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PDE:010117The Cauchy problem u_t+2u_x=-3u+6 with u(x,0)=1+e^{-x^2} has explicit profile 2+(e^{-(x-2t)^2}-1)e^{-3t}

Pde1 Linear Advection Reaction SourcePde1 Characteristics Integrating Factor

First-order equations and characteristicsPartial Differential Equations I · Mixed review

PDE:010118Nonconservative dilation u_t+x u_x=0 stretches the Lorentzian to mass pi e^t and L^2 energy (pi/2) e^t

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Pde1 Dilating Nonconservative TransportPde1 Mass Growth Under Dilation

First-order equations and characteristicsPartial Differential Equations I · Mixed review

PDE:010219The power x^{3/4} on (0,1) lies in H^1 with squared norm 61/40, while x^{1/2} fails logarithmically

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Pde1 One Dimensional Sobolev PowerPde1 Critical Alpha Logarithmic

Distributions, weak derivatives, and Sobolev spacesPartial Differential Equations I · Mixed review

PDE:010220The radial power r^{1/2} on the unit disk is in H^1 with squared norm 7pi/6, while r^{-1/2} is only L^2

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Pde1 Radial Sobolev Power DiskPde1 Polar Jacobian H1

Distributions, weak derivatives, and Sobolev spacesPartial Differential Equations I · Mixed review

PDE:010321The mode sin(pi x)sin(2 pi y) on the unit square has L^2 mass 1/4 and Dirichlet energy 5 pi^2/4

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Pde1 Dirichlet Laplacian Manufactured ModePde1 Variational Energy Identity

Elliptic equations and variational methodsPartial Differential Equations I · Mixed review

PDE:010322On the unit square, u=x(1-x)y(1-y) has max 1/16, source 1 at the center, and Dirichlet energy 1/45

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Pde1 Polynomial Poisson SourcePde1 Dirichlet Green Energy

Elliptic equations and variational methodsPartial Differential Equations I · Mixed review

PDE:010323The mode sin(2πx)sin(3πy) has eigenvalue 13π², L2 mass 1/4, and six nodal rectangles

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Pde1 Rectangle Dirichlet EigenvaluePde1 Rayleigh Nodal Rectangles

Elliptic equations and variational methodsPartial Differential Equations I · Mixed review

PDE:010424The forced heat flow ut=uxx+sin x, u0=0, is (1-e^{-t})sin x with t=ln 2 mass π/8

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Pde1 Forced Heat Variation OF ParametersPde1 Single Mode Heat Norms

Heat-type equations, regularity, and maximum principlesPartial Differential Equations I · Mixed review

PDE:010425The data 3sin x-2sin 2x decays as 3e^{-t}sin x-2e^{-4t}sin 2x with t=ln 2 mass 145π/128

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Pde1 Dirichlet Heat Quadratic DecayPde1 Two Mode Heat Masses

Heat-type equations, regularity, and maximum principlesPartial Differential Equations I · Mixed review

PDE:010426The Neumann datum 2+3cos x-cos 2x keeps mean 2 and at t=ln 2 has traces 55/16 and 7/16

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Pde1 Neumann Heat Conserved MeanPde1 Neumann Cosine Mode Traces

Heat-type equations, regularity, and maximum principlesPartial Differential Equations I · Mixed review

PDE:010527The resonant forced wave u_tt-u_xx=sin(2x)cos(2t) has amplitude (t/4)sin(2t) and energy π(4+π²)/256 at t=π/4

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Pde1 Forced Wave Resonant ModePde1 Dirichlet Wave Energy Forced

Wave equations, energy methods, and finite propagationPartial Differential Equations I · Mixed review

PDE:010528The underdamped 2-mode of u_tt+2u_t-u_xx=0 is e^{-t}[cos(√3 t)+(1/√3)sin(√3 t)] with first zero 2π/(3√3)

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Pde1 Damped Wave Underdamped ModePde1 Damped Wave Energy Dissipation

Wave equations, energy methods, and finite propagationPartial Differential Equations I · Mixed review

PDE:010629Burgers data u_L=-1, u_R=1 open a centered rarefaction whose L1 distance from sign is t

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Pde1 Burgers Centered RarefactionPde1 Rarefaction L1 Distance

Nonlinear conservation laws, shocks, and entropy conditionsPartial Differential Equations I · Mixed review

PDE:010630Conservative dilation u_t+(x u)_x=0 carries the Lorentz profile with mass π and L2²=(π/2)e^{-t}

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Pde1 Conservative Dilating TransportPde1 Dilation Mass L2 Identities

Nonlinear conservation laws, shocks, and entropy conditionsPartial Differential Equations I · Mixed review

PDE:010631Linear acoustics with left state (4,1) and right rest has middle state (3,3/2) between x=±2t

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Pde1 Linear Acoustics Riemann InvariantsPde1 Acoustic Rankine Hugoniot Jumps

Nonlinear conservation laws, shocks, and entropy conditionsPartial Differential Equations I · Mixed review

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PDE:010117The Cauchy problem u_t+2u_x=-3u+6 with u(x,0)=1+e^{-x^2} has explicit profile 2+(e^{-(x-2t)^2}-1)e^{-3t}

