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In this paper, a mathematical model of steady-state reaction-diffusion (RD) model for estimating the concentration of species is discussed. We have applied a new wavelet-based operational matrix of derivative method to obtain the approximate solutions for nonlinear RDEs. The proposed method is a powerful and easy-to-use analytical tool for linear and nonlinear problems. Some illustrative examples are given to validate our results with exact solutions. Satisfactory agreement with the exact solution is noticed. Moreover, the use of Legendre wavelets is found to be simple, accurate, efficient and requires small computation costs.

In this paper, we have developed an efficient wavelet based approximation method to biofilm model under steady state arising in enzyme kinetics. Chebyshev wavelet based approximation method is successfully introduced in solving nonlinear steady state biofilm reaction model. To the best of our knowledge, until now there is no rigorous wavelet based solution has been addressed for the proposed model. Analytical solutions for substrate concentration have been derived for all values of the parameters δ and SL. The power of the manageable method is confirmed. Some numerical examples are presented to demonstrate the validity and applicability of the wavelet method. Moreover the use of Chebyshev wavelets is found to be simple, efficient, flexible, convenient, small computation costs and computationally attractive.

A mathematical model of amperometric biosensors has been developed. The model is based on non-stationary diffusion equation containing a nonlinear term related to non-Michaelis–Menten kinetics of the reaction. In this paper, an efficient Chebyshev wavelet based approximation method is introduced for solving the steady state reaction–diffusion equations. Illustrative examples are given to demonstrate the validity and applicability of the method. The power of the manageable method is confirmed. Moreover the use of Chebyshev wavelet method is found to be simple, flexible, efficient, small computation costs and computationally attractive.

In this chapter, an efficient wavelet-based approximation method is established to nonlinear singular boundary value problems. To the best of our knowledge, until now there is no rigorous shifted second kind Chebyshev wavelet (S2KCWM) solution has been addressed for the nonlinear differential equations in population biology. With the help of shifted second kind Chebyshev wavelet operational matrices, the nonlinear differential equations are converted into a system of algebraic equations. The convergence of the proposed method is established. The power of the manageable method is confirmed. Finally, we have given some numerical examples to demonstrate the validity and applicability of the proposed wavelet method.

In this paper, a wavelet-based approximation method is introduced for solving the Newell-Whitehead (NW) and Allen-Cahn (AC) equations. To the best of our knowledge, until now there is no rigorous Legendre wavelets solution has been reported for the NW and AC equations. The highest derivative in the differential equation is expanded into Legendre series, this approximation is integrated while the boundary conditions are applied using integration constants. With the help of Legendre wavelets operational matrices, the aforesaid equations are converted into an algebraic system. Block pulse functions are used to investigate the Legendre wavelets coefficient vectors of nonlinear terms. The convergence of the proposed methods is proved. Finally, we have given some numerical examples to demonstrate the validity and applicability of the method.

The mathematical model of Rahamathunissa and Rajendran (J Math Chem 44:849-861, 2008) in an amperometric biosensor response is discussed. In this paper, we have applied the shifted second kind Chebyshev wavelets (CW) to obtain the numerical solutions of reaction-diffusion equations containing a nonlinear term related to Michaelis-Menton kinetics of the enzymatic reaction. The application of the shifted second kind CW operational matrices for solving initial and boundary value problems is presented. The obtained numerical results demonstrate efficient and applicability of the proposed method. The power of the manageable method is confirmed. Moreover the use of shifted second kind CW method is found to be simple, efficient, accurate, small computation cost, and computationally attractive.

In this chapter, we have applied the wavelet-based coupled method for finding the numerical solution of the Murray equation. To the best of our knowledge, until now there is no rigorous Legendre wavelets solution has been reported for the Murray equation. The highest derivative in the differential equation is expanded into Legendre series, and this approximation is integrated while the boundary conditions are applied using integration constants. With the help of Legendre wavelets’ operational matrices, the Murray equation is converted into an algebraic system. Block-pulse functions are used to investigate the Legendre wavelets coefficient vectors of nonlinear terms. The convergence of the proposed method is proved. Finally, we have given a numerical example to demonstrate the validity and applicability of the method. Moreover, the use of proposed wavelet-based coupled method is found to be simple, efficient, less computation costs, and computationally attractive.

