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Application of the Haar wavelet transform to solving integral and differential equations

A survey on the use of the Haar wavelet method for solving nonlinear integral and differential equations is presented. This approach is applicable to different kinds of integral equations (Fredholm, Volterra, and integro-differential equations). Application to partial differential equations is exemplified by solving the sine-Gordon equation. All these problems are solved with the aid of collocation techniques. Computer simulation is carried out for problems the exact solution of which is known. This allows us to estimate the precision of the obtained numerical results. High accuracy of the results even in the case of a small number of collocation points is observed.