Linz P. — Analytical and numerical methods for Volterra equations
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Название: Analytical and numerical methods for Volterra equations
Автор: 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.
-stability111 -splines209 Abel’s equation, generalized5 Abel’s equation, inversion formulas74 Abel’s equation, numerical solution165 Abel’s equation, simple4 Accumulated error, consistency104 Accumulated error, starting104 Adams — Moulton methods182 Aitken’s method134 Asymptotic behavior of solutions426189 Beltyukov methods122 Block-by-block methods, for Abel’s equation169 Block-by-block methods, for equations of the first kind154 Block-by-block methods, for equations of the second kind114 Block-by-block methods, for integrodifferential equations185 Block-by-block methods, using product integration136 Causality17 Characteristic polynomial of stability112 Comparison theorems404758 Consistency condition, for integral equations101 Consistency condition, for integrodifferential equations183 Consistency error, accumulated104 Consistency error, local101 Consistency error, order101 Contraction mapping argument32 Convergence, and stability185 Convergence, of an approximation method100 Convergence, order of101 Conversion, differential equation to integral equation13 Conversion, first kind equation to second kind6768 Conversion, integral equation into differential equation778 Conversion, integrodifferential equation to integral Equation5 Convolution equation, linear1877 Convolution equation, nonlinear77 Convolution theorem84 Dahlquist theory177 Degenerate kernel approximation methods126 Delay differential equation189 Difference kernel37 Differential resolvent83 Differentiation methods for first kind equations158 Direct methods for equations of the first kind143 Discretization error100 Effect of data errors, on Abel’s equation204 Effect of data errors, on equations of the first kind204 Elongation of a filament204 Error estimates103 Error expansion105121 Euler’s method133144 Existence of a solution, Abel’s equation7375 Existence of a solution, equations with unbounded kernels48 Existence of a solution, integrodifferential equations49 Existence of a solution, linear first kind equations67 Existence of a solution, linear second kind equations29 Existence of a solution, nonlinear first kind equations69 Existence of a solution, nonlinear second kind equations5255 Existence of a solution, systems of the second kind48 Finite rank kernel8 First kind equations, nonlinear69160 First kind equations, numerical solution143 First kind equations, with smooth kernels67 Fourier transform19 Functional equations188 Gregory methods, for equations of the first kind151 Gregory methods, for equations of the second kind98 Heard method29 Heat conduction equation19 ill-posed problems162 Initial value problems and Volterra equations7 Instability in numerical computation103 Integral equations, of Fredholm type3 Integral equations, of Volterra type3 Integral equations, singular3 Integral equations, weakly singular3 Integrodifferential equations, numerical methods177 Integrodifferential equations, reduction to integral Equations5 Interchanging order of integration6 Inversion formulas, for Fourier transforms19 Inversion formulas, for Laplace transform84 Iterated kernels36 Kernel, degenerate8 Kernel, finite rank8 Kernel, Lipschitz continuous51 Kernel, monotonicity properties58 Kernel, of an integral equation4 Kernel, unbounded4 Laplace transforms84
Linear multistep methods182 Lipschitz condition, for equations of the first kind69 Lipschitz condition, for equations of the second kind52 Lipschitz condition, for integrodifferential equations177 Method of continuation32 Method of successive approximation29 Midpoint method, for Abel’s equation166 Midpoint method, for equations of the first kind144 Milne — Simpson methods182 Newton — Cotes methods, for first kind equations152 Newton — Cotes methods, for second kind equations98 Numerical stability, equations of the first kind149 Numerical stability, equations of the second kind108110 Numerical stability, integrodifferential equations186 Pascal program, midpoint method for first kind equations197 Pascal program, product midpoint method for first kind Equations198 Pascal program, product trapezoidal method for second kind Equations196 Pascal program, trapezoidal method for second kind Equations195 Perturbation, effect, on first kind equations71162 Perturbation, effect, on second kind equations45 Piecewise polynomial approximations118157 Polymer rheology example204 Population equation15 Pouzet methods122 Predictor-corrector methods125 Product integration130 Product trapezoidal method, equations of the first kind168 Product trapezoidal method, equations of the second kind135 Radiating source23 Reactor dynamics equation21 Renewal, density14 Renewal, equation14 Repetition factor106 Resolvent equation, difference kernels38 Resolvent equation, linear problems36 Resolvent equation, nonlinear problems63 Resolvent kernel35 Richardson’s extrapolation133 Roundoff error, in first kind equations161 Roundoff error, in second kind equations121 Runge — Kutta methods, explicit122 Runge — Kutta methods, implicit114 Runge — Kutta methods, of Beltyukov type122 Runge — Kutta methods, of Pouzet type122 Second kind equations, linear29 Second kind equations, nonlinear51 Second kind equations, numerical solution95 Second kind equations, product integration methods129 Semicirculant matrix172 Simpson’s methods99 Stability, definition104183 Stability, interval111 Starting error104 Starting values98 Stereology example24 System identification18 Systems of integral equations546 Systems theory15 Systems, causal17 Systems, evolutionary3 Systems, feedback18 Systems, history-dependent13 Systems, linear16 Systems, nonlinear16 Systems, time-invariant17 Systems, with memory13 Tauberian theorems91 Trapezoidal method, Abel’s equation166 Trapezoidal method, equations of the first kind144 Trapezoidal method, integrodifferential equations178 Uniqueness of a solution, Abel’s equation7375 Uniqueness of a solution, equations with an unbounded Kernel4862 Uniqueness of a solution, integrodifferential equation49 Uniqueness of a solution, linear first kind equation67 Uniqueness of a solution, linear second kind equation29 Uniqueness of a solution, nonlinear first kind equation69 Uniqueness of a solution, nonlinear second kind equation5255 Uniqueness of a solution, systems of the second kind4662 Unit-impulse function16 Unit-response function17 Volterra equations, applications13 Volterra equations, classification3 Volterra equations, linear4 Volterra equations, of first kind4 Volterra equations, of second kind3 Volterra population equation15 Weights for numerical integration98 Well-posed problems45
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