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267 | 267 | # generate fractional Laplacian
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268 | 268 | Δfrac = AbsLaplacianPower(axes(WZ,1),β)
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269 | 269 | Δ_Zfrac = Z \ (Δfrac * WZ)
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270 |
| - # define function whose fractional Laplacian is a known constant |
| 270 | + # define function whose fractional Laplacian is known |
271 | 271 | u = @. (1 - x^2 - y^2).^β
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272 | 272 | # explicit and computed solutions
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273 | 273 | fexplicit0(d,α) = 2^α*gamma(α/2+1)*gamma((d+α)/2)/gamma(d/2) # note that here, α = 2*β
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284 | 284 | # generate fractional Laplacian
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285 | 285 | Δfrac = AbsLaplacianPower(axes(WZ,1),β)
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286 | 286 | Δ_Zfrac = Z \ (Δfrac * WZ)
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287 |
| - # define function whose fractional Laplacian is a known constant |
| 287 | + # define function whose fractional Laplacian is known |
288 | 288 | u = @. (1 - x^2 - y^2).^β
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289 | 289 | # computed solution
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290 | 290 | f = Z*(Δ_Zfrac*(WZ \ u))
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300 | 300 | # generate fractional Laplacian
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301 | 301 | Δfrac = AbsLaplacianPower(axes(WZ,1),β)
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302 | 302 | Δ_Zfrac = Z \ (Δfrac * WZ)
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303 |
| - # define function whose fractional Laplacian is a known constant |
| 303 | + # define function whose fractional Laplacian is known |
304 | 304 | u = @. (1 - x^2 - y^2).^β
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305 | 305 | # computed solution
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306 | 306 | f = Z*(Δ_Zfrac*(WZ \ u))
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316 | 316 | # generate fractional Laplacian
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317 | 317 | Δfrac = AbsLaplacianPower(axes(WZ,1),β)
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318 | 318 | Δ_Zfrac = Z \ (Δfrac * WZ)
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319 |
| - # define function whose fractional Laplacian is a known constant |
| 319 | + # define function whose fractional Laplacian is known |
320 | 320 | u = @. (1 - x^2 - y^2).^(β+1)
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321 | 321 | # explicit and computed solutions
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322 | 322 | fexplicit1(d,α,x) = 2^α*gamma(α/2+2)*gamma((d+α)/2)/gamma(d/2)*(1-(1+α/d)*norm(x)^2) # α = 2*β
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335 | 335 | # generate fractional Laplacian
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336 | 336 | Δfrac = AbsLaplacianPower(axes(WZ,1),β)
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337 | 337 | Δ_Zfrac = Z \ (Δfrac * WZ)
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338 |
| - # define function whose fractional Laplacian is a known constant |
| 338 | + # define function whose fractional Laplacian is known |
339 | 339 | u = @. (1 - x^2 - y^2).^(β+1)
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340 | 340 | # explicit and computed solutions
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341 | 341 | f = Z*(Δ_Zfrac*(WZ \ u))
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354 | 354 | # generate fractional Laplacian
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355 | 355 | Δfrac = AbsLaplacianPower(axes(WZ,1),β)
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356 | 356 | Δ_Zfrac = Z \ (Δfrac * WZ)
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357 |
| - # define function whose fractional Laplacian is a known constant |
| 357 | + # define function whose fractional Laplacian is known |
358 | 358 | u = @. (1 - x^2 - y^2).^(β)*x
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359 | 359 | # explicit and computed solutions
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360 | 360 | fexplicit2(d,α,x) = 2^α*gamma(α/2+1)*gamma((d+α)/2+1)/gamma(d/2+1)*x[1] # α = 2*β
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373 | 373 | # generate fractional Laplacian
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374 | 374 | Δfrac = AbsLaplacianPower(axes(WZ,1),β)
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375 | 375 | Δ_Zfrac = Z \ (Δfrac * WZ)
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376 |
| - # define function whose fractional Laplacian is a known constant |
| 376 | + # define function whose fractional Laplacian is known |
377 | 377 | u = @. (1 - x^2 - y^2).^(β)*y
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378 | 378 | # explicit and computed solutions
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379 | 379 | fexplicit3(d,α,x) = 2^α*gamma(α/2+1)*gamma((d+α)/2+1)/gamma(d/2+1)*x[2] # α = 2*β
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393 | 393 | # generate fractional Laplacian
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394 | 394 | Δfrac = AbsLaplacianPower(axes(WZ,1),β)
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395 | 395 | Δ_Zfrac = Z \ (Δfrac * WZ)
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396 |
| - # define function whose fractional Laplacian is a known constant |
| 396 | + # define function whose fractional Laplacian is known |
397 | 397 | u = @. (1 - x^2 - y^2).^(β+1)*x
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398 | 398 | # explicit and computed solutions
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399 | 399 | fexplicit4(d,α,x) = 2^α*gamma(α/2+2)*gamma((d+α)/2+1)/gamma(d/2+1)*(1-(1+α/(d+2))*norm(x)^2)*x[1] # α = 2*β
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412 | 412 | # generate fractional Laplacian
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413 | 413 | Δfrac = AbsLaplacianPower(axes(WZ,1),β)
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414 | 414 | Δ_Zfrac = Z \ (Δfrac * WZ)
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415 |
| - # define function whose fractional Laplacian is a known constant |
| 415 | + # define function whose fractional Laplacian is known |
416 | 416 | u = @. (1 - x^2 - y^2).^(β+1)*y
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417 | 417 | # explicit and computed solutions
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418 | 418 | fexplicit5(d,α,x) = 2^α*gamma(α/2+2)*gamma((d+α)/2+1)/gamma(d/2+1)*(1-(1+α/(d+2))*norm(x)^2)*x[2] # α = 2*β
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434 | 434 | # generate fractional Laplacian
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435 | 435 | Δfrac = AbsLaplacianPower(axes(WZ,1),β)
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436 | 436 | Δ_Zfrac = Z \ (Δfrac * WZ)
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437 |
| - # define function whose fractional Laplacian is a known constant |
| 437 | + # define function whose fractional Laplacian is known |
438 | 438 | uexplicit = @. (1 - x^2 - y^2).^(β+1)
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439 | 439 | uexplicitcfs = WZ \ uexplicit
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440 | 440 | # RHS
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454 | 454 | # generate fractional Laplacian
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455 | 455 | Δfrac = AbsLaplacianPower(axes(WZ,1),β)
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456 | 456 | Δ_Zfrac = Z \ (Δfrac * WZ)
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457 |
| - # define function whose fractional Laplacian is a known constant |
| 457 | + # define function whose fractional Laplacian is known |
458 | 458 | uexplicit = @. (1 - x^2 - y^2).^(β+1)*y
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459 | 459 | uexplicitcfs = WZ \ uexplicit
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460 | 460 | # RHS
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473 | 473 | # generate fractional Laplacian
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474 | 474 | Δfrac = AbsLaplacianPower(axes(WZ,1),β)
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475 | 475 | Δ_Zfrac = Z \ (Δfrac * WZ)
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476 |
| - # define function whose fractional Laplacian is a known constant |
| 476 | + # define function whose fractional Laplacian is known |
477 | 477 | uexplicit = @. (1 - x^2 - y^2).^(β+1)*x
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478 | 478 | uexplicitcfs = WZ \ uexplicit
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479 | 479 | # RHS
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