Skip to main content
main-content

Über dieses Buch

Fractals and wavelets are emerging areas of mathematics with many common factors which can be used to develop new technologies. This volume contains the selected contributions from the lectures and plenary and invited talks given at the International Workshop and Conference on Fractals and Wavelets held at Rajagiri School of Engineering and Technology, India from November 9-12, 2013. Written by experts, the contributions hope to inspire and motivate researchers working in this area. They provide more insight into the areas of fractals, self similarity, iterated function systems, wavelets and the applications of both fractals and wavelets. This volume will be useful for the beginners as well as experts in the fields of fractals and wavelets.

Inhaltsverzeichnis

Frontmatter

Fractal Theory

Frontmatter

Introduction to Fractals

This non-technical introduction tries to place fractal geometry into the development of contemporary mathematics. Fractals were introduced by Mandelbrot to model irregular phenomena in nature. Many of them were known before as mathematical counterexamples. The essential model assumption is self-similarity which makes it possible to describe fractals by parameters which are called dimensions or exponents. Most fractals are constructed from dynamical systems. Measures and probability theory play an important part in the study of fractals.

Christoph Bandt

Geometry of Self-similar Sets

Self-similar sets form a well-defined class of fractals which are relatively easy to study. This talk introduces their main features with a lot of examples. We explain the need of a separation condition for the tangential structure. Hausdorff measure is the natural concept of volume. Under certain conditions Hausdorff measures define also the “surface” of the boundary and the interior distance. A number of open problems are mentioned.

Christoph Bandt

An Introduction to Julia and Fatou Sets

We give an elementary introduction to the holomorphic dynamics of mappings on the Riemann sphere, with a focus on Julia and Fatou sets. Our main emphasis is on the dynamics of polynomials, especially quadratic polynomials.

Scott Sutherland

Parameter Planes for Complex Analytic Maps

In this paper we describe the structure of the parameter planes for certain families of complex analytic functions. These families include the quadratic polynomials

z

2

+

c

, the exponentials

λ

exp(

z

), and the family of rational maps

z

n

+

λ

z

n

$$z^{n} +\lambda /z^{n}$$

.These are, in a sense, the simplest polynomial, transcendental, and rational families, as each has essentially one critical orbit.

In this paper we give a brief overview of the structure of the parameter plane for three different families of complex analytic maps, namely quadratic polynomials (the Mandelbrot set), singularly perturbed rational maps, and the exponential family. The goal is to show how these objects allow us to understand almost completely the different dynamical behaviors that arise in these families as well as the accompanying bifurcations.

Robert L. Devaney

Measure Preserving Fractal Homeomorphisms

The basic theory of fractal transformations is recalled. For a fractal homeomorphism generated by a pair of affine iterated function systems (IFSs), a condition under which the transformation is measure (i.e. area, volume, etc.) preserving is established. Then three families of fractal homeomorphisms, two of them entirely new, generated by pairs of affine IFSs, are introduced. It is proved that they admit subfamilies that preserve

n

-dimensional Lebesque measure, where

n

is 2 or 3. Several examples are illustrated and applications to computer aided design and manufacture, via three-dimensional printing, are envisaged.

Michael F. Barnsley, Brendan Harding, Miroslav Rypka

The Dimension Theory of Almost Self-affine Sets and Measures

A self-affine IFS

=

f

i

(

x

)

=

A

i

x

+

t

i

i

=

1

m

$$\mathcal{F} = \left \{f_{i}(x) = A_{i}x + t_{i}\right \}_{i=1}^{m}$$

is a finite list of contracting affine maps on

d

$$\mathbb{R}^{d}$$

, for some

d

 ≥ 1. The attractor of

$$\mathcal{F}$$

is

0.1

Λ

=

n

=

1

i

1

,

,

i

n

f

i

1

f

i

n

(

B

)

,

$$\displaystyle{ \Lambda =\bigcap _{ n=1}^{\infty }\bigcap _{ i_{1},\ldots,i_{n}}f_{i_{1}} \circ \cdots \circ f_{i_{n}}(B), }$$

where

B

is a sufficiently large ball centered at the origin. In most cases we cannot compute the dimension of

Λ

$$\Lambda $$

. However, if we add an independent additive random error to each

f

i

k

$$f_{i_{k}}$$

in (0.1) then the dimension of this random perturbation (called almost self-affine system) is almost surely the so-called affinity dimension of the original deterministic system. The dimension theory of almost self-affine sets and measures were described in Jordan et al. (Commun. Math. Phys. 270(2):519–544, 2007). The multifractal analysis of almost self-affine measures has been studied in some recent papers (Falconer, Nonlinearity 23:1047–1069, 2010; Barral and Feng, Commun. Math. Phys. 318(2):473–504, 2013). In the second part of this note I give a survey of this field but first we review some results related to the dimension theory of self-affine sets.