Fractional reaction–diffusion equations serve as more relevant models for studying complex patterns in several fields of nonlinear sciences. In this paper, we have developed the wavelet methods to find the approximate solutions for the Fitzhugh–Nagumo (FN) and fractional FN equations. The proposed method techniques provide the solutions in rapid convergence series with computable terms. To the best of our knowledge, until now there is no rigorous wavelet solutions have been reported for the FN and fractional FN equations arising in gene propagation and model. With the help of Laplace operator and Legendre wavelets operational matrices, the FN equation is converted into an algebraic system. Finally, we have given some numerical examples to demonstrate the validity and applicability of the wavelet methods. The power of the manageable method is confirmed. Moreover, the use of the wavelet methods is found to be accurate, efficient, simple, low computation costs and computationally attractive.

In this paper, the Homotopy analysis method (HAM) is employed to obtain the analytical/numerical solutions for linear and nonlinear Kolmogorov Petrovskii-Piskunov (KPP) and fractional KPP equations. The proposed method is a powerful and easy-to-use analytical tool for linear and nonlinear problems. This method contains the auxiliary parameter \(h\), which provides us with a simple way to adjust and control the convergence region of solution series. Some illustrative examples are presented. Moreover the use of HAM is found to be accurate, simple, convenient, flexible and computationally attractive.

As the available fresh water is finite on earth, its demand is increasing day by day. The distillation is one of the important methods of getting potable water from brackish and sea water using the free energy supply from the sun. The problem of solar stills is the low productivity. In this context, an experimental work is conducted on a single slope solar still in which the still basin was built with copper sheet. The rate of heat transfer to water in the still made up of Copper is more and hence the increase in the water temperature and productivity. The attempts are also made to increase the productivity of water by painting black coating inside the Copper still basin. The theoretical results agree well with the experimental ones.

JPEG2000 Part 2 supports the use of multicomponent transforms (MCTs) to decorrelate multicomponent images along the component direction. Such point transforms can be performed on arbitrary subsets of components, known as "component collections." These Part 2 extensions have been used for compressing 3-D images in applications such as medical imaging and remote sensing. It is widely believed that the MCT extensions are applicable only to 3-D data. In this letter, we demonstrate their use for compressing N-D datasets for any N ges 3.

In this paper, we have applied an efficient wavelet-based approximation method for solving the Fisher's type and the fractional Fisher's type equations arising in biological sciences. To the best of our knowledge, until now there is no rigorous wavelet solution has been addressed for the Fisher's and fractional Fisher's equations. The highest derivative in the differential equation is expanded into Legendre series; this approximation is integrated while the boundary conditions are applied using integration constants. With the help of Legendre wavelets operational matrices, the Fisher's equation and the fractional Fisher's equation are converted into a system of algebraic equations. Block-pulse functions are used to investigate the Legendre wavelets coefficient vectors of nonlinear terms. The convergence of the proposed methods is proved. Finally, we have given some numerical examples to demonstrate the validity and applicability of the method.

In this paper, we have established an efficient Legendre wavelet based approximation method to solve film-pore diffusion model arising in engineering. Film-pore diffusion model is widely used to determine study the kinetics of adsorption systems. The use of Legendre wavelet based approximation method is found to be accurate, simple, fast, flexible, convenient, and computationally attractive. It is shown that film-pore diffusion model satisfactorily describe kinetics of methylene blue adsorption onto the three low-cost adsorbents, Guava, teak and gulmohar plant leaf powders, used in this study.

Wavelet analysis is a recently developed mathematical tool in applied mathematics. A wavelet method for a class of space-time fractional Klein–Gordon equations with constant coefficients is proposed, by combining the Haar wavelet and operational matrix together and efficaciously dispersing the coefficients. The behavior of the solutions and the effects of different values of fractional order α are graphically shown. The fundamental idea of the Haar wavelet method is to convert the fractional Klein–Gordon equations into a group of algebraic equations, which involves a finite number of variables. The examples are given to demonstrate that the method is effective, fast, and flexible; in the meantime, it is found that the difficulties of using the Daubechies wavelets for solving the differential equation, which need to calculate the correlation coefficients, are avoided.

Remote visualization of volumetric images has gained importance over the past few years in medical and industrial applications. Volume visualization is a computationally intensive process, often requiring hardware acceleration to achieve a real time viewing experience. One remote visualization model that can accomplish this would transmit rendered images from a server, based on viewpoint requests from a client. For constrained server-client bandwidth, an efficient compression scheme is vital for transmitting high quality rendered images. In this paper, we present a new view compensation scheme that utilizes the geometric relationship between viewpoints to exploit the correlation between successive rendered images. The proposed method obviates motion estimation between rendered images, enabling significant reduction to the complexity of a compressor. Additionally, the view compensation scheme, in conjunction with JPEG2000 performs better than AVC, the state of the art video compression standard.

