# What is Partial differential equations: Definition and 149 Discussions

In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to how x is thought of as an unknown number, to be solved for, in an algebraic equation like x2 − 3x + 2 = 0. However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations. Among the many open questions are the existence and smoothness of solutions to the Navier–Stokes equations, named as one of the Millennium Prize Problems in 2000.
Partial differential equations are ubiquitous in mathematically-oriented scientific fields, such as physics and engineering. For instance, they are foundational in the modern scientific understanding of sound, heat, diffusion, electrostatics, electrodynamics, fluid dynamics, elasticity, general relativity, and quantum mechanics. They also arise from many purely mathematical considerations, such as differential geometry and the calculus of variations; among other notable applications, they are the fundamental tool in the proof of the Poincaré conjecture from geometric topology.
Partly due to this variety of sources, there is a wide spectrum of different types of partial differential equations, and methods have been developed for dealing with many of the individual equations which arise. As such, it is usually acknowledged that there is no "general theory" of partial differential equations, with specialist knowledge being somewhat divided between several essentially distinct subfields.Ordinary differential equations form a subclass of partial differential equations, corresponding to functions of a single variable. Stochastic partial differential equations and nonlocal equations are, as of 2020, particularly widely studied extensions of the "PDE" notion. More classical topics, on which there is still much active research, include elliptic and parabolic partial differential equations, fluid mechanics, Boltzmann equations, and dispersive partial differential equations.

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1. ### I Maxwell's equations PDE interdependence and solutions

Hi, as in this thread Are maxwells equations linearly dependent I would like to better understand from a mathematical point of view the interdependence of Maxwell's equations. Maxwell's equations are solved assuming as given/fixed the charge density ##\rho## and the current density ##J## as...

22. ### I General solution of heat equation?

We know $$K(x,t) = \frac{1}{\sqrt{4\pi t}}\exp(-\frac{x^2}{4t})$$ is a solution to the heat equation: $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}$$ I would like to ask how to prove: $$u(x,t) = \int_{-\infty}^{\infty} K(x-y,t)f(y)dy$$ is also the solution to...
23. ### I How to solve this second order differential equation

Any idea how to solve this equation: ## \ddot \sigma - p e^\sigma - q e^{2\sigma} =0 ## Or ## \frac{d^2 \sigma}{dt^2} - p e^\sigma - q e^{2\sigma} =0 ## Where p and q are constants.Thanks.
24. ### Heat Diffusion Equation - Using BCs to model as an orthonormal system

I've tried to show b) by using the sine Fourier series on ##[0,2a]##, to get ##g_k = \Sigma_{n=0}^{2a} \sqrt\frac{2}{a} Sin(q_k x)## Therefore ##\sqrt\frac{2}{a} = \frac{1}{a} \int_0^{2a} Sin(q_kx)g_k dx## These are equal therefore it is an orthonomal basis. I'm not sure if this is correct so...
25. ### Heat Equation with Periodic Boundary Conditions

I'm solving the heat equation on a ring of radius ##R##. The ring is parameterised by ##s##, the arc-length from the 3 o'clock position. Using separation of variables I have found the general solution to be: $$U(s,t) = S(s)T(t) = (A\cos(\lambda s)+B\sin(\lambda s))*\exp(-\lambda^2 kt)$$...
26. ### Partial Derivative of a formula based on the height of a cylinder

The function should use (r,z,t) variables The domain is (0,H) Since U is not dependent on angle, then theta can be ignored in the expression for Laplacian in cylindrical coordinates(?)
27. ### Partial Differential Equations result -- How to simplify trig series?

Solve the boundary value problem Given u_{t}=u_{xx} u(0, t) = u(\pi ,t)=0 u(x, 0) = f(x) f(x)=\left\{\begin{matrix} x; 0 < x < \frac{\pi}{2}\\ \pi-x; \frac{\pi}{2} < x < \pi \end{matrix}\right. L is π - 0=π λ = α2 since 0 and -α lead to trivial solutions Let u = XT X{T}'={X}''T...
28. ### A How to get a converging solution for a second order PDE?

I have been struggling with a problem for a long time. I need to solve the second order partial differential equation $$\frac{1}{G_{zx}}\frac{\partial ^2\phi (x,y)}{\partial^2 y}+\frac{1}{G_{zy}}\frac{\partial ^2\phi (x,y)}{\partial^2 x}=-2 \theta$$ where ##G_{zy}##, ##G_{zx}##, ##\theta##...
29. ### How to apply the Fourier transform to this problem?

