Partial differential equations — Basic concepts and definitions, order, degree, and classification of first-order PDEs. Geometric interpretation of first-order PDEs, normal vectors $(p, q, -1)$, and tangent planes. Formation of PDEs by elimination of arbitrary constants and elimination of arbitrary functions. Four varieties of solutions: Complete Integral, Particular Integral, Singular Integral, and General Integral. Envelope theory and determination of Singular Integrals. Lagrange's linear partial differential equation $Pp + Qq = R$, geometric foundation, and Lagrange's auxiliary equations $\frac{dx}{P} = \frac{dy}{Q} = \frac{dz}{R}$ via method of grouping and multipliers. Integral surfaces passing through a given space curve (Cauchy problem for first-order quasilinear equations). Charpit's general method for non-linear first-order PDEs $f(x, y, z, p, q) = 0$, derivation of Charpit's auxiliary system. Special Standard Forms I ($f(p,q)=0$), II ($f(z,p,q)=0$), III ($f_1(x,p)=f_2(y,q)$), and IV (Clairaut's form $z=px+qy+f(p,q)$).
📝Module:Unit 2 — Higher-Order PDEs with Constant and Reducible Coefficients | 12
Linear differential operators $D = \frac{\partial}{\partial x}$ and $D' = \frac{\partial}{\partial y}$. Homogeneous linear PDEs with constant coefficients: general form, auxiliary equation, and determination of Complementary Function (CF) for distinct linear factors $(D - m D')z = 0$ and repeated linear factors $(D - m D')^r z = 0$. Particular Integral (PI) inverse operator definition $z = \frac{1}{F(D, D')} f(x, y)$. General method for Particular Integral via factor integration. Shortcut rules for Particular Integral: exponential functions $e^{ax + by}$ and polynomial resonance/failure cases, trigonometric functions $\sin(ax + by)$ and $\cos(ax + by)$, and polynomials $x^m y^n$ via binomial operator expansion. Non-homogeneous linear PDEs with constant coefficients factorable into linear factors $(D - m D' - c)$. Non-homogeneous linear PDEs with non-factorable operators: series solutions and trial function methods. Reducible linear PDEs with variable coefficients (Euler-Cauchy homogeneous equations in $x, y$ via logarithmic substitution $x = e^u, y = e^v$).
General second-order linear PDE in two independent variables: $Rr + Ss + Tt + Pp + Qq + Zz = F$. Classification of second-order linear PDEs via discriminant $\Delta = S^2 - 4RT$: Hyperbolic ($\Delta > 0$), Parabolic ($\Delta = 0$), and Elliptic ($\Delta < 0$). Invariance of the discriminant under non-singular coordinate transformations. Geometric and physical characterization of classification: domain partitions and disturbance propagation. Characteristic curves and characteristic roots from the quadratic equation $R (dy/dx)^2 - S (dy/dx) + T = 0$. Reduction of Hyperbolic equations to canonical forms ($u_{\xi\eta} = \Phi$ and $u_{\alpha\alpha} - u_{\beta\beta} = \Psi$). Reduction of Parabolic equations to canonical form ($u_{\eta\eta} = \Phi$). Reduction of Elliptic equations to canonical form ($u_{\alpha\alpha} + u_{\beta\beta} = \Phi$). Monge's Method for equations of Form I ($Rr + Ss + Tt = V$): Pfaffian differential relations and derivation of Monge's subsidiary equations, intermediate integrals $u = f(v)$. Monge's Method for equations of Form II ($Rr + Ss + Tt + U(rt - s^2) = V$): auxiliary system derivation, intermediate integrals, and applications to minimal surfaces.
📝Module:Unit 4 — Fourier Series & Boundary Value Problems (Wave, Heat, Laplace) | 14
Periodic functions, trigonometric system orthogonality on intervals of length $2\pi$, and definition of Fourier series. Dirichlet's conditions for Fourier series convergence (without proof) and behavior at jump discontinuities. Euler's formulae for Fourier coefficients $a_0, a_n, b_n$ on fundamental interval $(c, c + 2\pi)$. Fourier series on symmetric interval $(-\pi, \pi)$ for even functions (cosine series) and odd functions (sine series). Fourier series on arbitrary symmetric intervals $(-L, L)$ and general periodic intervals of length $2L$. Fourier series on arbitrary interval $(0, 2L)$ and general intervals $(c, c + 2L)$. Fourier Half-range sine series and cosine series on $(0, L)$ via odd and even periodic reflections. Parseval's identity and summation of numerical series via Fourier series point evaluations. The Method of Separation of Variables: product ansatz $u(x, t) = X(x) T(t)$ and separation constant sign analysis. 1D Wave equation $u_{tt} = c^2 u_{xx}$: physical modeling of vibrating string with fixed boundary conditions, normal modes, harmonic frequencies, and d'Alembert's traveling wave solution. 1D Heat conduction equation $u_t = k u_{xx}$: physical modeling, Dirichlet and Neumann boundary conditions, transient exponential temperature decay and steady-state limits. 2D Laplace equation $u_{xx} + u_{yy} = 0$: steady-state temperature in rectangular plates, Dirichlet and Neumann boundary value problems.
📝Textbooks
Sneddon, Ian N. (2006). Elements of Partial Differential Equations. Dover Publications. Indian Reprint.
Raisinghania, M. D. (2018). Advanced Differential Equations (19th ed.). S. Chand Publication.
Raisinghania, M. D. (2020). Ordinary and Partial Differential Equations (20th ed.). S. Chand Publication.
Amarnath, T. (2003). An Elementary Course in Partial Differential Equations (2nd ed.). Narosa Publication.
Sharma, J. N. & Singh, Kehar. Partial Differential Equations for Engineers and Scientists. Narosa / Pragati Prakashan.
02First-Order Partial Differential Equations and Charpit's MethodCH
03Higher-Order PDEs with Constant and Reducible CoefficientsCH
04Second-Order PDEs with Variable Coefficients, Canonical Forms, and Monge's MethodsCH
05Fourier Series and Classical Boundary Value ProblemsCH