{
  "id": "1408.2454",
  "version": "v2",
  "published": "2014-08-11T16:08:26.000Z",
  "updated": "2015-03-31T11:36:01.000Z",
  "title": "A modified integral equation method of the nonlinear elliptic equation with globally and locally Lipschitz source",
  "authors": [
    "Nguyen Huy Tuan",
    "Le Duc Thang",
    "Vo Anh Khoa"
  ],
  "comment": "28 pages, 23 figures, 2 tables",
  "categories": [
    "math.AP"
  ],
  "abstract": "The paper is devoted to investigating a Cauchy problem for nonlinear elliptic PDEs in the abstract Hilbert space. The problem is hardly solved by computation since it is severely ill-posed in the sense of Hadamard. We shall use a modified integral equation method to regularize the nonlinear problem with globally and locally Lipschitz source terms. Convergence estimates are established under priori assumptions on exact solution. A numerical test is provided to illustrate that the proposed method is feasible and effective. These results extend some earlier works on a Cauchy problem for elliptic equations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-11T16:08:26.000Z",
      "title": "Regularization of an ill-posed problem for elliptic equation with nonlinear source",
      "abstract": "This paper studies the Cauchy problem for elliptic equations with nonlinear source term. This semilinear problem arises in many application problems, such as Helmholtz equation, elliptic sine-Gordon equation, and other equations formed by, for example, Lane-Emden equation and Poisson equation. However, it is severely ill-posed in the sense of Hadamard. Therefore, we consider theoretical aspects of regularization of the problem by a method of integral equation. Under some priori assumptions on the exact solution, we obtain convergence estimates in many cases. A numerical test is presented that validate the applicability and efficiency of the theoretical result.",
      "comment": "26 pages, 23 figures, 2 tables",
      "journal": null,
      "doi": null,
      "authors": [
        "Nguyen Huy Tuan",
        "Vo Anh Khoa",
        "Le Duc Thang"
      ]
    },
    {
      "version": "v2",
      "updated": "2015-03-31T11:36:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A52",
      "35J61",
      "26D15"
    ],
    "keywords": [
      "elliptic equation",
      "ill-posed problem",
      "regularization",
      "nonlinear source term",
      "semilinear problem arises"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.2454T"
    }
  }
}