numerical and mathematical biology · dennis g. zill and patric d. shanahan- a first course in...

BT31 (2019-20)

Numerical and Mathematical Biology

Course code: BT31 Course Credits: 3:1:0

Prerequisite: Calculus Contact Hours: 42L + 14T

Course coordinator: Dr. Dinesh P. A. & Dr. M. S. Basavaraj

Course Objectives:

The student will

1. Learn to solve algebraic, transcendental and ordinary differential equations numerically

and find correlation between two variables.

2. Learn the concepts of finite differences, interpolation and their applications.

3. Understand the concepts of PDE and their applications to engineering and also able to

comment on consistency of system of linear equations.

4. Learn the concepts of Fourier transforms and finite element methods and their

applications to solve ordinary differential equations.

5. Understand basic concepts of fluid mechanics and some application models for

blood flow in different geometries.

Unit-I

Numerical solution of Algebraic and Transcendental equations: Method of false position,

Newton - Raphson method.

Numerical solution of Differential equations: Taylor’s series method, Euler’s & modified

Euler’s method, fourth order Runge-Kutta method.

Statistics: Curve fitting by the method of least squares, Fitting linear, quadratic and geometric

curves, Correlation and Regression lines.

Unit-II

Finite Differences and Interpolation: Forward and backward differences, Interpolation, Newton-

Gregory forward and backward interpolation formulae, Lagrange’s interpolation formula,

Newton’s divided difference interpolation formula (no proof).

Numerical Differentiation and Numerical Integration: Derivatives using Newton-Gregory

forward and backward interpolation formulae, Newton-Cote’s quadrature formula, Trapezoidal

Rule, Simpson’s (1/3) rd rule, Simpson’s (3/8) th rule.

Unit-III

Linear Algebra: Rank of a Matrix, Gauss elimination method, eigenvalues and eigenvectors,

similarity transformation.

Partial Differential Equations: Classification of second order PDE, Numerical solution of one-

dimensional heat and wave equations.

Unit-IV

Finite Element Method: Introduction, element shapes, nodes and coordinate systems, Shape

functions, Assembling stiffness equations- Galerkin’s method, Discretization of a structure,

Applications to solve ordinary differential equations.

Unit-V

Fourier Transforms: Infinite Fourier transform and properties.

Models of flows for other Bio-fluids: Introduction to fluid dynamics, continuity equation

for two and three dimensions in different coordinate systems, Navier-Stokes equation in

different coordinate systems, Hagen-Poiseuille flow, special characteristics of blood flows,

fluid flow in circular tubes, Stenosis and different types of stenosis, blood flow through

artery with mild stenosis.

Text Books

1. B.S. Grewal – Higher Engineering Mathematics – Khanna Publishers – 44thedition – 2017.

2. J.N. Kapur – Mathematical Models in Biology and Medicine – East-West Press Private

Ltd., New Delhi – 2010.

3. S.S. Bhavikatti – Finite Element Analysis –NewAge International Publishers – 2015.

Reference Books

1. Dennis G. Zill, Michael R. Cullen – Advanced Engineering mathematics – Jones

and Barlett Publishers Inc. – 3rd edition – 2009.

2. S.S. Sastry – Introductory methods of Numerical Analysis – Prentice Hall of India – 4th

edition – 2006.

3. B.V. Ramana – Higher Engineering Mathematics-Tata McGraw Hill Publishing Company

Ltd, New Delhi – 2008.

Course Outcomes:

At the end of the course, students will be able to

1. Apply numerical techniques to solve engineering problems and fit a least squares

curve to the given data. (PO-1,2 & PSO-1)

2. Find functional values, derivatives, areas and volumes numerically from a given data. (PO-1,2 & PSO-1)

3. Solve system of linear equations and partial differential equations. (PO-1,2 &PSO-1)

4. Solve ordinary differential equations using finite element method. (PO-1,2 & PSO-1)

5. Evaluate Fourier transform and discuss models of Bio-fluid flows. (PO-1,2 & PSO-1)

CH31 (2019-20)

Engineering Mathematics-III

Course Code: CH31 Course Credits: 3:1:0

Prerequisite: Calculus Contact Hours: 42 L + 14T

Course Coordinators: Dr. G. Neeraja & Mr. Vijaya Kumar

Course Objectives:

The students will

1. Learn to solve algebraic, transcendental and ordinary differential equations numerically.

2. Learn to fit a least squares curve and find correlation and regression for a statistical data.

3. Learn the concepts of consistency and solve linear system of equations and system of

ODE’s using matrix method.

