• Journal of applied mathematics & informatics
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• 제41권3호
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• pp.647-656
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• 2023

In this work, we are concerned by the problem of identification of noisy terms which arise in singular problem as for remote sensing problems, and which are modeled by a linear singular parabolic equation. For the reason of missing some data that could be arisen when using the traditional sentinel method, the later will be changed by a new sentinel method for attaining the same purpose. Such new method is a particular least square-like method which permits one to distinguish between the missing terms and the pollution terms. In particular, a sentinel method will be given here in its more realistic setting for singular parabolic problems, where in this case, the observation and the control have their support in different open sets. The problem of finding a new sentinel is equivalent to finding singular optimality system of the least square control for the parabolic equation that we solve.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제13권1호
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• pp.19-38
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• 2006

We analyze the spectral collocation approximation for a parabolic partial integrodifferential equations(PIDE) with a weakly singular kernel. The space discretization is based on the spectral collocation method and the time discretization is based on Crank-Nicolson scheme with a graded mesh. We obtain the stability and second order convergence result for fully discrete scheme.

• 대한수학회지
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• 제34권4호
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• pp.1049-1064
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• 1997

We study the existence and uniqueness of nonnegative singular solution u(x,t) of the semilinear parabolic equation $$ u_t = \Delta u - a \cdot \nabla(u^q) = u^p, $$ defined in the whole space $R^N$ for t > 0, with initial data $M\delta(x)$, a Dirac mass, with M > 0. The exponents p,q are larger than 1 and the direction vector a is assumed to be constant. We here show that a unique singular solution exists for every M > 0 if and only if 1 < q < (N + 1)/(N - 1) and 1 < p < 1 + $(2q^*)$/(N + 1), where $q^* = max{q, (N + 1)/N}$. This result agrees with the earlier one for N = 1. In the proof of this result, we also show that a unique singular solution of a diffusion-convection equation without absorption, $$ u_t = \Delta u - a \cdot \nabla(u^q), $$ exists if and only if 1 < q < (N + 1)/(N - 1).

• 대한수학회지
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• 제32권3호
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• pp.483-492
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• 1995

In this paper we discuss the existence and uniqueness of singular solutions for equations of the form $$ (F) u_t = u{xx} - $\mid$u$\mid$^{q-1} u_x - $\mid$u$\mid$^{p-1}u, p,q > 1, $$ in the domain $Q = {(x,t) : x \in R, t > 0}$. This equation represents a model of diffusion-convection with absorption.

• 대한수학회지
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• 제47권4호
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• pp.789-804
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• 2010

We here investigate an existence and uniqueness of the nontrivial, nonnegative solution of a nonlinear ordinary differential equation: $$[\mid(w^m)]'\mid^{p-2}(w^m)']'\;+\;{\beta}rw'\;+\;{\alpha}w\;+\;(w^q)'\;=\;0$$ satisfying a specific decay rate: $lim_{r\rightarrow\infty}\;r^{\alpha/\beta}w(r)$ = 0 with $\alpha$ := (p - 1)/[pd-(m+1)(p-1)] and $\beta$:= [q-m(p-1)]/[pd-(m+1)(p-1)]. Here m(p-1) > 1 and m(p - 1) < q < (m+1)(p-1). Such a solution arises naturally when we study a very singular solution for a doubly degenerate equation with nonlinear convection: $$u_t\;=\;[\mid(u^m)_x\mid^{p-2}(u^m)_x]_x\;+\;(u^q)x$$ defined on the half line.

• Journal of applied mathematics & informatics
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• 제39권5_6호
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• pp.655-676
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• 2021

A parameter uniform numerical scheme is proposed for solving singularly perturbed parabolic partial differential-difference convection-diffusion equations with a small delay and advance parameters in reaction terms and spatial variable. Taylor's series expansion is applied to approximate problems with the delay and advance terms. The resulting singularly perturbed parabolic convection-diffusion equation is solved by utilizing the implicit Euler method for the temporal discretization and finite difference method for the spatial discretization on a uniform mesh. The proposed numerical scheme is shown to be an ε-uniformly convergent accurate of the first order in time and second-order in space directions. The efficiency of the scheme is proved by some numerical experiments and by comparing the results with other results. It has been found that the proposed numerical scheme gives a more accurate approximate solution than some available numerical methods in the literature.

• 대한수학회지
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• 제60권3호
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• pp.565-586
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• 2023

We study the Dirichlet problem for the degenerate nonlocal parabolic equation u_t - a(||∇u||²_L²(Ω))∆u = C_b ||u||^β_L²(Ω) |u|^q(x,t)-2 u log |u| + f in Q_T, where Q_T := Ω × (0, T), T > 0, Ω ⊂ ℝ^N, N ≥ 2, is a bounded domain with a sufficiently smooth boundary, q(x, t) is a measurable function in Q_T with values in an interval [q^-, q⁺] ⊂ (1, ∞) and the diffusion coefficient a(·) is a continuous function defined on ℝ₊. It is assumed that a(s) → 0 or a(s) → ∞ as s → 0⁺, therefore the equation degenerates or becomes singular as ||∇u(t)||2 → 0. For both cases, we show that under appropriate conditions on a, β, q, f the problem has a global in time strong solution which possesses the following global regularity property: ∆u ∈ L²(Q_T) and a(||∇u||²_L²(Ω))∆u ∈ L²(Q_T ).