# Recent Lecture Notes:
9. Partial Differential Equations
1. Foundations & Introduction
2. Quasi-linear PDEs and Method of Characteristics
3. General Solution and Burgers' Equation
4. First order Non-Linear PDEs
5. Classification of 2nd order PDEs
6. Wave Equation part-1
7. Heat Equation part-1
8. Poisson Equation part-1
9. Power Series Solutions
10. Frobenius Method
Additional Read:
1. Solution for Burgers' equation
2. Method of Characteristics
3. Monge Cone
4. Canonical Forms
5. Wave Equation part-2
6. Heat Equation part-2
7. Poisson Equation part-2
8. Mathematics - II
1. Introduction to Complex Numbers
2. Limit, Continuity and Derivative
3. Analytic Functions and Cauchy-Riemann Equations
4. Elementary Complex Functions
5. Branches of Logarithm
6. Line Integral for Complex functions
7. Cauchy Integral Theorem and Formula
8. Power, Taylor and Laurent Series
9. Zeroes, Singularities and Residue Theorem
Additional Read:
1. Spherical Representation of Complex Numbers
2. Complex Logarithmic function
3. Logarithmic Functions
7. Numerical Linear Algebra
1. Introduction and Gauss Elimination Method, LU, Cholesky Factorization
2. Banded Matrix, Norms, Perturbation Theory, Floating Point Arithmetic
3. Iterative Methods for Linear Systems
4. Gram Schmidt, Householder Transformation and QR Factorization
5. Least Squares Problems, SVD Decomposition, Power Method
6. Krylov Subspace Method, Arnoldi, Lanczos iterations
6. Mathematical Methods / Numerical Techniques
1. Introduction
2. Methods for ODEs: IVP
3. Error Analysis
4. Methods for ODEs: BVP
5. Solving Linear systems: Direct Methods
6. Solving Linear systems: Iterative Methods
7. Numerical Methods for Elliptic PDEs
8. Numerical Methods for Parabolic PDEs
9. Numerical Methods for Hyperbolic PDEs
10. MATLAB Codes: a) 2D Laplace equation; b) 1D Heat equation; c) 1D Wave equation 11. Solving Non-linear systems: Newton Raphson Method
12. Solving Linear/Non-linear systems: Conjugate Gradient Method
13. MATLAB Codes: a) Conjugate Gradient Method; b) CG vs Gauss-Seidel
14. Techniques to find Maximum range of Eigen-Value of few Matrix
Additional Read:
1. Floating point representation 1 and 2
2. Lecture note 1 by Dr. Mark Trew
3. Lecture note 2 by Dr. Mark Trew
4. Non-linear Shooting Method
5. Summary of Jacobi and Gauss-Seidel
6. Local error vs Global error
7. LU decomposition
8. Gersch-Gorin Theorem (Page 36)
Tutorial Sheets
1. Tutorial 1
2. Tutorial 2
3. Tutorial 3
4. Tutorial 4
5. Tutorial 5
6. Tutorial 6
7. Tutorial 7
8. Tutorial 8
5. Mathematics - I
0. Introduction
1. Limit and Continuity
2. Differentiation and Mean Value Theorems
3. Mean Value Theorems Part 2
4. Indeterminate Forms
5. Convex-Concave function and Maxima-Minima
6. Asymptotes
7. Curve Sketching
8. Polar Co-ordinates
8a. Asymptotes in Polar Co-ordinates
9. Curvature, and Gamma and Beta Functions
10. Calculus of Several variables: Limit and Continuity
11. Partial Derivatives and Differentiability
12. Chain Rule and Homogeneous Functions
13. Harmonic Function and Taylor's Formula
14. Several variable Maxima-Minima and Lagrange Method
15. Multiple Integrals
16. Gradient, Divergence and Curl
17. Line Integrals and Green's Theorem
18. Surface Integrals and Stokes' and Divergence Theorems
19. First order ODEs: solution methods
20. Bernoulli Equation and 2nd Order Homogeneous ODEs
21. Operator and Undetermined Coefficients Methods
22. Variation of Parameters and Simultaneous ODEs
Additional Read:
0. Graph Plotter
1. Limit and Differentiation
2. Graphical representation of Limits
3. Convex, Concave and Point of Inflection
4. Sertöz Theorem on Multivariate limit (Link 1, Link2)
5. Video link for Partial Derivatives
6. 3D plot tool
7. Lagrange's Multiplier Method
8. Vector Calculus Formulae
9. Line and Surface Integrals Summary
10. Interpretation of Line integrals
11. Surface Integrals explanation
12. Curvature additional read
Tutorial Sheets:
0. Tutorial 0
1. Tutorial 1