4. Learn to test for convergence of positive terms and represent a periodic function in terms

of sine and cosine.

5. Understand the concepts of calculus of functions of complex variables.

Unit I



Numerical solution of Ordinary differential equations: Taylor’s series method, Euler’s &

modified Euler’s method, fourth order Runge-Kutta method.

Statistics: Curve fitting by the method of least squares, fitting linear, quadratic and geometric

curves, Correlation and Regression.

Unit II

Linear Algebra: Elementary transformations on a matrix, Echelon form of a matrix, rank of a

matrix, consistency of system of linear equations, Gauss elimination and Gauss – Seidel method

to solve system of linear equations, Eigen values and eigen vectors of a matrix, Rayleigh power

method to determine the dominant eigen value of a matrix, Diagonalization of a matrix, Solution

of system of ODE’s using matrix method.

Unit III

Fourier Series: Convergence and divergence of infinite series of positive terms, Periodic

functions, Dirichlet conditions, Fourier series of periodic functions of period 2π and arbitrary

period, half range Fourier series, Practical harmonic analysis.

Unit IV

Complex Variables - I: Functions of complex variables, Analytic function, Cauchy-Riemann

Equations in Cartesian and polar coordinates, Consequences of Cauchy-Riemann Equations,

Construction of analytic functions.

Transformations: Conformal transformation, Discussion of the transformations zew , 2zw

andz

azw

2

, 0z , Bilinear transformations.

Unit V

Complex Variables-II: Complex integration, Cauchy’s theorem, Cauchy’s integral formula,

Taylor’s & Laurent’s series (statements only), Singularities, poles and residues, Cauchy residue

theorem (statement only).

Text Books:

1. Erwin Kreyszig –Advanced Engineering Mathematics – Wiley publication – 10th edition-2015.

2. B. S. Grewal –Higher Engineering Mathematics – Khanna Publishers – 44th edition – 2017.

References:

1. David C. Lay – Linear Algebra and its Applications – Pearson Education – 3rd edition – 2011.

2. Glyn James – Advanced Modern Engineering Mathematics – Pearson Education – 4th

edition – 2010.

3. Dennis G. Zill and Patric D. Shanahan- A First Course in Complex Analysis with

Applications- Jones and Bartlett Publishers – 2nd edition–2009.

Course Outcomes


1. Apply numerical techniques to solve engineering problems and fit a least squares curve

to the given data. (PO-1,2 & PSO-2)

2. Test the system of linear equations for consistency and solve ODE’s using Matrix

method. (PO-1,2 & PSO-2)

3. Construct the Fourier series expansion of a function/tabulated data. (PO-1,2 & PSO-2)

4. Examine and construct analytic functions. (PO-1,2 & PSO-2)

5. Classify singularities of complex functions and evaluate complex integrals. (PO-1,2 &

PSO-2)

CS31 (2019-20)


Course Code: CS31 Course Credits: 3:1:0

Prerequisite: Calculus Contact Hours: 42 L+14T

Course Coordinators: Dr. N. L. Ramesh & Dr. A. Sreevallabha Reddy

Course Objectives:

The students will

1. Learn to solve algebraic, transcendental and ordinary differential equations numerically, fit

a least squares curve and find correlation, regression for a statistical data.

2. Learn the concepts of consistency, solve linear system of equations and system of

differential equations using matrix method.

3. Learn the concepts of orthogonal diagonalization, vector spaces and linear transformations.

4. Learn to represent a periodic function in terms of sines and cosines.

5. Learn the concepts of Fourier and Z transforms.

Unit I



Numerical solution of Ordinary differential equations: Taylor series method, Euler and

modified Euler method, fourth order Runge-Kutta method.



