Laplace Diffusion DE solve

8efa99e6-82a2-471a-a623-b65b8b4e6825 128 15 12 decimal

&[DATE] &[TIME] - &[FILENAME]

3 appVersion 0.98.6179 ≡

Solve/evaluate contour

Incremental discretizer ...
Mesh N  within the bounds L, H L H N xd X 0 : i 1 N range X i el L H L - N / i * + : for X 31 line :

a 816 π * sqrt * : 351.67021

x y Φ 1 - 3 / x 2^y 2^+ sqrt * exp 2 y 2^* - x 2^+ a / * - 0.005 + :

1. The upper limit

x y XY x y / - :

L 14 : H 14.2455293 : N 29 : 13 mat x L H N xd : 21 line

XY Φ 2 1 sys

n 1 x length range Sols n el y x n el y Φ vectorize y solve at : 11 line for

high x Sols augment :

high high rows row 14.2455293 0.00034955 12 mat

====================================

2. The lower limit

find low limit u 1.7856878 1.7857 1.78569 range :

Explain . . .

0. The solver is implicit-recursive

implicit: in the sense that it solves y given Φ(x,y)=0.

recursive: as it runs on the vectorized discrete x.

1. The low limit is searched first on a practical low range.

2. The upper limit is searched over a generously populated

range, starting from a solvable mid - limit.

3. steps [1, 2] are trial/error.

4. Pairwise solutions aren't true @ the limits.

5. The model document is unique for the given Φ(x,y

]]>

n 1 u length range Sol n el y u n el y Φ vectorize y solve at : 11 line for

low u Sol augment :

low 1 row 1.7856878 0.00094163 12 mat

reasonable zeros
the [2] lobes Xzeros 1.7856878 ± 14.2455293 ± at

Local pairwise solutions

# # at

# # # at

L < u < H ... escape end-points u 10 :

x pairwise user x y Φ y solve : x user 12 mat 21 line :

Draghilev 1.78568577382905 14.24552938284325 21 sys :

pair wise u pairwise 10 4.1749 - 4.1749 21 mat 12 mat

Solving for the green red lobes

]]>

a 816 π * sqrt * : 351.67021

x y f 1 - 3 / x 2^y 2^+ sqrt * exp 2 y 2^* - x 2^+ a / * - 0.0025 - :

About the Electron function . . .

A closed lobe that has no symbolic expression

of the ± branches. It complicates more familiar

expansion of f(x,y) but delights Smath.

]]>

x y Φ 1 - 3 / x 2^y 2^+ sqrt * exp 2 y 2^* - x 2^+ a / * - 0.005 + :

x y axes x y / - :

solving 6 values u 5 :

axes Φ f u 25 u 25 - 2 2 mat 14.2455293 5 14.2455293 5 - 2 2 mat 5 1 sys

x green user x y f y solve : x user 12 mat 21 line :

axes Φ f u 25 u 25 - 22 mat 14.24553 5 14.24553 5 - 22 mat 51 sys

pair wise u green 5 19.58772 - 4.029687 - 4.029687 19.58772 41 mat 12 mat

x red user x y Φ y solve : x user 12 mat 21 line :

pair wise u red 5 2.597823 - 2.597823 21 mat 12 mat

===============================================

solving  2 values green lobe ω 10 :

solving 4 values red lobe v 2.5 :

axes Φ f ω v 5 1 sys

y Green user x y f x solve : y user 12 mat 21 line :

pair wise ω Green 10 10.045492 - 10.045492 21 mat 12 mat

y Red user x y Φ x solve : y user 12 mat 21 line :

pair wise v Red 2.5 13.461052 - 4.821716 - 4.821716 13.461052 41 mat 12 mat

# exp

r 0.25 - :

given Y [u], solving  X circle u 0.375 :

x y q x 2^y 2^+ r + :

x circle user x y q y solve : user 21 line :

x y f x 4^2 x 2^* y * + 3 x * y 2^* - y 3^+ :

pair wise X u circle : 0.330719 - 0.330719 21 mat

xy u X 2 el augment :

f q xy 3 1 sys

x y Clear 1

Possible export pairwise

r 0.25 - : x y c x 2^y 2^+ r + : 21 sys

rejects ? 0.3 0.45 0.31 range

solving  X circle ω 0.25 0.45 0.261 range :

profitable for in-situ  pair(s) x circle collect x y c y 1 - 1 solve : collect 21 line :

pair wise pairs ω circle vectorize : 0.433013 - 0.433013 21 mat 0.426473 - 0.426473 21 mat 0.419543 - 0.419543 21 mat 31 mat

