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Integral Calculus - Solving Functional Differential Equations

  • In this lecture, we will be discussing techniques of solving functional differential equations using integral calculus. slide 2

What are Functional Differential Equations?

  • A functional differential equation (FDE) is an equation involving an unknown function and its derivatives, where the derivatives depend on both the current and past values of the function.
  • It can be represented as: F(y(t), y'(t), y''(t), ..., y^(n)(t), y(t-h), y(t-2h), ... , y(t-nh)) = 0 slide 3

Solving FDEs using Integral Calculus

  • One way to solve functional differential equations is by using integral calculus.
  • We can find the solution by evaluating definite integrals involving the function and its derivatives. slide 4

Example 1:

  • Consider the functional differential equation: y'(t) = y(t-1)
  • We can solve this equation using integral calculus.
  • Taking the integral of both sides, we get: ∫ y'(t) dt = ∫ y(t-1) dt
  • Simplifying, we have: y(t) = ∫ y(t-1) dt + C slide 5

Example 2:

  • Let’s solve another functional differential equation: y''(t) = ty(t-2)
  • Integrating both sides, we obtain: ∫ y''(t) dt = ∫ ty(t-2) dt
  • By applying integration by parts, we get: y'(t) = ∫ t y(t-2) dt - ∫ y'(t-2) dt slide 6

Example 3:

  • Now, let’s consider a more complex functional differential equation: y'''(t) = t^2 y(t-1)
  • Integrating both sides, we have: ∫ y'''(t) dt = ∫ t^2 y(t-1) dt
  • By applying multiple integrations by parts, we can obtain the final solution for y(t). slide 7

Properties of FDE Solutions

  • The solutions of functional differential equations possess some interesting properties.
  • Delay property: The solutions depend on previous values of the function, resulting in delayed responses.
  • Stability property: The stability of FDE solutions can be analyzed using stability analysis techniques. slide 8

Advantages and Applications

  • Solving functional differential equations using integral calculus provides a systematic approach.
  • It allows us to model and analyze systems with delayed responses, such as biological processes, control systems, and physics phenomena.
  • The solutions obtained provide valuable insights into the behavior of the system. slide 9

Example 4:

  • Consider a population growth model with time delay: N'(t) = rN(t - τ) * (1 - N(t) / K) Where:
    • N(t) represents the population at time t.
    • r is the growth rate.
    • K is the carrying capacity.
    • τ is the time delay.
  • By solving this FDE, we can predict the population dynamics in the presence of time delays. slide 10

Conclusion

  • Solving functional differential equations using integral calculus is a powerful tool in understanding systems with delayed responses.
  • It allows us to find solutions and analyze the properties of these equations.
  • Solving FDEs provides valuable insights into the behavior of various systems and has broad applications in various fields. slide 11

Solving FDEs using Integral Calculus

  • We can solve FDEs using integral calculus by expressing the derivative terms as integrals.
  • This allows us to evaluate definite integrals involving the function and its derivatives.
  • In some cases, we may need to apply techniques like integration by parts to simplify the equation.
  • The solution can be found by integrating both sides of the equation and solving for the unknown function.
  • It is important to consider any initial conditions or boundary conditions when solving FDEs. slide 12

Example 1: First-order functional differential equation

  • Consider the following FDE: y'(t) = ∫ y(t-1) dt
  • To solve this equation, we can differentiate both sides to eliminate the integral: y''(t) = y(t-1)
  • Now we have a regular differential equation that can be solved using other methods. slide 13

Example 2: Second-order functional differential equation

  • Let’s solve another FDE: y''(t) = ∫ y(t-1) dt
  • Differentiating both sides gives: y'''(t) = y(t-1)
  • Again, we obtain a regular differential equation to solve. slide 14

Example 3: Higher-order functional differential equation

  • Now let’s consider a higher-order FDE: y'''(t) = ∫ t^2 y(t-1) dt
  • Differentiating both sides multiple times, we can reduce the equation to a regular differential equation.
  • Solving the reduced equation will give us the solution to the original FDE. slide 15

Delay Property of FDE Solutions

  • One notable property of FDE solutions is the delay property.
  • FDE solutions depend on previous values of the function, leading to delayed responses.
  • The delay corresponds to the time delay or history dependence in the system.
  • This delay property can result in unique and interesting behavior in the solutions. slide 16

Example 4: Delayed response in population growth

  • Consider a population growth model with time delay: N'(t) = rN(t - τ) * (1 - N(t) / K)
  • In this equation, the population growth rate depends on the population at a previous time t - τ.
  • The delayed response can have significant impacts on the dynamics of the population.
  • By solving this FDE, we can study the effects of time delays on population growth and stability. slide 17

Stability Analysis of FDE Solutions

  • Stability analysis is an essential aspect of studying FDEs.
  • It helps us understand the behavior of solutions over time.
  • Stability analysis techniques include examining the stability of equilibrium points, studying the eigenvalues of the linearized system, and analyzing Lyapunov functions.
  • By determining the stability properties of FDE solutions, we can make predictions about the long-term behavior of the system. slide 18