Unit II

Linear Algebra I: Elementary transformations on a matrix, Echelon form of a matrix, Rank of a

matrix, Consistency of system of linear equations, Gauss elimination and Gauss – Seidel method

to solve system of linear equations, Eigen values and eigen vectors of a matrix, Rayleigh power

method to determine the dominant eigen value of a matrix, Diagonalization of a matrix, Solution

of system of ODEs using matrix method.

Unit-III

Linear Algebra II: Symmetric matrices, Orthogonal diagonalization and Quadratic forms, Vector

Spaces, Linear combination and span, Linearly independent and dependent vectors, Basis and

Dimension, Linear transformations, Composition of matrix transformations, Rotation about the

origin, Dilation, Contraction and Reflection, Kernel and Range, Change of basis.

Unit IV

Fourier Series: Convergence and divergence of infinite series of positive terms. Periodic

functions, Dirichlet’s conditions, Fourier series of periodic functions of period 2π and arbitrary

period, half range Fourier series, Practical harmonic analysis.

Unit V

Fourier Transforms: Infinite Fourier transform, Fourier sine and cosine transform, Properties,

Inverse transform.

Z-Transforms: Definition, Standard Z-transforms, Single sided and double sided, Linearity

property, Damping rule, Shifting property, Initial and final value theorem, Inverse Z-transform,

Application of Z-transform to solve difference equations.

Text Books:

1. Erwin Kreyszig-Advanced Engineering Mathematics-Wiley-India publishers- 10th edition-

2015.

2. B.S. Grewal - Higher Engineering Mathematics - Khanna Publishers – 44th edition-2017.

Reference Books:

1. David C. Lay – Linear Algebra and its Applications – Jones and Bartlett Press – 3rd edition

– 2011.

2. Peter V. O’Neil – Advanced Engineering Mathematics – Thomson Brooks/Cole – 7th

edition – 2011.

3. Gareth Williams – Linear Algebra with Applications, Jones and Bartlett Press – 8rd edition

– 2012.

Course Outcomes

At the end of the Course, students will be able to

1. Apply numerical techniques to solve engineering problems and fit a least

squares curve to the given data. (PO-1,2 & PSO-2)

2. Test the system of linear equations for consistency and solve system of ODE’s using matrix


3. Diagonalize a given matrix orthogonally and find kernel and range of linear transformation.

(PO-1,2 & PSO-2)

4. Construct the Fourier series expansion of a function/ tabulated data. (PO-1,2 & PSO-2)

5. Evaluate Fourier transforms and use Z-transforms to solve difference equations.

(PO-1,2 & PSO-2)

CV31 (2019-20)


Course Code: CV31 Course Credits: 3:1:0

Prerequisite: Calculus Contact Hours: 42L +14T

Course Coordinator: Dr. G. Neeraja & Dr. Monica Anand

Course Objectives

The students will

1) Learn the concepts of consistency and solve linear system of equations and system of


2) Learn the concepts of orthogonal diagonalization and linear transformation through matrix

algebra.

3) Learn to solve one dimensional heat, wave and two dimensional Laplace equations by

numerical methods.

4) Understand the method of finding the extremal of a functional using analytical technique.

5) Learn to fit a least squares curve and find correlation and regression for a statistical data.

Unit-I


matrix, Consistency of system of linear equations, Gauss elimination and Gauss-Seidel method to

solve system of linear equations, Eigen values and Eigen vectors of a matrix, Rayleigh’s power

method to determine the dominant Eigen value of a matrix, Diagonalization of a matrix, Solutions


Unit-II

Linear Algebra II: Symmetric matrices, Orthogonal diagonalization and Quadratic forms.Vector

Spaces, Linear Combination and Span, Linearly Independent and Dependent vectors, Basis and

Dimension, Linear Transformations, Composition of matrix transformations, Rotation about the

origin, Dilation, Contraction and Reflection, Kernel and Range, Change of basis.