0.3 circle 0.4 - 0.4 21 mat

r 0.25 - : x y c x 2^y 2^+ r + : 21 sys

x λ 1 r x ≤ (x r - ≤ (& cases :

solving  X circle ω 0 0.5 0.01 range :

Implicit 'roots' ≠ more generous x circle collect x y c y roots : collect 21 line :

π 2^0.3 0.4 / * 7.40220330081702

pair wise single ω ω circle vectorize augment :

x circle x circle - 2 1 sys x circle x λ * x circle - ( x λ * 1.85 x * 1.85 x * - 4 1 sys

single 0 0.5 0.01 0.4999 0.02 0.4996 0.03 0.4991 0.04 0.4984 0.05 0.4975 0.06 0.4964 72 mat

Typical solvable multibranches f(x,y)

x y Clear 1

x y Φ3 x 2^y 2^+ (2^1 x 3^3 x * y 2^* - (* - : x y Φ6 x 2^y 2^+ (4^1 x 3^3 x * y 2^* - (2^* - : x C 1 x 2^- sqrt : 31 line

x f x y Φ3 0 ≡ y solve maple :

junction 0.5625 - 0.5625 - f 1 el + 15 black 15 mat :

plot x f 1 el x f 2 el x f 3 el x f 4 el 41 sys :

plot x C Re x C Re - junction 4 1 sys

Solving for the junction ... value in the Trifolium domain x f 1 el x f 3 el - abs 0 ≡ x 0.563 - 0.562 - solve 0.5625 - f 1 el 21 mat 0.562500000000000 - 0.726184377413890 21 mat

θ rot θ cos θ sin - θ sin θ cos 22 mat :

Trifolium j 0.5625 - : u j 0 j 0.005 + (range : v j 1 j 0.005 + (range : i 1 v rows range V i el v i el f 1 el eval : for i 1 u rows range U i el u i el f 3 el eval : for red v V - augment : magenta u U - augment : blue v V augment : green u U augment : blue red green magenta stack 101 line :

T Trifolium π 3 / - rot * :

Hub 00.15 red 15 mat :

Trifolium T 2 1 sys

θ 0 :

T t Eol Trifolium θ π 30 / t * + rot * eval :

x re x C Re 1 - x ≤ (x 1 ≤ (& cases :

x im x C Re - 1 - x ≤ (x 1 ≤ (& cases :

step1 060 range :

Smath "Wind Mill" project

T t Eol x re x im Hub 4 1 sys

x y Clear 1

r 0.25 - : x y c x 2^y 2^+ r + : 21 sys

Solve at(2) . . . didactic

]]>

t0 1 time :

lo 0.25 : hi 0.475 : N 19 : 13 mat

find low limit ζ lo hi lo hi lo - N / + (range :

1 variable ... 2 values # u j el y c y solve at

ζ rows 19

C1 j 1 ζ length range at j el Ω ζ j el y c y solve 1 el at : 11 line for at 21 line :

C2 j 1 ζ length range at j el Ω ζ j el y c y solve 2 el at : 11 line for at 21 line :

ζ C1 augment ζ C2 augment 2 1 sys

1 time t0 - 6 s *

More profitable "roots" at(2) . . . examples

]]>

lo 0.0001 : hi 0.4999 : N 51 : 13 mat

find low limit u lo hi lo hi lo - N / + (range :

t0 1 time :

