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CONTENTS xv

16. Integral Equations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 801
16.1. Linear Integral Equations of the First Kind with Variable Integration Limit . . . . . . . . . 801
16.1.1. Volterra Equations of theFirst Kind .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 801
16.1.2. Equations with Degenerate Kernel: K(x, t)= g 1 (x)h 1 (t)+ ··· + g n (x)h n (t) . . 802
16.1.3. Equations with Difference Kernel: K(x, t)= K(x – t) ... .. .. .. .. .. .. .. .. 804
16.1.4. Reduction of Volterra Equations of the First Kind to Volterra Equations of the
Second Kind .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 807
16.1.5. Method of Quadratures . .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 808
16.2. Linear Integral Equations of the Second Kind with Variable Integration Limit . . . . . . . 810
16.2.1. Volterra Equations of theSecond Kind .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 810
16.2.2. Equations with Degenerate Kernel: K(x, t)= g 1 (x)h 1 (t)+ ··· + g n (x)h n (t) . . 811
16.2.3. Equations with Difference Kernel: K(x, t)= K(x – t) ... .. .. .. .. .. .. .. .. 813
16.2.4. Construction of Solutions of Integral Equations with Special Right-Hand Side 815
16.2.5. Method of Model Solutions .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 818
16.2.6. Successive Approximation Method . . . . . . . . . . . . . . . ... .. .. .. .. .. .. .. .. 822
16.2.7. Method of Quadratures . .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 823
16.3. Linear Integral Equations of the First Kind with Constant Limits of Integration . . . . . . 824
16.3.1. Fredholm Integral Equations of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . 824
16.3.2. Method of Integral Transforms .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 825
16.3.3. Regularization Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 827
16.4. Linear Integral Equations of the Second Kind with Constant Limits of Integration . . . . 829
16.4.1. Fredholm Integral Equations of the Second Kind. Resolvent . . . . . . . . . . . . . . 829
16.4.2. Fredholm Equations of the Second Kind with Degenerate Kernel . . . . . . . . . . 830
16.4.3. Solution as a Power Series in the Parameter. Method of Successive
Approximations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 832
16.4.4. Fredholm Theorems and the Fredholm Alternative . . . . . . . . . . . . . . . . . . . . . . 834
16.4.5. Fredholm Integral Equations of the Second Kind with Symmetric Kernel . . . . 835
16.4.6. Methods of Integral Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 841
16.4.7. Method of Approximating a Kernel by a Degenerate One . . . . . . . . . . . . . . . . 844
16.4.8. Collocation Method .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 847
16.4.9. Method of LeastSquares . .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 849
16.4.10. Bubnov–Galerkin Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 850
16.4.11. Quadrature Method .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 852
16.4.12. Systems of Fredholm Integral Equations of the Second Kind . . . . . . . . . . . . . 854
16.5. Nonlinear Integral Equations .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 856
16.5.1. Nonlinear Volterra and Urysohn Integral Equations .. ... .. .. .. .. .. .. .. .. 856
16.5.2. Nonlinear Volterra Integral Equations . . . . . . . . . . . . . ... .. .. .. .. .. .. .. .. 856
16.5.3. Equations with Constant Integration Limits .. .. .. .. ... .. .. .. .. .. .. .. .. 863
References for Chapter 16 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 871
17. Difference Equations and Other Functional Equations . . . . . ... .. .. .. .. .. .. .. .. 873
17.1. Difference Equations of Integer Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873
17.1.1. First-Order Linear Difference Equations of Integer Argument . . . . . . . . . . . . . 873
17.1.2. First-Order Nonlinear Difference Equations of Integer Argument . . . . . . . . . . 874
17.1.3. Second-Order Linear Difference Equations with Constant Coefficients .. .. .. 877
17.1.4. Second-Order Linear Difference Equations with Variable Coefficients .. .. .. 879
17.1.5. Linear Difference Equations of Arbitrary Order with Constant Coefficients . . 881
17.1.6. Linear Difference Equations of Arbitrary Order with Variable Coefficients . . . 882
17.1.7. Nonlinear Difference Equations of Arbitrary Order .. ... .. .. .. .. .. .. .. .. 884