations 188 Gregory methods, for equations of the first kind 151 Gregory methods, for equations of the second kind 98 Heard method 29 Heat conduction equation 19 ill-posed problems 162 Initial value problems and Volterra equations 7 Instability in numerical computation 103 Integral equations, of Fredholm type 3 Integral equations, of Volterra type 3 Integral equations, singular 3 Integral equations, weakly singular 3 Integrodifferential equations, numerical methods 177 Integrodifferential equations, reduction to integral Equations 5 Interchanging order of integration 6 Inversion formulas, for Fourier transforms 19 Inversion formulas, for Laplace transform 84 Iterated kernels 36 Kernel, degenerate 8 Kernel, finite rank 8 Kernel, Lipschitz continuous 51 Kernel, monotonicity properties 58 Kernel, of an integral equation 4 Kernel, unbounded 4 Laplace transforms 84 Linear multistep methods 182 Lipschitz condition, for equations of the first kind 69 Lipschitz condition, for equations of the second kind 52 Lipschitz condition, for integrodifferential equations 177 Method of continuation 32 Method of successive approximation 29 Midpoint method, for Abel’s equation 166 Midpoint method, for equations of the first kind 144 Milne — Simpson methods 182 Newton — Cotes methods, for first kind equations 152 Newton — Cotes methods, for second kind equations 98 Numerical stability, equations of the first kind 149 Numerical stability, equations of the second kind 108 110 Numerical stability, integrodifferential equations 186 Pascal program, midpoint method for first kind equations 197 Pascal program, product midpoint method for first kind Equations 198 Pascal program, product trapezoidal method for second kind Equations 196 Pascal program, trapezoidal method for second kind Equations 195 Perturbation, effect, on first kind equations 71 162 Perturbation, effect, on second kind equations 45 Piecewise polynomial approximations 118 157 Polymer rheology example 204 Population equation 15 Pouzet methods 122 Predictor-corrector methods 125 Product integration 130 Product trapezoidal method, equations of the first kind 168 Product trapezoidal method, equations of the second kind 135 Radiating source 23 Reactor dynamics equation 21 Renewal, density 14 Renewal, equation 14 Repetition factor 106 Resolvent equation, difference kernels 38 Resolvent equation, linear problems 36 Resolvent equation, nonlinear problems 63 Resolvent kernel 35 Richardson’s extrapolation 133 Roundoff error, in first kind equations 161 Roundoff error, in second kind equations 121 Runge — Kutta methods, explicit 122 Runge — Kutta methods, implicit 114 Runge — Kutta methods, of Beltyukov type 122 Runge — Kutta methods, of Pouzet type 122 Second kind equations, linear 29 Second kind equations, nonlinear 51 Second kind equations, numerical solution 95 Second kind equations, product integration methods 129 Semicirculant matrix 172 Simpson’s methods 99 Stability, definition 104 183 Stability, interval 111 Starting error 104 Starting values 98 Stereology example 24 System identification 18 Systems of integral equations 5 46 Systems theory 15 Systems, causal 17 Systems, evolutionary 3 Systems, feedback 18 Systems, history-dependent 13 Systems, linear 16 Systems, nonlinear 16 Systems, time-invariant 17 Systems, with memory 13 Tauberian theorems 91 Trapezoidal method, Abel’s equation 166 Trapezoidal method, equations of the first kind 144 Trapezoidal method, integrodifferential equations 178 Uniqueness of a solution, Abel’s equation 73 75 Uniqueness of a solution, equations with an unbounded Kernel 48 62 Uniqueness of a solution, integrodifferential equation 49 Uniqueness of a solution, linear first kind equation 67 Uniqueness of a solution, linear second kind equation 29 Uniqueness of a solution, nonlinear first kind equation 69 Uniqueness of a solution, nonlinear second kind equation 52 55 Uniqueness of a solution, systems of the second kind 46 62 Unit-impulse function 16 Unit-response function 17 Volterra equations, applications 13 Volterra equations, classification 3 Volterra equations, linear 4 Volterra equations, of first kind 4 Volterra equations, of second kind 3 Volterra population equation 15 Weights for numerical integration 98 Well-posed problems 45
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