C3 j 1 u length range rho j el ρ u j el y c y roots at : 11 line for rho 21 line :

u C3 augment u C3 - augment u - C3 augment u - C3 - augment 4 1 sys

1 time t0 - 5.1 s *

C3 1 row C3 C3 rows row 21 mat 0.5 11 mat 0.01 11 mat 21 mat

x y Clear 1

x y Φ3 x 2^y 2^+ (2^1 x 3^3 x * y 2^* - (* - : x C 1 x 2^- sqrt : 21 sys

junction 0.5625 - 0.5625 - f 1 el + 15 black 15 mat :

x f x y Φ3 0 ≡ y solve maple :

plot x f 1 el x f 2 el x f 3 el x f 4 el 41 sys :

plot x C Re x C Re - junction 4 1 sys

Maple  symbolic x q1 x 32 x * + 916 x * + sqrt - (* - sqrt 2 sqrt / : x q2 x 32 x * + 916 x * + sqrt - (* - sqrt 2 sqrt / - : 21 sys

lo 0.56245 - : hi 0.99 : N 99 : 13 mat

Hi 0.001 - :

find low limit u lo hi lo hi lo - N / + (range :

find low limit v lo Hi lo Hi lo - 51 / + (range :

t0 1 time :

T j 1 u length range rho j el dum u j el y Φ3 y roots at : 11 line for rho 21 line :

u T augment u T - augment v v q1 vectorize augment v v q2 vectorize augment 4 1 sys

1 time t0 - 27 s *

u T augment u T - augment v v q1 vectorize augment v v q2 vectorize augment 41 sys

Trifolium u T augment u T - augment v v q1 vectorize augment v v q2 vectorize augment stack :

Trifolium

Discrete differential "roots" at(3) . . . example

]]>

r 0.125 - :

x y Clear 1

x y q x 2^y 2^+ r + :

x y f 1 x 4^(* 2 x 2^* y * + 3 x * y 2^* - 1 y 3^(* + :

lo 0.35 : hi 0.36 : N 1000 : 13 mat

f q 0.35355 0.04526 - 1 2 mat 3 1 sys

find low limit u lo hi lo hi lo - N / + (range :

t0 1 time :

last value "on error" ... NO real root. data j 1 u length range rho j el at3 u j el y f y roots u j el y q y roots at : break try for rho 21 line :

data ListCollect collect 'select'.. select 345356 range : ct 1 : i select a ct el data i 1 el : b ct el data i 2 el : ct ct 1 + : 31 line for a b augment 51 line :

"real root" exists in the positive region left of 0.35355. On the other side, the differential region is negative ... thus => "NO real root".

data rows 356

k 1 data rows range : junction u k el data augment : junction junction : 31 sys

u 0.35355 : Yjunction u y q u y f ≡ y roots : 21 sys

1 time t0 - min 1.5

junction ListCollect 0.35344 0.059879 0.35345 0.059477 0.35346 0.059056 0.35347 0.058612 0.35348 0.058139 0.35349 0.057633 0.3535 0.057085 0.35351 0.056483 0.35352 0.055805 0.35353 0.055016 0.35354 0.054029 0.35355 0.052503 122 mat

Yjunction 0.04526 -

junction 0.35355 0.5 0.35355 0.05 - 2 2 mat 0.35 0.0525 0.36 0.0525 2 2 mat 3 1 sys

real root

NO real root at(3)

Export Laplace Diffusion . . . disabled

]]> Laplace Diffusion DE solve Stem(data)

vx vy a b c plotG a char ≡ b size ≡ c color ≡ 13 mat n vx length : plot vx 1 el vy 1 el a b c augment : i 2 n range plot plot vx i el vy i el a b c augment stack : for plot 51 line :

Spectrum stem plot for QuickPlot unicolor data Stem stemX 1 data rows range : i 1 stemX rows range stem i el data i 1 el data i 2 el data i 1 el 022 mat : for i 1 stem rows 1 - range stem i el stem i el stem i 1 + (1 el 1 el 012 mat stack : for V stem 1 el : j 2 stem rows 1 - range V V stem j el stack : for V V data data rows row stack : 71 line :