Example 5: Stability analysis in a delayed predator-prey model

  • Consider a delayed predator-prey model: x'(t) = αx(t) - βxy(t - τ) y'(t) = δxy(t - τ) - γy(t)
  • By linearizing the system around the equilibrium point, we can analyze the stability of the solutions.
  • The delay term introduces additional complexity and potential stability changes compared to the non-delayed case.
  • Stability analysis helps us understand the coexistence or extinction of predator and prey populations with time delay. slide 19

Applications of FDEs in Various Fields

  • Functional differential equations have widespread applications in various fields.
  • They are used to model and analyze real-world systems that exhibit time delays.
  • Examples of applications include biology, physics, chemistry, economics, and control systems.
  • FDEs enable us to understand delayed responses, dynamic behavior, and stability properties in these systems.
  • By solving FDEs using integral calculus and analyzing their solutions, we can make informed predictions and decisions. slide 20

Summary

  • Solving FDEs using integral calculus involves expressing derivatives as integrals and then solving the resulting equations.
  • The delay property of FDE solutions leads to delayed responses and interesting behavior.
  • Stability analysis helps in understanding the long-term behavior of FDE solutions.
  • FDEs find applications in a wide range of fields, providing insights into systems with time delays.
  • Understanding and solving FDEs using integral calculus contribute to a deeper understanding of dynamic systems. slide 21

Integral Calculus - Solving Functional Differential Equations

  • Solving FDEs using integral calculus involves expressing derivatives as integrals and then solving the resulting equations.
  • The delay property of FDE solutions leads to delayed responses and interesting behavior.
  • Stability analysis helps in understanding the long-term behavior of FDE solutions.
  • FDEs find applications in a wide range of fields, providing insights into systems with time delays.
  • Understanding and solving FDEs using integral calculus contribute to a deeper understanding of dynamic systems. slide 22

Example 1: Solving a first-order functional differential equation

  • Consider the FDE: y'(t) = ∫ y(t-1) dt
  • We can differentiate both sides to eliminate the integral: y''(t) = y(t-1)
  • Now we have a regular differential equation that can be solved. slide 23

Example 2: Solving a second-order functional differential equation

  • Let’s solve the FDE: y''(t) = ∫ y(t-1) dt
  • By differentiating both sides, we get: y'''(t) = y(t-1)
  • This reduces to a regular differential equation which can be solved like any other. slide 24

Example 3: Solving a higher-order functional differential equation

  • Now let’s consider a higher-order FDE: y'''(t) = ∫ t^2 y(t-1) dt
  • By differentiating multiple times, we can reduce this equation to a regular differential equation.
  • Solving the reduced equation will give us the solution to the original FDE. slide 25

Delay Property in FDE Solutions

  • One notable property of FDE solutions is the delay property.
  • The solutions depend on previous values of the function, leading to delayed responses.
  • This delay corresponds to the time delay or history dependence in the system.
  • The delay property can result in unique and interesting behavior in the solutions.
  • It is important to consider the effects of time delays when solving and analyzing FDEs. slide 26

Example 4: Delayed response in a population growth model

  • Consider a population growth model with time delay: N'(t) = rN(t - τ) * (1 - N(t) / K)
  • In this equation, the population growth rate depends on the population at a previous time t - τ, introducing a delay.
  • The delayed response can have significant impacts on the dynamics and stability of the population.
  • By solving this FDE, we can study the effects of time delays on population growth and stability.
  • The delayed response may result in oscillations or other behaviors not seen in non-delayed models. slide 27

Stability Analysis of FDE Solutions

  • Stability analysis is an essential aspect of studying FDEs.
  • It helps us understand the behavior and long-term stability of solutions over time.
  • Stability analysis techniques include examining the stability of equilibrium points, studying the eigenvalues of the linearized system, and analyzing Lyapunov functions.
  • By determining the stability properties of FDE solutions, we can make predictions about the long-term behavior of the system.
  • The presence of delays may affect the stability of the system, requiring specific analysis techniques. slide 28

Example 5: Stability analysis in a delayed predator-prey model

  • Consider a delayed predator-prey model: x'(t) = αx(t) - βxy(t - τ) y'(t) = δxy(t - τ) - γy(t)
  • By linearizing the system around the equilibrium point, we can analyze the stability of the solutions.
  • The delay term introduces additional complexity and potential stability changes compared to the non-delayed case.
  • Stability analysis helps us understand the coexistence or extinction of predator and prey populations with time delay.
  • Delayed responses in the predator-prey model may result in different dynamics and stability behavior compared to non-delayed models. slide 29

Applications of FDEs in Various Fields

  • Functional differential equations have widespread applications in various fields.
  • They are used to model and analyze real-world systems that exhibit time delays.
  • Examples of applications include biology, physics, chemistry, economics, and control systems.
  • FDEs enable us to understand delayed responses, dynamic behavior, and stability properties in these systems.
  • Modeling and solving FDEs using integral calculus provide valuable insights and predictions in various fields. slide 30

Summary

  • Solving functional differential equations using integral calculus involves expressing derivatives as integrals and solving the resulting equations.
  • The delay property in FDE solutions leads to delayed responses and unique behaviors.
  • Stability analysis plays a crucial role in understanding the long-term behavior of FDE solutions.
  • FDEs find applications in biology, physics, economics, and other fields, helping to predict and analyze systems with time delays.
  • Understanding and solving FDEs using integral calculus deepen our understanding of dynamic systems.