ADVANCED MATHEMATICS FOR ENGINEERS: SPECIAL COURSES
1 Ordinary Differential Equations 1 11 Preliminaries 1 12 Separable Equations 5 121 Exercises 9 13 Exact Differential Equations 10 131 Integrating factor 12 132 Exercises 13 14 Linear Differential Equations 14 141 Exercises 15 15 Homogeneous Differential Equations 16 151 Exercises 19 16 Bernoulli Equation 20 161 Exercises 21 17 Higher-Order Differential Equations 21 171 Exercises 25 18 Equations with Constant Coefficients 25 181 Exercises 30 19 Nonhomogeneous Equations 30 191 Exercises 32 110 Wronskian Method 33 1101 Exercises 35 111 Cauchy-Euler Equation 36 1111 Exercises 39 2 Partial Differential Equations 40 21 Introduction 40 211 Exercises 44 22 Linear Equations 45 221 Linear equations with constant coefficients 45 222 Exercises 49 223 Equations with variable coefficients 50 224 Exercises 53 23 Quasi-Linear Equations 53 231 Exercises 61 24 Burger’s Equation 63 241 Shock path 69 242 Exercises 72 25 Second-Order PDEs 73 251 Exercises 78 26 Wave Equation and D’Alembert’s Solution 79 261 Exercises 90 262 Vibrating string with fixed ends 93 263 Exercises 97 27 Heat Equation 99 271 Solution of the heat equation 104 272 Heat equation on semi-infinite domain: Dirichlet condition 111 273 Heat equation on semi-infinite domain: Neumann condition 115 274 Exercises 117 28 Wave Equation on Semi-Infinite Domain 120 281 Exercises 123 3 Matrices and Systems of Linear Equations 126 31 Systems of Equations and Gaussian Elimination 126 32 Homogeneous Systems 130 321 Exercises 131 33 Matrices 133 331 Exercises 135 34 Determinants and Inverse of Matrices 136 341 Application to least square fitting 147 342 Exercises 150 35 Vector Spaces 152 351 Exercises 158 36 Eigenvalues-Eigenvectors 160 361 Exercises 167 37 Inner Product Spaces 167 371 Exercises 175 38 Diagonalization 177 381 Exercises 181 39 Quadratic Forms 182 391 Exercises 193 310 Functions of Symmetric Matrices 195 3101 Exercises 201 4 Calculus of Variations 203 41 Introduction 203 42 Euler-Lagrange Equation 206 421 Exercises 216 43 Impact of y? on Euler-Lagrange Equation 220 431 Exercises 222 44 Necessary and Sufficient Conditions 223 441 Exercises 238 45 Applications 239 451 Exercises 245 46 Generalization of Euler-Lagrange Equation 246 461 Exercises 250 47 Natural Boundary Conditions 252 48 Impact of y?? on Euler-Lagrange Equation 261 481 Exercises 261 49 Discontinuity in Euler-Lagrange Equation 263 491 Exercises 266 410 Transversality Condition 267 4101 Problem of Bolza 271 4102 Exercises 273 411 Corners and Broken Extremal 275 4111 Exercises 282 412 Variational Problems with Constraints 282 4121 Exercises 286 413 Isoperimetric Problems 287 4131 Exercises 295 414 Sturm-Liouville Problem 298 4141 The First Eigenvalue 302 4142 Exercises 308 415 Rayleigh Ritz Method 310 4151 Exercises 312 416 Multiple Integrals 313 4161 Exercises 319 5 Integral Equations 321 51 Introduction and Classifications 321 511 Exercises 323 52 Connection between Ordinary Differential Equations and Integral Equations 324 521 Exercises 331 53 The Green’s Function 334 531 Exercises 339 54 Fredholm Integral Equations and Green’s Function 341 541 Exercises 342 542 Beam problem 343 543 Exercises 347 55 Fredholm Integral Equations with Separable Kernels 348 551 Exercises 354 56 Symmetric Kernel 356 561 Exercises 363 57 Iterative Methods and Neumann Series 365 571 Exercises 376 58 Approximating Non-Degenerate Kernels 378 581 Exercises 380 59 Laplace Transform and Integral Equations 381 591 Frequently used Laplace transforms 387 592 Exercises 388 510 Odd Behavior 390 5101 Exercises 396