Laplace linear diffusion x t f t diff x t f x 2 diff ≡

F α 0.25 : β 20 : 13 mat f => spatial initial conditions i 141 range f 1 i el 0 : f 1 β el 1 : 21 line for recursive finite differences 'x' sol's t 150 range x 240 range f t 1 + x el f t x el (α f t x 1 - el 2 f t x el * - f t x 1 + el + (* + : for for f 61 line :

CWD C:\SmathFile CurrentDirectory :

F CWD Diffusion wfile 1

D CWD Diffusion rfile :

t Diffusion j 141 range : j D t row transpose augment 21 line :

τ 140 range :

1 Diffusion Stem 2 Diffusion Stem 10 Diffusion Stem 3 1 sys t 1 ≡ t diffusion t Diffusion Stem : t diffusion if 1 Diffusion Stem 2 1 sys

M Low High Scale High Low - (M M Min - M Max M Min - 106 -^+ / (* Low + :

D D 01024 Scale :

iVBORw0KGgoAAAANSUhEUgAAACkAAAAzCAIAAACFT7ZcAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAAW2SURBVFhHvde5T1ZdEAZwUBTEDURRxJXN4IYaBXcbXAFBxS3uG7hFjZEQXKIJiSaaQEGjhVTQECtjZywtLG1s7Wz5Ayzwl5w3Nyef5jO8XDzFZHjmuedhcmbmnDcn6zU2Npbx/vEiHFbm73+29uzZQ/Xp06csP4P+gzV37twLFy6Mjo5SZfmQTGxS19SpU5cvX3737t3v378vXLiQ5UPgGcbkrRkzZqxdu7a7u/vr168/f/5k+RB4hjFJa8qUKUVFRfX19Y8ePfr8+fOPHz9YPgQumuFNxpo2bVppaenu3bufPHny8ePHb9++sXwIXDTDS33l5uYWFBQsWbKksbGR3vv37798+cLyIXBRnAw73aWaZs6cuXLlygMHDjx+/HhkZOTTp08sHwIXnZSKk1BeXp5eqq6ubmpqcsZDQ0MfPnxg+RC4KE76qdtx+vTp8+bNq62tPXz4cE9Pz9u3b9+9e8fyIXBRnPS11bDjXLBggXZqbW3VV69fvx4eHmb5ELgoTsrVLhUHWVhYaJjU1dUdOXKkq6trYGBgcHCQ5UPgojiYaaZuLweplBYvXrxp0yZKDx486Ovre/PmDcuHwEVxUj5ye+nd2bNna6TNmzcfPXr0/v37L1++lDTLh8BFcTDT1HaEikgZL1u2bMuWLZTu3bv3/Pnz/v5+lg+Bi+JgpnbkknCE+fn5xcXFK1asaGhooHTnzp3e3t5Xr16xfAhcFAcztSMP2gpYC1VUVGzdupXS7du3nz179uLFC5YPgYviYKaprXwU8Pz586uqqrZt20bp1q1bRqmkWT4ELoqDmVq5OTzlo4C1r+G1fft2Sjdu3Hj48KGkWT4ELoqDiZ/OkQftWbNmualqamp27NhB6fr160aKpFk+BC6Kg5mOdjhspat5Fi1atGrVqqDd0dFhpEia5QdtURxM/BSOPGgr3Tlz5pSVlRnaO3fuNEmuXbumrSXN8iFwURzMdEo9aCtdjWtsJdpXrlzR1pJm+Yk2DmY6pe7YFK29PInKy8tXr169a9eutra2y5cva2tJs3wIXBQHE99XEzpy/3jQ9g40NIzMNWvWBO2LFy9qa0mz/KAtioOJH7SzTz1oK1ota2gsXbrURUnDjelNfvPmTUmzfAhcFAcTP5T6hLQdm6LVsiUlJcb1unXrgvb58+e1lqRZftAWxcHEn2ip/0fbDwC7e5J6pZw7d66zs1PSLB8CF8VJU1vDGBeGpati/fr1Qfvs2bNay0Rj+UFbFAcTf6JtFgrNLsaFYZlot7S0nDlz5urVq5Jm+ZBEGxPfV9mXun85aGsYexmWXsGJ9unTp7WWpFl+oo2DiZ+0WTapx9pGlR1dkR5lNJqbm0+dOnXp0iVJs3wIXBQHEz8Fba2iWY0q78BY++TJk1pL0iw/1sbE91X2beYbxZJouyQqKys3bNigl/wSOHHihNaSNMuHwEVxMBPtLMst0TYojMlY+9ChQ8ePH9dakmb5kFgbP4yXCWlr06DtgvIsibW1lqTZWBsHM2hn2eI+cFRB26AwomPtgwcPtre3ay1Js3xIrI2fjJdxH3nQVqja1C5GtMsx1j527JjWkjTLj7Ux8X2VtHj22oZU0PYc27hxIw2/eD1UtJakWT4ELooTtMNoS0HbiHYxx9oeC1pL0iw/1sbET0E7DLVY2/tk//79LmytJWmWD4HH2tmPNmxlokmCtuvBo8AzNNZW3pJmY20cTPygnU2bJdpGxO/a+/btc2Erb0mzfMjv2tmMF9RY23C2lwdJrO3SpCpplh9rY+L7Kta2Mrv//8JzSL7RoEHbtWhHz++gvXfvXheX1iLM8iFBGwcTP2iPu8VjbePJcI61/QYI2lqLNhu04bG2r5LRNm5tJZpouxY9xGJtF5fWkjTLj7Ux8RPt8ZX6X7UbGxtdXMpb0iwfkrK24WA0uhj+qK28Jc3+UdtXfx+rOTm/AH98R8ks+P2ZAAAAAElFTkSuQmCC D