Károly Simon

Countable Alphabet Non-autonomous Self-affine Sets

We extend Falconer’s formula from Falconer (Math. Proc. Camb. Philos. Soc. 103:339–350, 1988) by identifying the Hausdorff dimension of the limit sets of almost all contracting affine iterated function systems to the case of an infinite alphabet, non-autonomous choice of iterating matrices, and time-dependent random choice of translations.

Mariusz Urbański

On Transverse Hyperplanes to Self-similar Jordan Arcs

We consider self-similar Jordan arcs

γ

in

d

$$\mathbb{R}^{d}$$

, different from a line segment and show that they cannot be projected to a line bijectively. Moreover, we show that the set of points

x

 ∈ 

γ

, for which there is a hyperplane, intersecting

γ

at the point

x

only, is nowhere dense in

γ

.

Andrey Tetenov

Fractals in Product Fuzzy Metric Space

The purpose of this paper is to prove the Hutchinson–Barnsley operator on the product fuzzy metric space is fuzzy B-contraction. We also present the fuzzy B-contraction properties of HB operator in product fuzzy metric space. The notion of product fuzzy fractal is introduced in product fuzzy metric space in the sense of the fuzzy B-contraction.

R. Uthayakumar, A. Gowrisankar

Some Properties on Koch Curve

Many physical problems on fractal domains lead to nonlinear models involving reaction–diffusion equations, problems on elastic fractal media or fluid flow through fractal regions, etc. The prevalence of fractal-like objects in nature has led both mathematicians and physicists to study various processes on fractals. In recent years there has been an increasing interest in studying nonlinear partial differential equations on fractals, also motivated and stimulated by the considerable amount of literature devoted to the definition of a Laplace-type operator for functions on fractal domains. The energy of a function defined on a post critically finite (p.c.f) self-similar fractal can be written as a sum of directional energies. A general concept of graph energy defined on a finite connected graph is given. A work about the graph energy is mainly concerned on a Koch curve. First graphs on this Koch curve are built. These graphs produced from the initial graph by iteration repeatedly. Find the energy renormalization constant. Second we find the non-normalized and Normalized Laplacian of a Koch Curve. With the help of this we examine the Laplacian Renormalization constant and forbidden eigenvalues. Finally we develop the Spectral decimation function of Koch Curve.

R. Uthayakumar, A. Nalayini Devi

Projections of Mandelbrot Percolation in Higher Dimensions

We consider the fractal percolations which are one of the most well-studied examples of random Cantor sets. Rams and Simon studied the projections of fractal percolation sets on the plane. We extend the scope of their theorem and generalize it to higher dimensions. An extended version of this note is avaible on the arxiv [

7

].

Károly Simon, Lajos Vágó

Some Examples of Finite Type Fractals in Three-Dimensional Space

By choosing the contraction functions in the Iterated Function System we extend the construction from two-dimensional space to three-dimensional space to build self-similar sets in 3-space. We also extend the neighbor map concept to Iterated Function Systems with different contraction factors in order to identify examples with finite type. Some interesting examples of self-similar sets in three-dimensional space are given.

Mai The Duy

Fractals in Partial Metric Spaces

Partial metric space is a generalisation of metric space due to non zero self-distance. In this paper, we discuss the nature of fractals in a partial metric space.

S. Minirani, Sunil Mathew

Wavelet Theory

Frontmatter

Frames and Extension Problems I

In this article we present a short survey of frame theory in Hilbert spaces. We discuss Gabor frames and wavelet frames and set the stage for a discussion of various extension principles; this will be presented in the article

Frames and extension problems II

(joint with H.O. Kim and R.Y. Kim).

Ole Christensen

Frames and Extension Problems II

This article is a follow-up on the article

Frames and Extension Problems I.

Here we will go into more recent progress on the topic and also present some open problems.