Remote visualization of volumetric data has gained importance over the past few years in order to realize the full potential of tele-radiology. Volume rendering is a computationally intensive process, often requiring hardware acceleration to achieve real time visualization. Hence a remote visualization model that is well-suited for high speed networks would be to transmit rendered images from the server (with dedicated hardware) based on view point requests from clients. In this regard, a compression scheme for the rendered images is vital for efficient utilization of the server-client bandwidth. Also, the complexity of the decompressor should be considered so that a low end client workstation can decode images at the desired frame rate. We present a scalable low complexity image coder that has good compression efficiency and high throughput.

Although magnetic resonance imaging (MRI) is routinely used in clinical practice, long acquisition times limit its practical utility in many applications. To increase the data acquisition speed of MRI, parallel MRI (pMRI) techniques have recently been proposed. These techniques utilize multi-channel receiver arrays and are based on simultaneous acquisition of data from multiple receiver coils. Recently, a novel framework called Compressed Sensing (CS) was introduced. Since this new framework illustrates how signals can be reconstructed from much fewer samples than suggested by the Nyquist theory, it has the potential to significantly accelerate data acquisition in MRI. This paper illustrates that CS and pMRI techniques can be combined and such joint processing yields results that are superior to those obtained from independent utilization of each technique.

Remote visualization of volumetric images has gained importance over the past few years in medical and industrial applications. Volume visualization is a computationally intensive process, often requiring hardware acceleration to achieve a real time viewing experience. One remote visualization model that can accomplish this would transmit rendered images (with dedicated hardware) from the server based on view-point requests from a client. For a given server-client bandwidth, an efficient compression scheme is vital for transmitting high quality rendered images. In this paper, we present a new compression scheme that utilizes the geometric relationship between view-points to exploit the correlation between successive rendered images. The proposed method performs better than AVC, the state of the art video compression standard. Additionally, our scheme obviates motion estimation between rendered images, enabling significant reduction to the complexity of the encoder.

Summary form only given. JPEG2000 Part 2 by Taubman, D.S. and Marcellin, M.W. (2002) supports the use of multi-component transforms (MCT) to decorrelate multi-component image data along the component direction. Three types of point transforms are allowed: linear block transforms, dependency transforms and wavelet transforms. Besides these transforms, Part 2 also allows grouping of arbitrary subsets of components into 'component collections.' Any of the three transform types can be used for each such collection. These Part 2 extensions have been used for compressing 3-D images in many applications such as medical, hyper-spectral and video. It is widely believed that the MCT extensions are applicable only to 3-D data. In this work, we demonstrate their use for compressing N-datasets for any N ges 3. Let (M1, M2,..., MN-2, X, Y) denote the dimensions (size) of an N-dataset. The data can then be interpreted as a collection of 2-D (X, Y) images (or components). The number of such components is T = pi k=1 N-2 Mk. Each component is indexed by an N - 2 dimensional vector: [m1,m2,...,mN-2] where mk isin {1, 2,..., Mk}. To compress an N-D dataset, the T components are processed by N - 2 stages of MCT. Specifically, each stage has T input components and produces T output components, which in turn are used as inputs to the next stage. Consider the kth (k isin {1, 2,..., N - 2}) stage of the MCT which is used to decorrelate the data along the k dimension. The input components are first reordered to form collections. A collection is formed by grouping all components having the same value of the N - 3 dimensional index [m1, m2,...,mk+1,...mN-2]. There are then T/Mk collections, each containing Mk components. For each collection, a point transform is applied, resulting in decorrelation of the data along the kth dimension. The output components of the final MCT stage (N - 2) are compressed using the 2-D wavelet transform and block coding of JPEG2000. It should be noted that the encoding procedure described above will result in a code-stream that is entirely compliant with JPEG2000 Part 2. Two major applications of multi-dimensional sources are medical images (fMRI, 4-D cardiac and ultrasound) and multi-view video. We have applied the proposed methodology to a 5-D (X, Y, slice, time, trial) fMRI dataset. A PSNR (Peak Signal to Noise Ratio) gain of 8 to 14.9 dB is observed at various bit rates, when 3 stages of MCT (5-D) are used as opposed to the usual single stage of MCT (3-D). The proposed scheme can support scalability in all N dimensions. Additionally, incremental processing can be used to achieve considerable savings in memory usage.

