**Numerical solution of singular integral equations**

In this Chapter the numerical ways for the answer of 2 teams of singular integral equations are represented. These equations arise from the formulation of the mixed boundary price issues in applied physics and engineering. particularly, they play a crucial role within the answer of an excellent kind of contact and crack issues in solid mechanics. within the 1st cluster of integral equations the kernels have an easy Cauchy-type singularity.** [1]**

**Rapid solution of integral equations of classical potential theory**

An formula is represented for speedy resolution of classical boundary worth issues (Dirichlet associate degree Neumann) for the astronomer equation supported iteratively determination integral equations of potential theory. central processor time needs for antecedently printed algorithms of this sort are proportional to n2, wherever n is that the range of nodes within the discretization of the boundary of the region. The central processor time needs for the formula of this paper are proportional to n, creating it significantly additional sensible for giant scale issues. **[2]**

**INTEGRAL EQUATIONS GOVERNING RADIATION EFFECTS. (NOTES ON ATOMIC COLLISIONS, III)**

A theoretica study is formed of harm effects by particle radiations in mamma, and their dependence on energy, mass,, ANd charge variety of an incoming particle, in addition as on the composition of the medium. Typical samples of injury effects square measure the quantity of particle pairs fashioned during a gas, or the quantity of vacancies created during a crystal. The study is petition between energy transfer to atomic electrons ANd to translatory motion on of an atoms as an entire. For these functions, common integril equations square measure developed and studied. The study treats primarily average effects ensuing from AN atomic particle with given energy, however additionally their average fluctuation and chance distribution. As a very important example, the division of the whole energy dissipation, E, into energy given to recoiling atoms, nu , and energy given to electrons, E- nu , is studied. many radiation effects square measure accounted for from information regarding E and letter.** [3]**

**Newtonian flow inside carbon nanotube with permeable boundary taking into account van der Waals forces**

Here, water flow within massive radii semi-infinite carbon nanotubes is investigated. semipermeable wall taking under consideration the molecular interactions between water and a fullerene, and therefore the slip condition are going to be thought-about. what is more, interactions among molecules are approximated by the time approximation. incompressible and Newtonian fluid is assumed, and therefore the Navier-Stokes equations, when sure assumptions, transformations and derivations, is reduced into 2 initial integral equations. In conjunction with the straight line growth technique, we have a tendency to are able to derive the radial and axial velocities analytically, capturing the result of the water leak, wherever each delicate and exceptionally massive leakages are going to be thought-about. **[4]**

**Numerical Solution of Volterra-Fredholm Integral Equations Using Hybrid Orthonormal Bernstein and Block-Pulse Functions**

We have projected associate degree economical numerical technique to unravel a category of mixed Volterra-Fredholm integral equations (VFIE’s) of the second kind, numerically supported Hybrid Orthonormal Bernstein and Block-Pulse Functions (OBH). The aim of this paper is to use OBH technique to get approximate solutions of nonlinear Fuzzy Fredholm Integro-differential Equations. initial we have a tendency to introduce properties of Hybrid Orthonormal Bernstein and Block-Pulse Functions, we have a tendency to used it to remodel the integral equations to the system of linear pure mathematics equations then associate degree reiterative approach is projected to get approximate answer of sophistication of linear pure mathematics equations, a numerical examples is conferred maybe the projected technique. The error estimates of the projected technique is given. **[5]**

**Reference**

**[1]** Erdogan, F., Gupta, G.D.A. and Cook, T.S., 1973. Numerical solution of singular integral equations. In Methods of analysis and solutions of crack problems (pp. 368-425). Springer, Dordrecht. (Web Link)

**[2]** Rokhlin, V., 1985. Rapid solution of integral equations of classical potential theory. Journal of computational physics, 60(2), (Web Link)

**[3]** Lindhard, J., Nielsen, V., Scharff, M. and Thomsen, P.V., 1963. Integral equations governing radiation effects.(notes on atomic collisions, iii). Kgl. Danske Videnskab., Selskab. Mat. Fys. Medd., 33(10). (Web Link)

**[4]** Newtonian flow inside carbon nanotube with permeable boundary taking into account van der Waals forces

Yue Chan, Shern-Long Lee, Wenjian Chen, Lian Zheng, Yong Shi & Yong Ren

Scientific Reports volume 9, Article number: 12121 (2019) (Web Link)

**[5]** A. Ramadan, M. and R. Ali, M. (2017) “Numerical Solution of Volterra-Fredholm Integral Equations Using Hybrid Orthonormal Bernstein and Block-Pulse Functions”, Asian Research Journal of Mathematics, 4(4), (Web Link)