Fredholm Integral Equations of First Kind

Abstract

A unique variation of the inverse problem is the first type of Fredholm integral equation. To address the computing issue, inverse mathematical physics problems have been converted into the first type of Fredholm integral equation. We also use the Landweber iteration as an alternative to the well-known Tikhonov regularization technique , which has been shown to be most effective in solving ill-posed inverse problems. The Landweber iteration is a straight-forward and effective technique that exhibits convergence towards the accurate solution given specific conditions. Consequently, it serves as a valuable instru-ment for resolving inverse problems across diverse domains, including signal processing and geophysics. Following the examination of the properties of uniqueness and existence pertaining to solutions of integral equations of the first kind, the aforementioned equations are resolved through the utilization of the collocation method. The trapezoidal rule is widely utilized in numerical integration due to its straight-forward implementation and computational efficiency. However, it may not be appropriate for integrals with significant oscillatory behavior. In instances of this nature, it may be imperative to employ more sophisticated numerical integration methods, such as Gaussian quadrature or adaptive quadrature, in order to attain precise outcomes. For weakly singular integrals that appear in formulations of integral equations of potential problems in domains with corners and edges, we provide n-points Gaussian quadrature procedures which are particularly useful in numerical integration problems where the integral is difficult to evaluate. The accuracy of the method depends on the number of points used in the procedure, with higher order rules providing more accurate results.

Fredholm bütünsel eşitliğinin ilk türü, ters sorunun özel bir türüdür. Matematik fiziğinin tersi sorunları, hesaplama sorunu çözmek için ilk tip Fredholm bütünsel eşitliğine çevrildi. Bütünsel eşitliğin çözümü için tahmin etmeye çalıştığımız proje yöntemini ve kolokasyon yöntemini kullanırız. Ayrıca Tikhonov düzenleme yöntemi iyi bilinir ve alternatif olarak, Landweber iterasyonunu kullanırız. Trapezoidal kural, sürekli çekirdeklerle bütünleşen bütünsel operatörlerin sayısal entegrasyonu için kullanılırken, zayıf singular çekirgeler başka bir yöntem kullanılarak sayısal entegrasyonda kullanılır. Metodun doğruluğunu kontrol etmek için farklı test durumları dikkate alınır ve yaklaşım ve hata sonuçlarının sırası sayısal örneklerle gösterilir.