Discrete differential "roots" at(3) . . . productive

]]>

x y Clear 1

x q x : x f x 3^: 21 sys

lo 1.125 - : hi 1.125 : N 100 : 13 mat u lo hi lo hi lo - N / + (range : 21 sys

data rho 0 : j 1 u length range rho j el at3 u j el f x - 0 ≡ x roots u j el q x - 0 ≡ x roots at : for u rho augment 31 line :

x q x f data 3 1 sys plotG, discrete poly

vx vy char size color plotG n vx length : plot vx 1 el vy 1 el char size color augment : i 2 n range plot plot vx i el vy i el char size color augment stack : for plot 41 line :

Low High step x fnct inc U Low High step range : i 1 U rows range f i el U i el fnct eval : for U f augment 31 line :

X data 1 col : Y data 2 col : 21 line

x φ x x 3^12 mat transpose :

Cholesky companion solver j 1 x φ length range i 1 X length range M i j el X i el φ j el : for for Φ M transpose M * : d M transpose Y * : b Φ 1 -^eval d * : 41 line

b 11 - 21 mat

x fit b x φ * :

Poly 1.25 - 1.25 1.245 - x fit inc :

data Poly 2 1 sys

Plot-solve "roots" canvas . . . productive

]]>

D1 50 :

D2 60 mm * 100 mm * 1 m * 13 mat ≡

D1 50 mm * ≡

T2 773 K * ≡

A1 π D1 * m * ≡

q 20 :

°K T2 773 :

q 20 kW m 3^/ * ≡

Stefan Boltzmann σ 5.670400 108 -^* W m 2^K 4^* / * ≡

Stefan Boltzmann constant: σ 5.670400 108 -^* :

A1 π D1 * : 157.08

Unknown μ 1 :

Setup to plot/solve on the canvas. x D2 sol σ °K 4^T2 4^- (* 1 x - x A1 * / 1 μ A1 * / + 1 x - x π * D2 * / (+ / q π D1 2^* 4 / (* - 0 ≡ °K 1 roots :

λ 0.01 x ≤ (x 0.375 ≤ (* :

user 0.0625 :

solutions [°K] over each D2
given => user canvas 'x' User user 60 sol user 100 sol user 1000 sol 31 mat : 83482481031 mat

t0 1 time :

