A First Course in Integral Equations by Abdul-Majid Wazwaz

Author:Abdul-Majid Wazwaz
Language: eng
Format: epub, azw3, pdf
Publisher: World Scientific Publishing Co Pte Ltd
Published: 2015-05-03T16:00:00+00:00

It is clear from the examples given above that the unknown function u(x) or one of its derivatives appear under the integral sign, and other derivatives of u(x) appear outside the integral sign as well. Therefore, the above given equations involve the derivatives and the integral operators in the same equation, and consequently the term integro-differential equations has been used for problems involving this combination of operators.

Examining the limits of integrals in equations (1)-(2) and following the classification concept used in Chapter 1 allow us to use the classification Volterra integro-differential equations to equations (1) and (2). In addition, it is also interesting to know that equations (1)-(2) are linear Volterra integro-differential equations, and this is related to the linearity occurrence of the unknown function u(x) and its derivatives in the equations above. However, nonlinear integro-differential equations also arise in many scientific and engineering problems. Our concern in this chapter will be focused on the linear Volterra integro-differential equations.

To determine a solution for the Volterra integro-differential equation, the initial conditions should be given, and this may be clearly seen as a result of involving u(x) and its derivatives. The initial conditions are needed to determine the constants of integration.

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