partial differential equations

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1: 32.13 Reductions of Partial Differential Equations

§32.13 Reductions of Partial Differential Equations

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2: Peter A. Clarkson

… ► Kruskal, he developed the “direct method” for determining symmetry solutions of partial differential equations in New similarity reductions of the Boussinesq equation (with M. …

3: Ronald F. Boisvert

… ►His research interests include numerical solution of partial differential equations, mathematical software, and information services that support computational science. …

4: Howard S. Cohl

… ►His research interests include fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre and Jacobi functions, generalized and basic hypergeometric functions, eigenfunction expansions in separable coordinate systems, generating functions, $q$-series, and orthogonal polynomials in the Askey and $q$-Askey schemes. …

5: Bonita V. Saunders

… ►Her research interests include numerical grid generation, numerical solution of partial differential equations, and visualization of special functions. …

6: 23.21 Physical Applications

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§23.21(ii) Nonlinear Evolution Equations

►Airault et al. (1977) applies the function $\mathrm{\wp }$ to an integrable classical many-body problem, and relates the solutions to nonlinear partial differential equations. … ►

23.21.2 $$(\eta -\zeta )(\zeta -\xi )(\xi -\eta ){\nabla }^2=(\zeta -\eta )f(\xi )f^{\prime }(\xi )\frac{\partial }{\partial \xi }+(\xi -\zeta )f(\eta )f^{\prime }(\eta )\frac{\partial }{\partial \eta }+(\eta -\xi )f(\zeta )f^{\prime }(\zeta )\frac{\partial }{\partial \zeta },$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\xi$: ellipsoidal coordinate, $\eta$: ellipsoidal coordinate, $\zeta$: ellipsoidal coordinate and $f(\rho )$: function

Permalink:

http://dlmf.nist.gov/23.21.E2

Encodings:

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See also:

Annotations for §23.21(iii), §23.21 and Ch.23

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23.21.5 $$\left(\mathrm{\wp }\left(v\right)-\mathrm{\wp }\left(w\right)\right)\left(\mathrm{\wp }\left(w\right)-\mathrm{\wp }\left(u\right)\right)\left(\mathrm{\wp }\left(u\right)-\mathrm{\wp }\left(v\right)\right){\nabla }^2=\left(\mathrm{\wp }\left(w\right)-\mathrm{\wp }\left(v\right)\right)\frac{{\partial }^2}{{\partial u}^2}+\left(\mathrm{\wp }\left(u\right)-\mathrm{\wp }\left(w\right)\right)\frac{{\partial }^2}{{\partial v}^2}+\left(\mathrm{\wp }\left(v\right)-\mathrm{\wp }\left(u\right)\right)\frac{{\partial }^2}{{\partial w}^2}.$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $𝕃$: lattice, $u$: change of variable, $v$: change of variable and $w$: change of variable

Permalink:

http://dlmf.nist.gov/23.21.E5

Encodings:

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See also:

Annotations for §23.21(iii), §23.21 and Ch.23

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7: 36.15 Methods of Computation

… ►For numerical solution of partial differential equations satisfied by the canonical integrals see Connor et al. (1983).

8: 16.14 Partial Differential Equations

§16.14 Partial Differential Equations

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§16.14(i) Appell Functions

… ►In addition to the four Appell functions there are $24$ other sums of double series that cannot be expressed as a product of two ${}_2{}^{}F_1^{}$ functions, and which satisfy pairs of linear partial differential equations of the second order. …

9: 35.7 Gaussian Hypergeometric Function of Matrix Argument

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§35.7(iii) Partial Differential Equations

… ►Subject to the conditions (a)–(c), the function $f(𝐓)={}_2{}^{}F_1^{}(a,b;c;𝐓)$ is the unique solution of each partial differential equation ►

35.7.9 $$t_j(1-t_j)\frac{{\partial }^{2}F}{{\partial t_j}^2}-\frac{1}{2}\sum _{\begin{array}{c}k=1\\ k\ne j\end{array}}^m\frac{t_k(1-t_k)}{t_j-t_k}\frac{\partial F}{\partial t_k}+\left(c-\frac{1}{2}(m-1)-\left(a+b-\frac{1}{2}(m-3)\right)t_j+\frac{1}{2}\sum _{\begin{array}{c}k=1\\ k\ne j\end{array}}^m\frac{t_j(1-t_j)}{t_j-t_k}\right)\frac{\partial F}{\partial t_j}=abF,$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $a$: complex variable, $t_j$: eigenvalues of $𝐓$ , $b$: complex variable, $c$: complex variable, $j$: nonnegative integer, $k$: nonnegative integer and $m$: positive integer

Referenced by:

§35.7(iii)

Permalink:

http://dlmf.nist.gov/35.7.E9

Encodings:

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See also:

Annotations for §35.7(iii), §35.7 and Ch.35

… ►Systems of partial differential equations for the ${}_0{}^{}F_1^{}$ (defined in §35.8) and ${}_1{}^{}F_1^{}$ functions of matrix argument can be obtained by applying (35.8.9) and (35.8.10) to (35.7.9). …

10: 9.16 Physical Applications

… ►These first appeared in connection with the equation governing the evolution of long shallow water waves of permanent form, generally called solitons, and are predicted by the Korteweg–de Vries (KdV) equation (a third-order nonlinear partial differential equation). …