The and diffusion processes are key applications of Fourier series. These equations describe how heat or substances spread over time, using partial differential equations that can be solved with Fourier techniques.
Initial and boundary conditions are crucial for finding unique solutions to heat and diffusion problems. By applying Fourier methods like , we can solve these equations and model real-world phenomena like temperature changes or chemical diffusion.
Heat and Diffusion Equations
Fundamental Equations Describing Heat and Diffusion
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Heat equation describes the distribution of heat (or variation in temperature) in a given region over time
Derived from the principle of and of thermal conduction
Expressed as a second-order partial differential equation: ∂t∂u=α∇2u, where u is temperature, t is time, and α is
describes the behavior of the concentration of a substance in a medium over time
Derived from the principle of and Fick's law of diffusion
Expressed as a second-order partial differential equation: ∂t∂ϕ=D∇2ϕ, where ϕ is concentration, t is time, and D is the
is a physical property that measures a material's ability to conduct heat
Represents the rate at which heat is transferred through a material by conduction (metals like copper and silver have high thermal conductivity)
Appears in the heat equation as part of the thermal diffusivity term: α=ρcpk, where k is thermal conductivity, ρ is density, and cp is specific heat capacity
Initial and Boundary Conditions
Specifying Conditions for Unique Solutions
Initial conditions specify the state of the system at the beginning (t=0)
Describe the initial temperature distribution in the heat equation (e.g., u(x,0)=f(x))
Describe the initial concentration distribution in the diffusion equation (e.g., ϕ(x,0)=g(x))
Boundary conditions specify the behavior of the system at the boundaries of the domain
: specifies the value of the function at the boundary (e.g., u(0,t)=u(L,t)=0)
: specifies the value of the derivative of the function at the boundary (e.g., ∂x∂u(0,t)=∂x∂u(L,t)=0)
is the solution that does not change with time
Obtained by setting the time derivative term to zero in the heat or diffusion equation (e.g., ∇2u=0)
Represents the long-term behavior of the system (temperature distribution in a well-insulated room)
is the solution that varies with time
Obtained by solving the heat or diffusion equation with the time derivative term included (e.g., ∂t∂u=α∇2u)
Describes how the system evolves from the initial state to the steady state (cooling of a hot metal rod)
Solution Methods
Techniques for Solving Heat and Diffusion Equations
Separation of variables is a method for solving partial differential equations by assuming the solution can be written as a product of functions, each depending on only one variable
Assume the solution has the form u(x,t)=X(x)T(t)
Substitute into the heat or diffusion equation and separate the variables to obtain ordinary differential equations for X(x) and T(t)
Solve the ordinary differential equations and combine the solutions to obtain the general solution (e.g., u(x,t)=∑n=1∞Ansin(Lnπx)e−α(Lnπ)2t)
is obtained by representing the solution as an infinite series of sine and cosine functions
Assume the solution has the form u(x,t)=∑n=1∞An(t)sin(Lnπx) or u(x,t)=∑n=0∞Bn(t)cos(Lnπx)
Substitute into the heat or diffusion equation and solve for the coefficients An(t) or Bn(t)
Determine the coefficients using initial conditions and orthogonality properties of sine and cosine functions (e.g., An=L2∫0Lf(x)sin(Lnπx)dx)