Cauchy Problem For Parabolic Equation Research Articles

Численный метод решения некоторых обратных задач теплопроводности с неизвестными начальными условиями

The article deals with some inverse Cauchy problems for parabolic equations with unknown initial conditions. The principal possibility of construction the numerical solution of these problems in this domain is showed. The computational scheme is proposed which allows to construct the numerical solution of the inverse problem both on the whole domain and on the boundary under unknown initial conditions. To evaluate the efficiency of the proposed method the computational experiment was carried out. The results of the experiments show the sufficient stability of numerical solutions and the advantages of the proposed method.

The quasi-reversibility method to solve the Cauchy problems for parabolic equations

In this paper, on the one hand, we take the conventional quasi-reversibility method to obtain the error estimates of approximate solutions of the Cauchy problems for parabolic equations in a sub-domain of Q T with strong restrictions to the measured boundary data. On the other hand, weakening the conditions on the measured data, then combining the duality method in optimization with the quasi-reversibility method, we solve the Cauchy problems for parabolic equations in the presence of noisy data. Using this method, we can get the proper regularization parameter ɛ that we need in the quasi-reversibility method and obtain the convergence rate of approximate solutions as the noise of amplitude δ tends to zero.

Asymptotics of solutions to the Dirichlet–Cauchy problem for parabolic equations in domains with edges

Asymptotics of solutions to the Dirichlet–Cauchy problem for parabolic equations in domains with edges

On existence and uniqueness classes for the Cauchy problem for parabolic equations of the p-Laplace type

We prove the existence and uniqueness of global solutions to the Cauchy problem for a class of parabolic equations of the p-Laplace type. In the singular case $p 2$ we impose a restriction on growth of solutions at infinity to obtain global existence and uniqueness. This restriction is given in terms of weighted energy classes with power-like weights.

