How to find the general solution of a differential equation with a shift, in the following form?

$$frac{partial}{partial t}g(x,t)=g(x-Delta,t)+frac{partial^2}{partial x^2} g(x,t)$$

where $Delta > 0$. And what about the following?

$$frac{partial}{partial t}g(x,t)=g(x,t-Delta)+frac{partial^2}{partial x^2} g(x,t)$$

Edit1: Here are few follow-up details about my question. Is there a “nice" way to represent the solution in $x$-space, as opposed to e.g., Fourier? Is the solution real + positive + normalizable? Does it have the correct properties of a probability density function?

MathOverflow Asked on November 22, 2021

1 AnswersFourier transform $G(k,t)=int_{-infty}^infty e^{ikx} g(x,t)dx$ with respect to $x$, then $$frac{partial}{partial t}G(k,t)=e^{ikDelta}G(k,t)-k^2 G(k,t),$$ hence $$G(k,t)=expleft(te^{ikDelta}-tk^2right)G(k,0).$$ For the second differential equation you would similarly Fourier transform with respect to $t$.

Answered by Carlo Beenakker on November 22, 2021

0 Asked on January 27, 2021 by lo-brunswic

0 Asked on January 26, 2021 by alkan

0 Asked on January 26, 2021 by aghostinthefigures

1 Asked on January 26, 2021 by lye012

1 Asked on January 24, 2021 by zachary-vance

0 Asked on January 24, 2021 by sbastien-loisel

combinatorial optimization compressed sensing convexity nonlinear optimization regularization

0 Asked on January 24, 2021 by xin-fu

1 Asked on January 24, 2021 by sam-roberts

2 Asked on January 24, 2021 by user114331

6 Asked on January 23, 2021 by truebaran

1 Asked on January 23, 2021 by m-rahmat

at algebraic topology gn general topology path connected real analysis

1 Asked on January 21, 2021 by ofra

ct category theory gr group theory homotopy theory limits and colimits

0 Asked on January 20, 2021 by ripon

eigenvalues expectation matrix analysis matrix theory random matrices

1 Asked on January 19, 2021 by adittya-chaudhuri

cech cohomology ct category theory grothendieck topology nonabelian cohomology

0 Asked on January 17, 2021 by harry-gindi

0 Asked on January 17, 2021 by manfred-weis

1 Asked on January 16, 2021 by learning-math

geometric probability pr probability probability distributions st statistics

18 Asked on January 16, 2021

big list mathematics education open problems soft question thesis

3 Asked on January 16, 2021 by jochen-glueck

fa functional analysis linear algebra matrix analysis oa operator algebras operator theory

Get help from others!

Recent Answers

- Gavin Ball on Invariant $SU(3)$ subgroup for ${bf 8}$ in ${bf 3}^* otimes {bf 3} ={bf 1} oplus {bf 8}$
- user10354138 on Proof that a continuous function with continuous right derivatives is differentiable.
- N. Bar on Finding the volume when a parabola is rotated about the line $y = 4$.
- Kavi Rama Murthy on Characteristic function of random variable is always integrable
- greg on Computing the matrix differential/derivative of the matrix$rightarrow$scalar function $log det(BCB^T)$

Recent Questions

- Computing the matrix differential/derivative of the matrix$rightarrow$scalar function $log det(BCB^T)$
- Characteristic function of random variable is always integrable
- Invariant $SU(3)$ subgroup for ${bf 8}$ in ${bf 3}^* otimes {bf 3} ={bf 1} oplus {bf 8}$
- Proof that a continuous function with continuous right derivatives is differentiable.
- Finding the volume when a parabola is rotated about the line $y = 4$.

© 2021 InsideDarkWeb.com. All rights reserved.