2. Tutorial 2
3. Tutorial 3
4. Tutorial 4
5. Tutorial 5
6. Tutorial 6
7. Tutorial 7
4. Numerical Analysis LAB
1. LabTask 0
2. LabTask1
3. LabTask2
4. LabTask3
5. LabTask4
6. LabTask5
7. LabTask6
8. LabTask7
3. Ordinary Differential Equations
1. Foundations & Introduction
2. Existence & Uniqueness Theorems-1
3. Existence Theorem-2 and Continuation
4. Second order ODEs
5. Techniques to Solve 2nd order ODEs
6. Power Series Solutions
7. Frobenius Method
8. Boundary Value Problems and Sturm-Liouville Theory
9. Separation and Comparison Theory
10. System of Linear Equations
Additional Read:
1. ODE Notes
2. Deduction of Legendre Equation
3. Sturm Liouville Theory 2
4. Phase Portrait
5. Phase Portrait 2
2. Optimization Techniques LAB
0. A general introduction to Optimization
1. Labtask 1
2. Labtask 2
3. Labtask 3
4. Labtask 4
5. Labtask 5
6. Labtask 6
7. Labtask 7
8. Labtask 8
9. Labtask 9
10. Labtask 10
1. Calculus-II (Michigan State University)
5.1-5.2. Area and Volume
5.4. Work
6.1. Inverse Functions
6.2-6.3. Natural Log and Exponential functions
6.4. General Log and Exponential functions
6.5-9.3. Separable equations
6.6. Inverse Trigonometric functions
6.7. Hyperbolic functions
6.8. Indeterminate forms and l'Hospital's Rule
7.1. Integration by parts
7.2. Trigonometric Integrals
7.3. Trigonometric Substitution
7.4. Integration of Partial Fraction
7.8. Improper Integrals
8.1. Arc Length
7.5. Strategic Integration
11.1. Sequence
11.2. Infinite Series
11.3. Integral Test
11.4. Comparison Test
11.5.-11.6. Alternating Series and Absolute Convergence
11.7. Strategy of Infinite Series
11.8.-11.9. Power Series
11.10.-11.11. Taylor and Maclaurin Series
10.1. Parametric Curves
10.2. Parametric Calculus
10.3. Polar Co-ordinates
10.4. Arc Length and Area in Polar Co-ordinates
Quiz Solutions: Calculus-II
1. Quiz 1
2. Quiz 2
3. Quiz 3
4. Quiz 4
5. Quiz 5
6. Quiz 6
7. Quiz 7
8. Quiz 8
9. Quiz 9
10. Quiz 10
11. Quiz 11
12. Quiz 12
# Mid Term-1 Question Paper and Solution
9. Partial Differential Equations
1. Foundations & Introduction
2. Quasi-linear PDEs and Method of Characteristics
3. General Solution and Burgers' Equation
4. First order Non-Linear PDEs
5. Classification of 2nd order PDEs
6. Wave Equation part-1
7. Heat Equation part-1
8. Poisson Equation part-1
9. Power Series Solutions
10. Frobenius Method
Additional Read:
1. Solution for Burgers' equation
2. Method of Characteristics
3. Monge Cone
4. Canonical Forms
5. Wave Equation part-2
6. Heat Equation part-2
7. Poisson Equation part-2
8. Mathematics - II
1. Introduction to Complex Numbers
2. Limit, Continuity and Derivative
3. Analytic Functions and Cauchy-Riemann Equations
4. Elementary Complex Functions
5. Branches of Logarithm
6. Line Integral for Complex functions
7. Cauchy Integral Theorem and Formula
8. Power, Taylor and Laurent Series
9. Zeroes, Singularities and Residue Theorem
Additional Read:
1. Spherical Representation of Complex Numbers
2. Complex Logarithmic function
3. Logarithmic Functions
7. Numerical Linear Algebra
1. Introduction and Gauss Elimination Method, LU, Cholesky Factorization
2. Banded Matrix, Norms, Perturbation Theory, Floating Point Arithmetic
3. Iterative Methods for Linear Systems
4. Gram Schmidt, Householder Transformation and QR Factorization
5. Least Squares Problems, SVD Decomposition, Power Method
6. Krylov Subspace Method, Arnoldi, Lanczos iterations
6. Mathematical Methods / Numerical Techniques
1. Introduction
2. Methods for ODEs: IVP
3. Error Analysis
4. Methods for ODEs: BVP
5. Solving Linear systems: Direct Methods
6. Solving Linear systems: Iterative Methods
7. Numerical Methods for Elliptic PDEs
8. Numerical Methods for Parabolic PDEs
9. Numerical Methods for Hyperbolic PDEs
10. MATLAB Codes: a) 2D Laplace equation; b) 1D Heat equation; c) 1D Wave equation 11. Solving Non-linear systems: Newton Raphson Method
12. Solving Linear/Non-linear systems: Conjugate Gradient Method