Unit – III

Partial Differential Equations: Classification of second order PDE’s, Numerical solution of one

dimensional heat equation using implicit and explicit finite difference methods. Numerical

solution of one dimensional wave equation, Two - dimensional Laplace and Poisson equations.

Unit-IV

Calculus of variation: Variation of a function and a functional, Extremal of a functional, Euler’s

equation, Standard variational problems, Geodesics, Minimal surface of revolution, Hanging cable

and Brachistochrone problems.

Unit-V


curves, Correlation and Regression analysis, Multiple correlation and regression.

Text Books:


2015.

2. B. S. Grewal – Higher Engineering Mathematics – Khanna Publishers – 44th edition – 2017.

References:

1. David C. Lay – Linear Algebra and its Applications – Jones and Bartlett Press – 3rd edition –

2011.

2. Peter V. O’Neil – Advanced Engineering Mathematics – Thomson Brooks/Cole – 7th edition

-2011

Course Outcomes


1. Test the system of linear equations for consistency and solve ODEs using matrix


2. Diagonalize the given matrix orthogonally and find kernel and range of linear

transformation. (PO-1,2 & PSO-1)

3. Solve partial differential equations numerically. (PO-1,2 & PSO-1)

4. Form functional as integral and find extremal curve using Euler-Lagrange equation.

(PO-1,2 & PSO-1)

5. Fit a least squares curve to the given data and interpret the correlation between

variables. (PO-1,2 & PSO-1)

EC31 (2019-20)


Course Code: EC31 Course Credits: 3:1:0


Course Coordinator: Dr. M.V.Govindaraju & Dr. M. Girinath Reddy

Course Objectives:

The students will


2. Learn to fit least squares curves and find correlation, regression for a statistical data.




5. Learn to represent a periodic function in terms of sine and cosine functions.

Unit I



Numerical solution of Ordinary differential equations: Taylor’s series method, Euler’s and



curves. Correlation and Regression.

Unit II



to solve system of linear equations, Eigen values and Eigen vectors of a matrix, Rayleigh power

method to determine the dominant Eigen value of a matrix, Diagonalization of a matrix, Solution


Unit III

Complex Variables-I: Functions of complex variables, Analytic function, Cauchy-Riemann

equations in Cartesian and polar coordinates, Consequences of Cauchy-Riemann equations,


Transformations: Conformal transformation, Discussion of the transformations - zewzw ,2

and )0(2

zz

azw , Bilinear transformation.

Unit IV

Complex Variables-II: Complex integration, Cauchy theorem, Cauchy integral formula, Taylor

and Laurent series (statements only), Singularities, Poles and residues, Cauchy residue theorem

(statement only).

Unit V

Fourier series: Convergence and divergence of infinite series of positive terms, Periodic function,

Dirichlet’s conditions, Fourier series of periodic functions of period 2 and arbitrary period. Half

range Fourier series. Fourier series for Periodic square wave, Half wave rectifier, Full wave

rectifier, Saw-tooth wave with graphical representation, Practical harmonic analysis.

Text Books



Reference Books

1. Glyn James – Advanced Modern Engineering Mathematics – Pearson Education – 4th edition –

2010.

2. Dennis G. Zill, Michael R. Cullen - Advanced Engineering Mathematics, Jones and Barlett

Publishers Inc. – 3rdedition – 2009.

3. Dennis G. Zill and Patric D. Shanahan- A first course in complex analysis with

applications- Jones and Bartlett publishers-second edition-2009.

Course Outcomes:

At the end of the course, students will be able to:

1. Apply numerical techniques to solve Engineering problems and fit a least squares curve to

the given data. (PO-1,2 & PSO-1,3)


method. (PO-1,2 & PSO-1,3)

3. Examine and construct the analytic functions. (PO-1,2 & PSO-1,3)


PSO-1,3)

5. Construct the Fourier series expansion of a function/tabulated data. (PO-1,2 & PSO-1,3)

EE31 (2019-20)


Course Code: EE31 Course Credits: 3:1:0


Course Coordinator: Dr. M.V.Govindaraju&Dr. M. Girinath Reddy

Course Objectives:

The students will







Unit I







Unit II






Unit III





and )0(2

zz


Unit IV

Complex Variables-II: Complex integration, Cauchy theorem, Cauchy integral formula. Taylor

and Laurent series (statements only). Singularities, Poles and residues, Cauchy residue theorem

(statement only).