PDE:010118Nonconservative dilation u_t+x u_x=0 stretches the Lorentzian to mass pi e^t and L^2 energy (pi/2) e^t

PDE:010219The power x^{3/4} on (0,1) lies in H^1 with squared norm 61/40, while x^{1/2} fails logarithmically

PDE:010220The radial power r^{1/2} on the unit disk is in H^1 with squared norm 7pi/6, while r^{-1/2} is only L^2

PDE:010321The mode sin(pi x)sin(2 pi y) on the unit square has L^2 mass 1/4 and Dirichlet energy 5 pi^2/4

PDE:010322On the unit square, u=x(1-x)y(1-y) has max 1/16, source 1 at the center, and Dirichlet energy 1/45

PDE:010323The mode sin(2πx)sin(3πy) has eigenvalue 13π², L2 mass 1/4, and six nodal rectangles

PDE:010424The forced heat flow ut=uxx+sin x, u0=0, is (1-e^{-t})sin x with t=ln 2 mass π/8

PDE:010425The data 3sin x-2sin 2x decays as 3e^{-t}sin x-2e^{-4t}sin 2x with t=ln 2 mass 145π/128

PDE:010426The Neumann datum 2+3cos x-cos 2x keeps mean 2 and at t=ln 2 has traces 55/16 and 7/16

PDE:010527The resonant forced wave u_tt-u_xx=sin(2x)cos(2t) has amplitude (t/4)sin(2t) and energy π(4+π²)/256 at t=π/4

PDE:010528The underdamped 2-mode of u_tt+2u_t-u_xx=0 is e^{-t}[cos(√3 t)+(1/√3)sin(√3 t)] with first zero 2π/(3√3)

PDE:010629Burgers data u_L=-1, u_R=1 open a centered rarefaction whose L1 distance from sign is t

PDE:010630Conservative dilation u_t+(x u)_x=0 carries the Lorentz profile with mass π and L2²=(π/2)e^{-t}

PDE:010631Linear acoustics with left state (4,1) and right rest has middle state (3,3/2) between x=±2t

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Problem results

PDE:010117The Cauchy problem u_t+2u_x=-3u+6 with u(x,0)=1+e^{-x^2} has explicit profile 2+(e^{-(x-2t)^2}-1)e^{-3t}

PDE:010118Nonconservative dilation u_t+x u_x=0 stretches the Lorentzian to mass pi e^t and L^2 energy (pi/2) e^t

PDE:010219The power x^{3/4} on (0,1) lies in H^1 with squared norm 61/40, while x^{1/2} fails logarithmically

PDE:010220The radial power r^{1/2} on the unit disk is in H^1 with squared norm 7pi/6, while r^{-1/2} is only L^2

PDE:010321The mode sin(pi x)sin(2 pi y) on the unit square has L^2 mass 1/4 and Dirichlet energy 5 pi^2/4

PDE:010322On the unit square, u=x(1-x)y(1-y) has max 1/16, source 1 at the center, and Dirichlet energy 1/45

PDE:010323The mode sin(2πx)sin(3πy) has eigenvalue 13π², L2 mass 1/4, and six nodal rectangles

PDE:010424The forced heat flow ut=uxx+sin x, u0=0, is (1-e^{-t})sin x with t=ln 2 mass π/8

PDE:010425The data 3sin x-2sin 2x decays as 3e^{-t}sin x-2e^{-4t}sin 2x with t=ln 2 mass 145π/128

PDE:010426The Neumann datum 2+3cos x-cos 2x keeps mean 2 and at t=ln 2 has traces 55/16 and 7/16

PDE:010527The resonant forced wave u_tt-u_xx=sin(2x)cos(2t) has amplitude (t/4)sin(2t) and energy π(4+π²)/256 at t=π/4

PDE:010528The underdamped 2-mode of u_tt+2u_t-u_xx=0 is e^{-t}[cos(√3 t)+(1/√3)sin(√3 t)] with first zero 2π/(3√3)

PDE:010629Burgers data u_L=-1, u_R=1 open a centered rarefaction whose L1 distance from sign is t

PDE:010630Conservative dilation u_t+(x u)_x=0 carries the Lorentz profile with mass π and L2²=(π/2)e^{-t}

PDE:010631Linear acoustics with left state (4,1) and right rest has middle state (3,3/2) between x=±2t

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PDE:010117The Cauchy problem u_t+2u_x=-3u+6 with u(x,0)=1+e^{-x^2} has explicit profile 2+(e^{-(x-2t)^2}-1)e^{-3t}View problem statement

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PDE:010117The Cauchy problem u_t+2u_x=-3u+6 with u(x,0)=1+e^{-x^2} has explicit profile 2+(e^{-(x-2t)^2}-1)e^{-3t}View problem statement

PDE:010117The Cauchy problem u_t+2u_x=-3u+6 with u(x,0)=1+e^{-x^2} has explicit profile 2+(e^{-(x-2t)^2}-1)e^{-3t}

PDE:010117The Cauchy problem u_t+2u_x=-3u+6 with u(x,0)=1+e^{-x^2} has explicit profile 2+(e^{-(x-2t)^2}-1)e^{-3t}