Integro-differential equation

Differential equations
Scope
Fields Natural sciences Engineering Astronomy Physics Chemistry Biology Geology Applied mathematics Continuum mechanics Chaos theory Dynamical systems Social sciences Economics Population dynamics List of named differential equations
Classification
Types Ordinary Partial Differential-algebraic Integro-differential Fractional Linear Non-linear By variable type Dependent and independent variables Autonomous Coupled / Decoupled Exact Nonhomogeneous Features Order Operator Notation
Relation to processes Difference (discrete analogue) Stochastic Stochastic partial Delay
Solution
Existence and uniqueness Picard–Lindelöf theorem Peano existence theorem Carathéodory's existence theorem Cauchy–Kowalevski theorem
General topics Initial conditions Boundary values Dirichlet Neumann Robin Cauchy problem Wronskian Phase portrait Asymptotic / Exponential stability Rate of convergence Series / Integral solutions Numerical integration Dirac delta function
Solution methods Inspection Method of characteristics Euler Exponential response formula Finite difference (Crank–Nicolson) Finite element Infinite element Finite volume Galerkin Petrov–Galerkin Green's function Integrating factor Integral transforms Perturbation theory Runge–Kutta Separation of variables Undetermined coefficients Variation of parameters
People
List Isaac Newton Gottfried Leibniz Jacob Bernoulli Leonhard Euler Joseph-Louis Lagrange Józef Maria Hoene-Wroński Joseph Fourier Augustin-Louis Cauchy George Green Carl David Tolmé Runge Martin Kutta Rudolf Lipschitz Ernst Lindelöf Émile Picard Phyllis Nicolson John Crank
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In mathematics, an integro-differential equation is an equation that involves both integrals and derivatives of a function.

The general first-order, linear (only with respect to the term involving derivative) integro-differential equation is of the form

${\displaystyle {\frac {d}{dx}}u(x)+\int _{x_{0}}^{x}f(t,u(t))\,dt=g(x,u(x)),\qquad u(x_{0})=u_{0},\qquad x_{0}\geq 0.}$

As is typical with

, obtaining a closed-form solution can often be difficult. In the relatively few cases where a solution can be found, it is often by some kind of integral transform, where the problem is first transformed into an algebraic setting. In such situations, the solution of the problem may be derived by applying the inverse transform to the solution of this algebraic equation.

Consider the following second-order problem,

${\displaystyle u'(x)+2u(x)+5\int _{0}^{x}u(t)\,dt=\theta (x)\qquad {\text{with}}\qquad u(0)=0,}$

where

${\displaystyle \theta (x)=\left\{{\begin{array}{ll}1,\qquad x\geq 0\\0,\qquad x<0\end{array}}\right.}$

is the Heaviside step function. The Laplace transform is defined by,

${\displaystyle U(s)={\mathcal {L}}\left\{u(x)\right\}=\int _{0}^{\infty }e^{-sx}u(x)\,dx.}$

Upon taking term-by-term Laplace transforms, and utilising the rules for derivatives and integrals, the integro-differential equation is converted into the following algebraic equation,

${\displaystyle sU(s)-u(0)+2U(s)+{\frac {5}{s}}U(s)={\frac {1}{s}}.}$

Thus,

${\displaystyle U(s)={\frac {1}{s^{2}+2s+5}}}$.

Inverting the Laplace transform using

then gives

${\displaystyle u(x)={\frac {1}{2}}e^{-x}\sin(2x)\theta (x)}$.

Alternatively, one can

("exponentially decaying sine wave") or recall from memory to proceed:

${\displaystyle U(s)={\frac {1}{s^{2}+2s+5}}={\frac {1}{2}}{\frac {2}{(s+1)^{2}+4}}\Rightarrow u(x)={\mathcal {L}}^{-1}\left\{U(s)\right\}={\frac {1}{2}}e^{-x}\sin(2x)\theta (x)}$.

Applications

Integro-differential equations model many situations from science and engineering, such as in circuit analysis. By Kirchhoff's second law, the net voltage drop across a closed loop equals the voltage impressed ${\displaystyle E(t)}$. (It is essentially an application of energy conservation.) An RLC circuit therefore obeys $${\displaystyle L{\frac {d}{dt}}I(t)+RI(t)+{\frac {1}{C}}\int _{0}^{t}I(\tau )d\tau =E(t),}$$ where ${\displaystyle I(t)}$ is the current as a function of time, ${\displaystyle R}$ is the resistance, ${\displaystyle L}$ the inductance, and ${\displaystyle C}$ the capacitance.^[1]

The activity of interacting

.

The Whitham equation is used to model nonlinear dispersive waves in fluid dynamics.^[2]

Epidemiology

Integro-differential equations have found applications in epidemiology, the mathematical modeling of epidemics, particularly when the models contain age-structure^[3] or describe spatial epidemics.^[4] The Kermack-McKendrick theory of infectious disease transmission is one particular example where age-structure in the population is incorporated into the modeling framework.

See also

• Delay differential equation
• Differential equation
• Integral equation
• Integrodifference equation

References

1. ISBN 978-1-111-82706-9
   . Chapter 7 concerns the Laplace transform.
2. ISBN 0-471-94090-9
   .
3. ISSN 0075-8434
   .
4. ^ Medlock, Jan (March 16, 2005). "Integro-differential-Equation Models for Infectious Disease" (PDF). Yale University. Archived from the original (PDF) on 2020-03-21.

Further reading

• Vangipuram Lakshmikantham, M. Rama Mohana Rao, “Theory of Integro-Differential Equations”, CRC Press, 1995

External links

• Interactive Mathematics
• Numerical solution of the example using Chebfun