Many medical imaging techniques available today generate 4D data sets. One such technique is functionalmagnetic resonance imaging (fMRI) which aims to determine regions of the brain that are activated due tovarious cognitive and/or motor functions or sensory stimuli. These data sets often require substantial resourcesfor storage and transmission and hence call for ecient compression algorithms.fMRI data can be seen as a time-series of 3D images of the brain. Many dierent strategies can be employedfor compressing such data. One possibility is to treat each 2D slice independently. Alternatively, it is alsopossible to compress each 3D image independently. Such methods do not fully exploit the redundancy presentin 4D data. In this work, methods using 4D wavelet transforms are proposed. They are compared to dierent2D and 3D methods. The proposed schemes are based on JPEG2000, which is included in the DICOM standardas a transfer syntax. Methodologies to test the eects of lossy compression on the end result of fMRI analysisare introduced and used to compare dierent compression algorithms.Keywords: JPEG2000, fMRI, 4D-Compression.

In this paper, we present a framework to determine Haar solutions of Bratu-type equations that are widely applicable in fuel ignition of the combustion theory and heat transfer. The method is proposed by applying Haar series for the highest derivatives and integrate the series. Several examples are given to confirm the efficiency and the accuracy of the proposed algorithm. The results show that the proposed way is quite reasonable when compared to exact solution., {"references":["U.M. Ascher, R. Matheij, R.D. Russell, Numerical solution of boundary\nvalue problems for ordinary differential equations, SIAM, Philadelphia,\nPA, 1995.","J.P. Boyd, Chebyshev polynomial expansions for simultaneous approximation\nof two branches of a function with application to the onedimensional\nBratu equation, Applied Mathematics and Computation 142\n(2003) 189-200.","J.P. Boyd, An analytical and numerical study of the two-dimensional\nBratu equation, Journal of Scientific Computing 1 (2) (1986) 183-206.","R. Buckmire, Investigations of nonstandard Mickens-type finitedifference\nschemes for singular boundary value problems in cylindrical\nor spherical coordinates, Numerical Methods for partial Differential\nequations 19 (3) (2003) 380-398.","R. Buckmire, Application of Mickens finite-difference scheme to the\ncylindrical Bratu Gelfand problem, doi:10.1002/num.10093.","C.F.Chen, C.H.Hsiao, Haar wavelet method for solving lumped and\ndistributed- parameter systems, IEE Proc.Pt.D 144(1) (1997) 87-94.","D.A. Frank-Kamenetski, Diffusion and Heat Exchange in Chemical\nKinetics, Princeton University Press, Princeton, NJ, 1955.","I.H.A.H. Hassan, V.S. Erturk, Applying differential transformation\nmethod to the One-dimensional planar Bratu problem, International\nJournal of Contemporary Mathematical Sciences 2 (2007) 1493-1504.","Hikmet Caglar, Nazan Caglar, Mehmet zer, Antonios Valaristo, Amalia\nN. Miliou, Antonios N. Anagnostopoulos, Dynamics of the solution of\nBratu-s Equation, Nonlinear Analysis, (Press).\n[10] C.H.Hsiao, Haar wavelet approach to linear stiff systems, Mathematics\nand Computers in simultion ,Vol 64, 2004, pp.561-567.\n[11] J. Jacobson, K. Schmitt, The Liouville-Bratu-Gelfand problem for radial\noperators, Journal of Differential Equations 184 (2002) 283-298.\n[12] U.Lepik, Numerical solution of evolution equations by the Haar wavelet\nmethod, Applied Mathematics and Computation 185 (2007) 695-704.\n[13] U.Lepik, Numerical solution of differential equations using Haar\nwavelets, Mathematics and Computers in Simulation 68 (2005) 127-143.\n[14] S. Li, S.J. Liao, An analytic approach to solve multiple solutions of a\nstrongly nonlinear problem, Applied Mathematics and Computation 169\n(2005) 854-865.\n[15] A.S. Mounim, B.M. de Dormale, From the fitting techniques to accurate\nschemes for the Liouville Bratu Gelfand problem, Numerical Methods\nfor Partial Differential Equations, doi: 10.1002/num.20116.\n[16] M.I. Syam, A. Hamdan, An efficient method for solving Bratu equations,\nApplied Mathematics and Computation 176 (2006) 704-713.\n[17] A.M. Wazwaz, A new method for solving singular initial value problems\nin the second order differential equations, Applied Mathematics and\nComputation 128 (2002) 47-57.\n[18] A.M. Wazwaz, Adomian decomposition method for a reliable treatment\nof the Bratu-type equations, Applied Mathematics and Computation 166\n(2005) 652-663."]}