Thermal Physics ... rod in cylinder. Abstract fron Originator

x 60 sol λ / x 100 sol λ / x 1000 sol λ / user User 1 el . 12 black 1 5 mat user User 2 el . 12 black 1 5 mat user User 3 el . 12 black 1 5 mat 6 1 sys

x 60 sol λ / x 100 sol λ / x 1000 sol λ / user User 1 el . 12 black 15 mat user User 2 el . 12 black 15 mat user User 3 el . 12 black 15 mat 61 sys

1 time t0 - min 2.4

A long uniform rod of 50-mm diameter with a thermal conductivity of 15 W/m-K is heated internally by volumetric energy generation of 20 kW/m3. The rod is positioned coaxially within a larger circular tube of 60-mm diameter whose surface is maintained at 500 degC. The annular region between the rod and the tube is evacuated, and their surfaces are diffuse and gray with an emissivity of 0.2. a) Determine the center and surface temperatures of the rod. b) Determine the center and surface temperatures of the rod if atmospheric air occupies the annular space. c) For tube diameters of 60, 100, and 1000 mm and for both the evacuated and atmospheric conditions, compute and plot the center and surface temperatures as a function of equivalent surface emissivities in the range from 0.1 to 1.0.

1. Export unitless K D2 Sol σ K 4^T2 4^- (* 1 x - x A1 * / 1 μ A1 * / + 1 x - x π * D2 * / (+ / q π D1 2^* 4 / (* - 0 ≡ K solve maple :

2. Assign global x D2 Analytical K D2 Sol :

3. Evaluate symbolic x D2 Analytical 1 el

D2 x q π * D1 2^* 4 σ * A1 * μ * T2 4^* + (* q D1 2^* μ * π * + q D1 2^* μ * π * x * - (* - q D1 2^* μ * A1 * 1 - x + (* + (σ 3^* x 3^* A1 3^* μ 3^* D2 3^* - 4 nthroot σ x * A1 * μ * D2 * 2 sqrt * /

x D2 Rod D2 x q π * D1 2^* 4 σ * A1 * μ * T2 4^* + (* q D1 2^* μ * π * + q D1 2^* μ * π * x * - (* - q D1 2^* μ * A1 * 1 - x + (* + (σ 3^* x 3^* A1 3^* μ 3^* D2 3^* - 4 nthroot σ x * A1 * μ * D2 * 2 sqrt * / :

Thermal Physics ... rod in cylinder. Abstract fron Originator

x 60 Rod λ / x 100 Rod λ / x 1000 Rod λ / 0.375 773 + 500 black 1 5 mat 4 1 sys

160 Rod 1100 Rod 11000 Rod 31 mat 77577577531 mat

Reconciliate the project . . . solution/units/export

. . . Export this section . . .

]]> Disabled ... export this section

D1 50 mm * :

D2 60 mm * 100 mm * 1 m * 13 mat :

<= Given:

q 20 kW m 3^/ * :

°K T2 773 K * :

A1 π D1 * : 0.157 m *

Unknown μ 1 :

Stefan Boltzmann constant: σ 5.670400 108 -^* W m 2^K 4^* / * :

L H N °K x 0 : dx H L - N 1 - / : 12 mat i 1 N range x i el L dx i * dx - (+ : for j 13 range i 1 x rows range T i j el q π D1 2^* 4 / (* 1 x i el - x i el A1 * / 1 μ A1 * / + 1 x i el - x i el π * D2 j el * / (+ (* σ / T2 4^+ 4 nthroot eval : for 11 line for x T K / augment 41 line :

Solution => Carlos Adapted => Jean

User: range/mesh T 0.01 1100 °K :

T #

Round OFF a matrix for convenient export M n Mround i 1 M rows range j 2 M cols range m i j 1 - el M i j el n round : for for M 1 col m augment 21 line :

collect T 1 col T 2 col T 3 col T 4 col augment : Export collect 0 Mround : 21 line

Export #