On Solving the Cauchy Problem for Parabolic Equation with Nonlocal Potential

On Solving the Cauchy Problem for Parabolic Equation with Nonlocal Potential

Embedding Operators in Vector-Valued Weighted Besov Spaces and Applications

The embedding theorems in weighted Besov-Lions type spaces <svg style="vertical-align:-4.30858pt;width:33.012501px;" id="M1" height="19.975" version="1.1" viewBox="0 0 33.012501 19.975" width="33.012501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.975)"> <g transform="translate(72,-56.02)"> <text transform="matrix(1,0,0,-1,-71.95,60.37)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝐵</tspan> </text> <text transform="matrix(1,0,0,-1,-62.75,66.1)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑙</tspan> <tspan style="font-size: 8.75px; " x="2.7831359" y="0">,</tspan> <tspan style="font-size: 8.75px; " x="4.9711361" y="0">𝑠</tspan> <tspan style="font-size: 8.75px; " x="-0.5" y="8.3900003">𝑝</tspan> <tspan style="font-size: 8.75px; " x="3.7797279" y="8.3900003">,</tspan> <tspan style="font-size: 8.75px; " x="5.9677281" y="8.3900003">𝑞</tspan> <tspan style="font-size: 8.75px; " x="9.9761438" y="8.3900003">,</tspan> <tspan style="font-size: 8.75px; " x="12.164144" y="8.3900003">𝛾</tspan> </text> </g> </g> </svg> (<svg style="vertical-align:-3.25793pt;width:54.5625px;" id="M2" height="14.75" version="1.1" viewBox="0 0 54.5625 14.75" width="54.5625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.75)"> <g transform="translate(72,-60.2)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">Ω</tspan> <tspan style="font-size: 12.50px; " x="9.3022318" y="0">;</tspan> <tspan style="font-size: 12.50px; " x="14.853564" y="0">𝐸</tspan> </text> <text transform="matrix(1,0,0,-1,-48.16,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0">0</tspan> </text> <text transform="matrix(1,0,0,-1,-43.29,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">,</tspan> <tspan style="font-size: 12.50px; " x="5.2012482" y="0">𝐸</tspan> </text> </g> </g> </svg>) in which <svg style="vertical-align:-3.25793pt;width:35.987499px;" id="M3" height="14.3875" version="1.1" viewBox="0 0 35.987499 14.3875" width="35.987499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.3875)"> <g transform="translate(72,-60.49)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝐸</tspan> </text> <text transform="matrix(1,0,0,-1,-63.02,60.66)"> <tspan style="font-size: 8.75px; " x="0" y="0">0</tspan> </text> <text transform="matrix(1,0,0,-1,-58.15,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0">,</tspan> <tspan style="font-size: 12.50px; " x="5.2012482" y="0">𝐸</tspan> </text> </g> </g> </svg> are two Banach spaces and <svg style="vertical-align:-3.25793pt;width:48.862499px;" id="M4" height="14.3875" version="1.1" viewBox="0 0 48.862499 14.3875" width="48.862499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.3875)"> <g transform="translate(72,-60.49)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝐸</tspan> </text> <text transform="matrix(1,0,0,-1,-63.02,60.66)"> <tspan style="font-size: 8.75px; " x="0" y="0">0</tspan> </text> <text transform="matrix(1,0,0,-1,-54.68,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0">⊂</tspan> <tspan style="font-size: 12.50px; " x="12.040389" y="0">𝐸</tspan> </text> </g> </g> </svg> are studied. The most regular class of interpolation space <svg style="vertical-align:-3.22282pt;width:18.075001px;" id="M5" height="14.35" version="1.1" viewBox="0 0 18.075001 14.35" width="18.075001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.35)"> <g transform="translate(72,-60.52)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝐸</tspan> </text> <text transform="matrix(1,0,0,-1,-63.02,60.66)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝛼</tspan> </text> </g> </g> </svg> between <svg style="vertical-align:-3.25793pt;width:17.375px;" id="M6" height="14.3875" version="1.1" viewBox="0 0 17.375 14.3875" width="17.375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.3875)"> <g transform="translate(72,-60.49)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝐸</tspan> </text> <text transform="matrix(1,0,0,-1,-63.02,60.66)"> <tspan style="font-size: 