Introduction to �Differential Equations

Dr. K. KRISHNAN Ph.D.

Research Supervisor

P. G. &RESEARCH DEPARTMENT OF MATHEMATICS

C.P.A.COLLEGE, BODINAYAKANUR

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How many ways we can divide this square into 4 equal parts?

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is 0.999… equal to 1?�

YES 0.999… = 1

FOR, LET X = 0.999…

10 X = 9.999….

10 X – X = 9

9 X = 9

X = 1

Differential Equations�

A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two.

Examples

•  In classical mechanics, the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow these variables to be expressed dynamically (given the position, velocity, acceleration and various forces acting on the body) as a differential equation for the unknown position of the body as a function of time.

Ordinary differential equations��

• An ordinary differential equation (ODE) is an equation containing an unknown function of one real or complex variable x, its derivatives, and some given functions of x. The unknown function is generally represented by a variable (often denoted y), which, therefore, depends on x. Thus x is often called the independent variable of the equation. The term "ordinary" is used in contrast with the term partial differential equation, which may be with respect to more than one independent variable.

Partial differential equations��

• A partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. (This is in contrast to ordinary differential equations, which deal with functions of a single variable and their derivatives.) 

Non-linear differential equations�

• Non-linear differential equations are formed by the products of the unknown function and its derivatives are allowed and its degree is > 1.

Second-Order Linear Differential Equations

Laplace Transform

Delay Differential Equations and Their Applications in Biology

Introduction

• ODEs have been used to model many different types of systems including biological ones
• But they often “cannot capture the rich variety of dynamics observed in natural systems” (Forde 1)
•  DDEs include a time delay which makes the systems more logical and accurate biologically
• Thesis: Delay differential equations include the time delays that inherently exist in biological processes, proving that delay differential equations more accurately model the life sciences

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ODEs vs. DDEs

• Main difference is in their initial conditions’
• ODEs: initial instant of time 🡪 x(t₀) = x₀
• DDEs: needs the values for all past values of time , in simple DDEs the initial time interval is: [t₀- τ, t₀]
• τ is a positive constant, that stands for a specific point in time
• t₀- τ stands for a “previous time”

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Solving: Method of Steps

• On the interval y(t) is given by p(t) so we can say it is solved for this interval and call it

• Note: when then becomes

• In order to solve DDE’s using Method of Steps, need a history function

Solving: Method of Steps Cont.

• This equation is now an ordinary differential equation because is known, it is

Lotka-Volterra Model with a Delay

• The Lotka-Volterra Predator-Prey Model with delays is shown below:
• x is the population of prey, y is the population of mature predators, y_j is the population of juvenile predators

Breast Cancer Model with DDEs

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• Golnar Newbury uses delay differential equations in his/her research in breast cancer.
• Uses two other models to create a new model that incorporates time delay for cells in the interphase stage, mitosis phase, as well as the ways in which Paclitaxel, a cycle specific drug, targets tumor cells.

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Conclusion

• DDEs were developed 10-20 years ago, compared to other math concepts this is new, therefore there is more to learn
• More difficult and complex to solve than ODEs, but luckily technology can be a great aid to solve DDEs
• The inclusion of delays in mathematical systems which are modeling biological phenomena is a theory and concept that will continue to develop over time, with more research done on them there should be an easier way to solve and analyze them

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��������THANK YOU���Questions ?