Haar wavelet method for solving Fisher's equation

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.

/pdf/application-of-the-haar-wavelet-transform-to-solving-4nxucuhtv8.pdf

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

On Haar wavelet operational matrix of general order and its application for the numerical solution of fractional Bagley Torvik equation

where \(\varGamma ,\varLambda \) are given constants and \(D_\gamma {}_\lambda \), f are prescribed functions.

Solving PDEs with the aid of two-dimensional Haar wavelets

Error analysis for numerical solution of fractional differential equation by Haar wavelets method

C.F.Chen Indexing terms: Haar wavelet, Matrix of integration, Dynamic systems Abstract: An operational matrix of integration based on Haar wavelets is established, and a procedure for applying the matrix to analyse lumped and distributed-parameters dynamic systems is formulated. The technique can be interpreted from the incremental and multiresolution viewpoint. Crude as well as accurate solutions can be obtained by changing the parameter m; in the mean time, the main features of the solution are preserved. Several nontrivial examples are included for demonstrating the fast, flexible and convenient capabilities of the new method. All previously mentioned orthogonal functions, how- ever, are supported on the whole interval a I t 5 b. This kind of global support is evidently a drawback for certain analysis work, particularly systems involving abrupt variations or a local function vanishing outside a short interval of time or space. Owing to cancella- tions of many terms being required in order to obtain a reasonable accuracy, the recently developed technique, wavelet analysis, could be a possible tool for solving the difficulty in physics, communication and image processing. This paper uses the simplest wavelet func- tion, or Haar wavelet, to explore the new direction in system analysis; i.e. we establish an operational matrix for integration via Haar wavelets. By using this new matrix, the drawbacks caused by the whole range sup- port situation are eliminated. We will apply this opera- tional matrix to lumped-parameter systems first, and then to distributed-parameter systems, to demonstrate the simplicity of the approach.

Haar wavelet method for solving lumped and distributed-parameter systems

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Haar wavelet method for solving lumped and distributed-parameter systems