8.75px; " x="0" y="0">0</tspan> </text> </g> </g> </svg> and <i>E</i> is found such that the mixed differential operator <svg style="vertical-align:-0.0pt;width:19.5px;" id="M7" height="11.2" version="1.1" viewBox="0 0 19.5 11.2" width="19.5" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.2)"> <g transform="translate(72,-63.04)"> <text transform="matrix(1,0,0,-1,-71.95,63.09)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝐷</tspan> </text> <text transform="matrix(1,0,0,-1,-61.89,68.09)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝛼</tspan> </text> </g> </g> </svg> is bounded from <svg style="vertical-align:-4.30858pt;width:33.012501px;" id="M8" height="19.975" version="1.1" viewBox="0 0 33.012501 19.975" width="33.012501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.975)"> <g transform="translate(72,-56.02)"> <text transform="matrix(1,0,0,-1,-71.95,60.37)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝐵</tspan> </text> <text transform="matrix(1,0,0,-1,-62.75,66.1)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑙</tspan> <tspan style="font-size: 8.75px; " x="2.7831359" y="0">,</tspan> <tspan style="font-size: 8.75px; " x="4.9711361" y="0">𝑠</tspan> <tspan style="font-size: 8.75px; " x="-0.5" y="8.3900003">𝑝</tspan> <tspan style="font-size: 8.75px; " x="3.7797279" y="8.3900003">,</tspan> <tspan style="font-size: 8.75px; " x="5.9677281" y="8.3900003">𝑞</tspan> <tspan style="font-size: 8.75px; " x="9.9761438" y="8.3900003">,</tspan> <tspan style="font-size: 8.75px; " x="12.164144" y="8.3900003">𝛾</tspan> </text> </g> </g> </svg> (<svg style="vertical-align:-3.25793pt;width:54.5625px;" id="M9" height="14.75" version="1.1" viewBox="0 0 54.5625 14.75" width="54.5625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.75)"> <g transform="translate(72,-60.2)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">Ω</tspan> <tspan style="font-size: 12.50px; " x="9.3022318" y="0">;</tspan> <tspan style="font-size: 12.50px; " x="14.853564" y="0">𝐸</tspan> </text> <text transform="matrix(1,0,0,-1,-48.16,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0">0</tspan> </text> <text transform="matrix(1,0,0,-1,-43.29,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">,</tspan> <tspan style="font-size: 12.50px; " x="5.2012482" y="0">𝐸</tspan> </text> </g> </g> </svg>) to <svg style="vertical-align:-4.77652pt;width:33.012501px;" id="M10" height="17.15" version="1.1" viewBox="0 0 33.012501 17.15" width="33.012501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.15)"> <g transform="translate(72,-58.28)"> <text transform="matrix(1,0,0,-1,-71.95,63.09)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝐵</tspan> </text> <text transform="matrix(1,0,0,-1,-62.75,68.09)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑠</tspan> <tspan style="font-size: 8.75px; " x="-0.5" y="8.1300001">𝑝</tspan> <tspan style="font-size: 8.75px; " x="3.7797279" y="8.1300001">,</tspan> <tspan style="font-size: 8.75px; " x="5.9677281" y="8.1300001">𝑞</tspan> <tspan style="font-size: 8.75px; " x="9.9761438" y="8.1300001">,</tspan> <tspan style="font-size: 8.75px; " x="12.164144" y="8.1300001">𝛾</tspan> </text> </g> </g> </svg> (<svg style="vertical-align:-3.22282pt;width:36.650002px;" id="M11" height="14.7125" version="1.1" viewBox="0 0 36.650002 14.7125" width="36.650002" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.7125)"> <g transform="translate(72,-60.23)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">Ω</tspan> <tspan style="font-size: 12.50px; " x="9.3022318" y="0">;</tspan> <tspan style="font-size: 12.50px; " x="14.853564" y="0">𝐸</tspan> </text> <text transform="matrix(1,0,0,-1,-48.16,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝛼</tspan> </text> </g> </g> </svg>) and Ehrling-Nirenberg-Gagliardo type sharp estimates are established. By using these results, the uniform separability of degenerate abstract differential equations with parameters and the maximal <i >B</i>-regularity of Cauchy problem for abstract parabolic equations are obtained. The infinite systems of the degenerate partial differential equations and Cauchy problem for system of parabolic equations are further studied in applications.

