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CONTENTS xxiii

T10.3. Nonlinear Systems of Two Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 1343
2 ∂w ∂ w
2
∂u
∂ u
T10.3.1. Systems of the Form = a 2 + F(u, w), = b 2 + G(u, w) . . . . . . 1343
∂t ∂x ∂t ∂x
T10.3.2. Systems of the Form ∂u = a ∂ x n ∂u + F(u, w),
∂t x n ∂x ∂x
∂w = b ∂ n ∂w + G(u, w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1357
∂t x n ∂x x ∂x
T10.3.3. Systems of the Form Δu = F(u, w), Δw = G(u, w) . . . . . . . . . . . . . . . . . . 1364
2 a ∂
∂ u
T10.3.4. Systems of the Form ∂t 2 = x n ∂x x n ∂u + F(u, w),
∂x
2
∂ w b ∂ n ∂w
∂t 2 = x n ∂x x ∂x + G(u, w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1368
T10.3.5. Other Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1373
T10.4. Systems of General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1374
T10.4.1. Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1374
T10.4.2. Nonlinear Systems of Two Equations Involving the First Derivatives in t . . 1374
T10.4.3. Nonlinear Systems of Two Equations Involving the Second Derivatives in t 1378
T10.4.4. Nonlinear Systems of Many Equations Involving the First Derivatives in t . 1381
References for Chapter T10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1382
T11. Integral Equations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 1385
T11.1. Linear Equations of the First Kind with Variable Limit of Integration . . . . . . . . . . . . . 1385
T11.2. Linear Equations of the Second Kind with Variable Limit of Integration . . . . . . . . . . . 1391
T11.3. Linear Equations of the First Kind with Constant Limits of Integration . . . . . . . . . . . . 1396
T11.4. Linear Equations of the Second Kind with Constant Limits of Integration . . . . . . . . . 1401
References for Chapter T11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406
T12. Functional Equations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 1409
T12.1. Linear Functional Equations in One Independent Variable . . . . . . . . . . . . . . . . . . . . . . 1409
T12.1.1. Linear Difference and Functional Equations Involving Unknown Function
with Two Different Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1409
T12.1.2. Other Linear Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1421
T12.2. Nonlinear Functional Equations in One Independent Variable . . . . . . . . . . . . . . . . . . . 1428
T12.2.1. Functional Equations with Quadratic Nonlinearity . . . . . . . . . . . . . . . . . . . . 1428
T12.2.2. Functional Equations with Power Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . 1433
T12.2.3. Nonlinear Functional Equation of General Form . . . . . . . . . . . . . . . . . . . . . 1434
T12.3. Functional Equations in Several Independent Variables . . . . . . . . . . . . . . . . . . . . . . . . 1438
T12.3.1. Linear Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1438
T12.3.2. Nonlinear Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1443
References for Chapter T12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1450
Supplement. Some Useful Electronic Mathematical Resources .. ... .. .. .. .. .. .. .. .. 1451

Index .. ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 1453