Application of Volterra Linear Integral Equations to the Numerical Solution of Vibration Problems-II

1954 ◽  
Vol 21 (6) ◽  
pp. 383-388
Author(s):  
J. L. BOGDANOFF ◽  
J. E. GOLDBERG ◽  
HSU LO
Keyword(s):  
Numerical Solution ◽  
Integral Equations ◽  
Vibration Problems ◽  
Linear Integral ◽  
Linear Integral Equations

A note on the numerical solution of certain non-linear integral equations

1956 ◽  
Vol 236 (1207) ◽  
pp. 529-534 ◽  
Keyword(s):  
Numerical Solution ◽  
Integral Equations ◽  
Asymptotic Form ◽  
Essential Singularity ◽  
Non Linear ◽  
Linear Integral ◽  
Hydrodynamical Problem ◽  
Linear Integral Equations

Methods are described for the numerical solution of two non-linear integral equations occurring in a hydrodynamical problem. In each case the existence of an essential singularity of the solution requires the application of special techniques. The asymptotic form of the solutions for large x is determined.

The numerical solution of non-singular linear integral equations

1953 ◽  
Vol 245 (902) ◽  
pp. 501-534 ◽  
Keyword(s):  
Finite Difference ◽  
Eigenvalue Problem ◽  
Error Analysis ◽  
Numerical Solution ◽  
Integral Equations ◽  
Numerical Quadrature ◽  
Iterative Processes ◽  
Upper Limit ◽  
Linear Integral ◽  
Linear Integral Equations

The integral equations discussed and illustrated are those of Fredholm, with fixed limits in the integral and including the eigenvalue problem, and of Volterra, with a variable upper limit in the integral. The methods are mostly based on finite-difference theory, the integrals being replaced by formulae for numerical quadrature. Computational details are given for several methods, and there is a discussion of error analysis for Volterra’s equation. Some methods are given for accelerating the convergence of classical iterative processes.

scholarly journals Numerical solution of linear integral equations system using the Bernstein collocation method

2013 ◽  
Vol 2013 (1) ◽  
pp. 123 ◽  
Author(s):  
Ahmad Jafarian ◽  
Safa A Measoomy Nia ◽  
Alireza K Golmankhaneh ◽  
Dumitru Baleanu
Keyword(s):  
Numerical Solution ◽  
Integral Equations ◽  
Collocation Method ◽  
Linear Integral ◽  
Linear Integral Equations

The numerical solution of integral equations using Chebyshev polynomials

1960 ◽  
Vol 1 (3) ◽  
pp. 344-356 ◽  
Author(s):  
David Elliott
Keyword(s):  
Numerical Solution ◽  
Integral Equations ◽  
Chebyshev Polynomials ◽  
Unknown Function ◽  
Chebyshev Series ◽  
Direct Expansion ◽  
Linear Integral ◽  
Polynomial Expansions ◽  
Linear Integral Equations

An investigation has been made into the numerical solution of non-singular linear integral equations by the direct expansion of the unknown function f(x) into a series of Chebyshev polynomials of the first kind. The use of polynomial expansions is not new, and was first described by Crout [1]. He writes f(x) as a Lagrangian-type polynomial over the range in x, and determines the unknown coefficients in this expansion by evaluating the functions and integral arising in the equation at chosen points xi. A similar method (known as collocation) is used here for cases where the kernel is not separable. From the properties of expansion of functions in Chebyshev series (see, for example, [2]), one expects greater accuracy in this case when compared with other polynomial expansions of the same order. This is well borne out in comparison with one of Crout's examples.

scholarly journals Approximate Solution of Volterra-Stieltjes Linear Integral Equations of the Second Kind with the Generalized Trapezoid Rule

2016 ◽  
Vol 2016 ◽  
pp. 1-6 ◽  
Author(s):  
Avyt Asanov ◽  
Elman Hazar ◽  
Mustafa Eroz ◽  
Kalyskan Matanova ◽  
Elmira Abdyldaeva
Keyword(s):  
Approximate Solution ◽  
Numerical Solution ◽  
Integral Equations ◽  
Stieltjes Integral ◽  
Linear Integral ◽  
Linear Integral Equations

The numerical solution of linear Volterra-Stieltjes integral equations of the second kind by using the generalized trapezoid rule is established and investigated. Also, the conditions on estimation of the error are determined and proved. A selected example is solved employing the proposed method.

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