Singular perturbation degenerate problems occurring in atmospheric dispersion of pollutants

Abstract The boundary value problems for the degenerate differential-operator equations with small parameters generated on all boundary are studied. Several conditions for the separability and the fredholmness in Banach-valued L p -spaces of are given. In applications, maximal regularity of degenerate Cauchy problem for parabolic equation arising in atmospheric dispersion of pollutants studied.

Homogenization of the parabolic cauchy problem in the Sobolev class H1(Rd)

Homogenization in the small period limit for the solution ue of the Cauchy problem for a parabolic equation in Rd is studied. The coefficients are assumed to be periodic in Rd with respect to the lattice ɛG. As ɛ → 0, the solution u ɛ converges in L2(Rd) to the solution u0 of the effective problem with constant coefficients. The solution u ɛis approximated in the norm of the Sobolev space H 1(Rd) with error O( ɛ); this approximation is uniform with respect to the L2-norm of the initial data and contains a corrector term of order ɛ. The dependence of the constant in the error estimate on time t is given. Also, an approximation in H 1(Rd) for the solution of the Cauchy problem for a nonhomogeneous parabolic equation is obtained.

Regular Degenerate Separable Differential Operators and Applications

The boundary value problems for differential-operator equations with variable coefficients, degenerated on all boundary are studied. Several conditions for the separability, fredholmness and resolvent estimates in L p -spaces are given. In applications degenerate Cauchy problem for parabolic equation, boundary value problems for degenerate partial differential equations and systems of degenerate elliptic equations on cylindrical domain are studied.

On existence and uniqueness of the Cauchy problem for parabolic equations with unbounded coefficients

A new kind of entropy solution to the Cauchy problem for strong degenerate parabolic equations with unbounded coefficients, ∂u/∂t= ΔA(u) + div(uE); A′(u) ≥ 0, is quoted. Suppose that u0 ∈ L∞(RN), E...

The Cauchy problem for parabolic equations in Zygmund spaces

The Cauchy problem for parabolic equations in Zygmund spaces

On the Feynman–Kac representation for solutions of the Cauchy problem for parabolic equations in divergence form

We consider the Cauchy problem for general second–order uniformly elliptic linear equation in divergence form. We give a stochastic representation of bounded weak solutions of the problem in terms of solutions of associated linear backward stochastic differential equations. Our representation may be considered as an extension of the classical Feynman–Kac formula.

Algorithm and mechanization to Cauchy problem of parabolic equation

The present paper discusses the algorithm and mechanization to the Cauchy problem of parabolic equation. The author uses the initial data to construct the solutions that satisfy the initial condition and obtain a brief algorithm to this problem. Thereafter, the algorithm is used in Maple to solve this Cauchy problem automatically.

Numerical analysis of Monte Carlo evaluation of Greeks by finite differences

An error analysis of approximation of deltas (derivatives of the solution to the Cauchy problem for parabolic equations) by finite differences is given, taking into account that the value of the hedging portfolio itself (the solution of the problem) is evaluated using weak-sense numerical integration of the corresponding system of stochastic differential equations together with the Monte Carlo technique. It is shown that finite differences are effective when the method of dependent realizations is exploited in the Monte Carlo simulations. Evaluation of other Greeks is also considered. Results of numerical experiments, including those with Heston stochastic volatility model, are presented.

On a necessary condition for analytic wellposedness of the Cauchy problem for parabolic equation

On a necessary condition for analytic wellposedness of the Cauchy problem for parabolic equation

On polynomials of Sheffer type arising from a Cauchy problem

A new sequence of eigenfunctions is developed and studied in depth. These theta polynomials are derived from a recent analytic solution of the canonical Cauchy problem for parabolic equations, namely, the inverse heat conduction problem. By appealing to the methods of the operator calculus, it is possible to categorize the new functions as polynomials of binomial and Sheffer types. The connection of the new set with the classical polynomials of Laguerre is carefully examined. Some integral relations involving the Laguerre polynomials and the theta polynomials are presented along with a number of binomial identities. The inverse heat conduction problem is revisited and an analytic solution depending on the generalized theta polynomials is presented.

Scaling in Nonlinear Parabolic Equations

The Cauchy problem for parabolic equations with quadratic nonlinearity is studied. We investigate the existence of global-in-time solutions and their large-time behavior assuming some scaling property of the equation as well as of the norm of the Banach space in which the solutions are constructed. Some applications of these abstract results are indicated.

An intrinsic metric approach to uniqueness of the positive Cauchy problem for parabolic equations

An intrinsic metric approach to uniqueness of the positive Cauchy problem for parabolic equations

Homogenization of attractors for semilinear parabolic equations on manifolds with complicated microstructure

An approach to a homogenized description of solutions of the Cauchy problem for parabolic equations on Riemannian manifolds with complicated microstructure is presented. This approach covers both linear and non-linear cases and makes it possible to establish a connection between global attractors of the initial problem of the homogenized one.

On stabilization of solutions of the Cauchy problem for parabolic equations

On stabilization of solutions of the Cauchy problem for parabolic equations