Unit V

Fourier series: Convergence and divergence of infinite series of positive terms. Periodic function,



rectifier, Saw-tooth wave with graphical representation. Practical harmonic analysis.

Text Books



Reference Books


2010.





Course Outcomes:








PSO-1,2)


EI31 (2019-20)


Course Code: EI31 Course Credits: 3:1:0


Course Coordinator: Dr. M.V. Govindaraju & Dr. M. Girinath Reddy

Course Objectives:

The students will







Unit I




modified Euler’s method, Fourth order Runge-Kutta method.



Unit II

Linear Algebra: Elementary transformations on a matrix, Echelon form of a matrix, Rank of a





Unit III





and )0(2

zz


Unit IV



(statement only).

Unit V

Fourier series: Convergence and divergence of infinite series of positive terms, Periodic function,




Text Books



Reference Books


2010.


Publishers Inc. – 3rd edition – 2009.



Course Outcomes:


1. Apply numerical techniques to solve engineering problems and fit a least squares curve to






PSO-1,3)


IM31 (2019-20)


Course Code: IM31 Course Credits: 3:1:0



Course Objectives:

The students will





4. Learn to test for convergence of positive terms and represent a periodic function in terms

of sine and cosine functions.


Unit I







Unit II






Unit III

Fourier Series: Convergence and divergence of infinite series of positive terms, Periodic


period, Half range Fourier series, Practical harmonic analysis.

Unit IV





andz

azw

2


Unit V

Complex Variables-II: Complex integration, Cauchy theorem, Cauchy integral formula. Taylor’s

& Laurent’s series (statements only). Singularities, poles and residues, Cauchy residue theorem

(statement only).

Text Books:


2. B. S. Grewal –Higher Engineering Mathematics – Khanna Publishers – 44thedition – 2017.

References Books:

1. David C. Lay – Linear Algebra and its Applications – Pearson Education – 3rdedition – 2011.


edition – 2010.


applications- Jones and Bartlett publishers – 2nd edition–2009.

Course Outcomes



curve to the given data. (PO-1,2 & PSO-1,2)

2. Test the system of linear equations for consistency and solve ODE’s using matrix


3. Construct the Fourier series expansion of a function/tabulated data. (PO-1,2 & PSO-

1,2)

4. Examine and construct analytic functions. (PO-1,2 & PSO-1,2)

5. Classify singularities of complex functions and evaluate complex integrals. (PO-1,2

& PSO-1,2)

IS31 (2019-20)


Course Code: IS31 Course Credits: 3:1:0


Course Coordinators: Dr. N. L. Ramesh & Dr. A. Sreevallabha Reddy

Course Objectives:

The students will

1. Learn to solve algebraic, transcendental and ordinary differential equations numerically,

fit a least squares curve and find correlation, regression for a statistical data.

2. Learn the concepts of consistency, solve linear system of equations and system of

differential equations using matrix method.

3. Learn the concepts of orthogonal diagonalization, vector spaces and linear transformations.

4. Learn to represent a periodic function in terms of sines and cosines.

5. Learn the concepts of Fourier and Z transforms.

Unit I

Numerical Solution of Algebraic and Transcendental Equations: Method of false position,


Numerical Solution of Ordinary Differential Equations: Taylor series method, Euler and

modified Euler method, fourth order Runge-Kutta method.



Unit II






Unit-III

Linear Algebra II: Symmetric matrices, Orthogonal diagonalization and Quadratic forms, Vector

Spaces, Linear combination and span, Linearly independent and dependent vectors, Basis and

dimension, Linear transformations, Composition of matrix transformations, Rotation about the

origin, Dilation, Contraction and Reflection, Kernel and range, Change of basis.