I am struggling to figure out how to approach this problem. I've only solved a homogenous heat equation $$u_t = u_{xx}$$ using a Fourier transform, where I can take the Fourier transform of both sides then solve the general solution in Fourier terms then inverse transform. However, since this...
30. ### I Separation of Variables for Partial Differential Equations

When using the separation of variable for partial differential equations, we assume the solution takes the form u(x,t) = v(x)*g(t). What is the justification for this?
31. ### A Determine PDE Boundary Condition via Analytical solution

I am trying to determine an outer boundary condition for the following PDE at ##r=r_m##: $$\frac{\sigma_I}{r} \frac{\partial}{\partial r} \left(r \frac{\partial z(r,t)}{\partial r} \right)=\rho_D gz(r,t)-p(r,t)-4 \mu_T \frac{\partial^2z(r,t)}{\partial r^2} \frac{\partial z(r,t)}{\partial t}$$...
32. ### A Laplace or Fourier Transform to solve a system of partial differential equations in thermoelasticity

I've a system of partial diff. eqs. in thermo-elasticity, I can solve it using normal mode analysis method but I need to solve it using laplace or Fourier

39. ### Solving Partial Differential Equation

1. The problem statement, all variables, and given/known data Task requires you to solve a partial differential equation $$u_{xy}=2yu_x$$ for ##u(x,y)##. A hint is given that a partial differential equation can be solved in terms of ordinary differential equations. According to the solution...
40. ### A How to simplify the solution of the following linear homogeneous ODE?

During solution of a PDE I came across following ODE ## \frac{d \bar h}{dt} + \frac{K}{S_s} \alpha^2 \bar h = -\frac{K}{S_s} \alpha H h_b(t) ## I have to solve this ODE which I have done using integrating factor using following steps taking integrating factor I=\exp^{\int \frac{1}{D} \alpha^2...
41. B

43. ### Extending function to determine Fourier series

In the following question I need to find the Fourier cosine series of the triangular wave formed by extending the function f(x) as a periodic function of period 2 $$f(x) = \begin{cases} 1+x,& -1\leq x \leq 0\\ 1-x, & 0\leq x \leq 1\\\end{cases}$$ I just have a few questions then I will be able...
44. S

### A How to find the partial derivatives of a composite function

Hello, dear colleague. Now I'm dealing with issues of modeling processes of heat and mass transfer in frozen and thawed soils. I am solving this problems numerically using the finite volume method (do not confuse this method with the finite element method). I found your article: "Numerical...
45. ### I Substitution in partial differential equation

Hello everybody. Consider $$\frac{\partial}{\partial t}f(x) + ax\frac{\partial }{\partial x}f(x) = b x^2\frac{\partial^2}{\partial x^2}f(x)$$ This is the equation (19) of...
46. ### A A system of partial differential equations with complex vari

Hi, I need to solve a system of first order partial differential equations with complex variables given by What software should I use for solving this problem..? The system includes 13 differential equations ...
47. ### I What are partial differential equations?

If the slope of the curve (derivative) at a given point is a number .
48. ### A Converting Partial Differential Equations to Frequency Domain

Hello All, I would like to convert a partial diff equation in time domain into frequency domain, however there is a term of the form: Re(∇(E1.E2*) exp(j[ω][/0]t)) where E1 and E2 are the magnitudes of the electric field and [ω][/0] is the angular frequency. Can someone please help me to...
49. ### I Help Solving an Equation with a Boundary Condition

Hello everybody. I'm about to take a final exam and I've just encountered with this exercise. I know it's simple, but i tried solving it by Separation of variables, but i couldn't reach the result Mathematica gave me. This is the equation: ∂u/∂x = ∂u/∂t Plus i have a condition...
50. ### What is the inverse Fourier transform of e^3iωt for solving ut+3ux=0?

Homework Statement Solve ut+3ux=0, where -infinity < x < infinity, t>0, and u(x,0)=f(x).Homework Equations Fourier Transform where (U=fourier transform of u) Convolution Theorem The Attempt at a Solution I've used Fourier transform to get that Ut-3iwU=0 and that U=F(w)e3iwt. However, I'm...