13. MATLAB Codes: a) Conjugate Gradient Method; b) CG vs Gauss-Seidel
14. Techniques to find Maximum range of Eigen-Value of few Matrix
Additional Read:
1. Floating point representation 1 and 2
2. Lecture note 1 by Dr. Mark Trew
3. Lecture note 2 by Dr. Mark Trew
4. Non-linear Shooting Method
5. Summary of Jacobi and Gauss-Seidel
6. Local error vs Global error
7. LU decomposition
8. Gersch-Gorin Theorem (Page 36)
Tutorial Sheets
1. Tutorial 1
2. Tutorial 2
3. Tutorial 3
4. Tutorial 4
5. Tutorial 5
6. Tutorial 6
7. Tutorial 7
8. Tutorial 8
5. Mathematics - I
0. Introduction
1. Limit and Continuity
2. Differentiation and Mean Value Theorems
3. Mean Value Theorems Part 2
4. Indeterminate Forms
5. Convex-Concave function and Maxima-Minima
6. Asymptotes
7. Curve Sketching
8. Polar Co-ordinates
8a. Asymptotes in Polar Co-ordinates
9. Curvature, and Gamma and Beta Functions
10. Calculus of Several variables: Limit and Continuity
11. Partial Derivatives and Differentiability
12. Chain Rule and Homogeneous Functions
13. Harmonic Function and Taylor's Formula
14. Several variable Maxima-Minima and Lagrange Method
15. Multiple Integrals
16. Gradient, Divergence and Curl
17. Line Integrals and Green's Theorem
18. Surface Integrals and Stokes' and Divergence Theorems
19. First order ODEs: solution methods
20. Bernoulli Equation and 2nd Order Homogeneous ODEs
21. Operator and Undetermined Coefficients Methods
22. Variation of Parameters and Simultaneous ODEs
Additional Read:
0. Graph Plotter
1. Limit and Differentiation
2. Graphical representation of Limits
3. Convex, Concave and Point of Inflection
4. Sertöz Theorem on Multivariate limit (Link 1, Link2)
5. Video link for Partial Derivatives
6. 3D plot tool
7. Lagrange's Multiplier Method
8. Vector Calculus Formulae
9. Line and Surface Integrals Summary
10. Interpretation of Line integrals
11. Surface Integrals explanation
12. Curvature additional read
Tutorial Sheets:
0. Tutorial 0
1. Tutorial 1
2. Tutorial 2
3. Tutorial 3
4. Tutorial 4
5. Tutorial 5
6. Tutorial 6
7. Tutorial 7
4. Numerical Analysis LAB
1. LabTask 0
2. LabTask1
3. LabTask2
4. LabTask3
5. LabTask4
6. LabTask5
7. LabTask6
8. LabTask7
3. Ordinary Differential Equations
1. Foundations & Introduction
2. Existence & Uniqueness Theorems-1
3. Existence Theorem-2 and Continuation
4. Second order ODEs
5. Techniques to Solve 2nd order ODEs
6. Power Series Solutions
7. Frobenius Method
8. Boundary Value Problems and Sturm-Liouville Theory
9. Separation and Comparison Theory
10. System of Linear Equations
Additional Read:
1. ODE Notes
2. Deduction of Legendre Equation
3. Sturm Liouville Theory 2
4. Phase Portrait
5. Phase Portrait 2
2. Optimization Techniques LAB
0. A general introduction to Optimization
1. Labtask 1
2. Labtask 2
3. Labtask 3
4. Labtask 4
5. Labtask 5
6. Labtask 6
7. Labtask 7
8. Labtask 8
9. Labtask 9
10. Labtask 10
1. Calculus-II (Michigan State University)
5.1-5.2. Area and Volume
5.4. Work
6.1. Inverse Functions
6.2-6.3. Natural Log and Exponential functions
6.4. General Log and Exponential functions
6.5-9.3. Separable equations
6.6. Inverse Trigonometric functions
6.7. Hyperbolic functions
6.8. Indeterminate forms and l'Hospital's Rule
7.1. Integration by parts
7.2. Trigonometric Integrals
7.3. Trigonometric Substitution
7.4. Integration of Partial Fraction
7.8. Improper Integrals
8.1. Arc Length
7.5. Strategic Integration
11.1. Sequence
11.2. Infinite Series
11.3. Integral Test
11.4. Comparison Test
11.5.-11.6. Alternating Series and Absolute Convergence
11.7. Strategy of Infinite Series
11.8.-11.9. Power Series
11.10.-11.11. Taylor and Maclaurin Series
10.1. Parametric Curves
10.2. Parametric Calculus
10.3. Polar Co-ordinates
10.4. Arc Length and Area in Polar Co-ordinates
Quiz Solutions: Calculus-II
1. Quiz 1
2. Quiz 2
3. Quiz 3
4. Quiz 4
5. Quiz 5
6. Quiz 6
7. Quiz 7
8. Quiz 8
9. Quiz 9
10. Quiz 10
11. Quiz 11
12. Quiz 12
# Mid Term-1 Question Paper and Solution