Unit IV

Fourier series: Convergence and divergence of infinite series of positive terms. Periodic



Unit V

Fourier Transforms: Infinite Fourier transform, Fourier sine and cosine transform, Properties,

Inverse transform.

Z-Transforms: Definition, Standard Z-transforms, Single sided and double sided, Linearity

property, Damping rule, Shifting property, Initial and final value theorem, Inverse Z-transform,

Application of Z-transform to solve difference equations.

Text Books:


2015.

2. B.S. Grewal - Higher Engineering Mathematics - Khanna Publishers – 44th edition-2017.

Reference Books:

1. David C. Lay – Linear Algebra and its Applications – Jones and Bartlett Press – 3rd edition

– 2011.

2. Peter V. O’Neil – Advanced Engineering Mathematics – Thomson Brooks/Cole – 7th

edition – 2011.

3. Gareth Williams – Linear Algebra with Applications, Jones and Bartlett Press – 8rd edition

– 2012.

Course Outcomes

At the end of the Course, students will be able to


curve to the given data. (PO-1,2 & PSO-2)



3. Diagonalize a given matrix orthogonally and find kernel and range of a linear

transformation. (PO-1,2 & PSO-2)


5. Evaluate Fourier transforms and use Z-transforms to solve difference equations.

(PO-1,2 & PSO-2)

ME31 (2019-20)


Course Code: ME31 Course Credits: 3:1:0



Course Objectives:

The students will





4. Learn to test for convergence of positive terms and to represent a periodic function in

terms of sine and cosine functions.


Unit I




modified Euler’s method, Fourth order Runge-Kutta method.



Unit II






Unit III

Fourier Series: Convergence and divergence of infinite series of positive terms. Periodic



Unit IV





andz

azw

2


Unit V

Complex Variables-II: Complex integration, Cauchy theorem, Cauchy integral formula. Taylor’s

& Laurent’s series (statements only). Singularities, poles and residues, Cauchy residue theorem

(statement only).

Text Books:


2. B. S. Grewal –Higher Engineering Mathematics – Khanna Publishers – 44th edition – 2017.

References:

1. David C. Lay – Linear Algebra and its Applications – Pearson Education – 3rd edition – 2011.


edition – 2010.


applications- Jones and Bartlett publishers – 2nd edition–2009.

Course Outcomes



curve to the given data. (PO-1,2 & PSO-1,2)

2. Test the system of linear equations for consistency and solve ODE’s using matrix


3. Construct the Fourier series expansion of a function/tabulated data. (PO-1,2 & PSO-

1,2)

4. Examine and construct analytic functions. (PO-1,2 & PSO-1,2)

5. Classify singularities of complex functions and evaluate complex integrals. (PO-1,2

& PSO-1,2)

ML31 (2019-20)


Course Code: ML31 Course Credits: 3:1:0


Course Coordinator: Dr. M.V. Govindaraju & Dr. M. Girinath Reddy

Course Objectives:

The students will







Unit I







Unit II






Unit III





and )0(2

zz


Unit IV



(statement only).

Unit V




rectifier, Saw-tooth wave with graphical representation, Practical harmonic analysis.

Text Books



Reference Books


2010.





Course Outcomes:


1. Apply numerical techniques to solve engineering problems and fit a least squares curve to

the given data. (PO-1,2 & PSO-1)



3. Examine and construct the analytic functions. (PO-1,2 & PSO-1)


PSO-1)


TC31 (2019-20)


Course Code: TC31 Course Credits: 3:1:0


Course Coordinator: Dr. M.V.Govindaraju & Dr. M. Girinath Reddy

Course Objectives:

The students will







Unit I







Unit II






Unit III





and )0(2

zz


Unit IV



(statement only).

Unit V





Text Books



Reference Books


2010.





Course Outcomes:



the given data. (PO-1,2 & PSO-1)



3. Examine and construct the analytic functions. (PO-1,2 & PSO-1)


PSO-1)


numerical and mathematical biology · dennis g. zill and patric d. shanahan- a first course in...

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