Differential Eqns
@diff_eq
Tweets on ordinary and partial differential equations from @JohnDCook
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One reason fractional differential equations can be useful is that fractional derivatives contain non-local information.
Differential equations in the complex plane johndcook.com/blog/2023/07/0…
If f(x + iy) = u(x,y) + i v(x, y) is analytic, then u and v are harmonic, i.e. u_xx + u_yy = 0 and v_xx + v_yy = 0.
If f(x + iy) = u(x,y) + i v(x, y) is analytic, then u and v satisfy the Cauchy-Riemann equations: u_x = v_y, v_x = -u_y.
Estimating local error with Runge-Kutta-Fehlberg, a.k.a. RKF45 johndcook.com/blog/2020/02/1…
Robin boundary conditions specify a linear combination of solution and normal derivative values on the boundary.
'The fact is that PDEs, in particular those that are nonlinear, are too subtle to fit into a too general scheme. On the contrary, each important PDE seems to be a world in itself.' -- Sergiu Klainerman
Stabilizing the Spherical Pendulum from Scratch community.wolfram.com/groups/-/m/t/3…
Third-order ODEs are uncommon, and usually nonlinear, such as the Blasius equation. kitchingroup.cheme.cmu.edu/blog/2013/03/1…
Bernoulli's equation: y' + p(x) y = q(x) y^n. Can be turned into a linear equation by the substitution u = y^(1-n).
Dirichlet boundary conditions specify solution values on boundary. Neumann boundary conditions specify normal derivatives of solution.
There are important non-linear PDEs with closed-form solutions johndcook.com/blog/2024/04/2…
“Good theory is (almost) as useful as exact formulas.” — Lawrence Evans re PDEs
Conservative vector fields and exact ODEs johndcook.com/blog/2022/12/0…
A solution corresponding to an eigenvalue with positive real part grows exponentially.
If a linear ODE has complex eigenvalues, solutions oscillate. If real parts are negative, oscillations decay exponentially.
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