Ole Christensen, Hong Oh Kim, Rae Young Kim

Local Fractal Functions and Function Spaces

We introduce local iterated function systems (IFSs) and present some of their basic properties. A new class of local attractors of local IFSs, namely local fractal functions, is constructed. We derive formulas so that these local fractal functions become elements of various function spaces, such as the Lebesgue spaces

L

p

, the smoothness spaces

C

n

, the homogeneous Hölder spaces

Ċ

s

$$\dot{C}^{s}$$

, and the Sobolev spaces

W

m

, 

p

.

Peter R. Massopust

Some Historical Precedents of the Fractal Functions

With this short text, we want to pay tribute to the scientists of older generations who, through their research, led to the current state of the knowledge of the fractal functions. We review the fundamental milestones of the origin and evolution of self-similar curves that, in some cases, agree with continuous and nowhere differentiable functions, but they are not exhausted by them. Our main interest is to emphasize the lesser known examples, due to a deficient or late publication (Bolzano’s map for instance). We describe different ways of defining self-similar curves. We recall the first functions without tangent, but also some fractal functions having derivative almost everywhere. Most of the models studied may seem quite paradoxical (“monsters” in the words of poincaré) as, for instance, curves with a fractal dimension of two and having tangent at every point. These instances suggest that the classification and even the defintion of fractal functions are far from being established.

M. A. Navascués, M. V. Sebastián

A New Class of Rational Quadratic Fractal Functions with Positive Shape Preservation

Fractal interpolation functions (FIFs) developed through iterated function systems prove more general than their classical counterparts. However, the theory of fractal interpolation functions in the domain of

shape preserving interpolation

is not fully explored. In this paper, we introduce a new kind of iterated function system (IFS) involving rational functions of the form

p

n

(

x

)

q

n

(

x

)

$$\frac{p_{n}(x)} {q_{n}(x)}$$

, where

p

n

(

x

) are quadratic polynomials determined through the interpolation conditions of the corresponding FIF and

q

n

(

x

) are preassigned quadratic polynomials involving one free shape parameter. The presence of the scaling factors in our rational FIF adds a layer of flexibility to its classical counterpart and provides fractality in the derivative of the interpolant. The uniform convergence of the rational quadratic FIF to the original data generating function is established. Suitable conditions on the rational IFS parameters are developed so that the corresponding rational quadratic fractal interpolant inherits the positivity property of the given data.

A. K. B. Chand, P. Viswanathan, M. A. Navascués

Interval Wavelet Sets Determined by Points on the Circle

Having observed that an interval wavelet set corresponds to the points in a circle, we obtain points in the circle which characterize two-interval wavelet sets and also those points which characterize three-interval wavelet sets for dilation

d

 ≥ 2. Further points in the circle characterizing one-interval and two-interval

H

2

-wavelet sets for dilation

d

 ≥ 2 are obtained. In addition, we discuss three-interval wavelet sets of

$$\mathbb{R}$$

in respect of being associated with a multiresolution analysis (MRA).

Divya Singh

Inverse Representation Theorem for Matrix Polynomials and Multiscaling Functions

Wavelet analysis provides suitable bases for the class of

L

2

functions. The function to be represented is approximated at different resolutions. The desirable properties of a basis are orthogonality, compact supportedness and symmetricity. In the scalar case, the only wavelet with these properties is Haar wavelet. Theory of multiwavelets assumes significance since it offers symmetric, compactly supported, orthogonal bases for

L

2

(

R

). The properties of a multiwavelet are determined by the corresponding Multiscaling Function. A multiscaling function is characterized by its symbol function which is a matrix polynomial in complex exponential. The inverse representation theorem of matrix polynomials provides a method to construct a matrix polynomial from its Jordan pair. Our objective is to find the properties that characterize a Jordan pair of a symbol function of a multiscaling function with desirable properties.

M. Mubeen, V. Narayanan

A Remark on Reconstruction of Splines from Their Local Weighted Average Samples

In this paper, we study the reconstruction of cardinal spline functions from their weighted local average samples

y

n

=

f

h

(

n

)

,

n

$$y_{n} = f \star h(n),n \in \mathbb{Z}$$

, where the weight function

h

(

t

) has support in

[

1

2

,

1

2

]

$$[-\frac{1} {2}, \frac{1} {2}]$$

. We prove that there exists a unique solution for the following problem: for the given data

y

n

and given degree

d

, find a cardinal spline

f

(

t

) of degree d satisfying

y

n

=

f

h

(

n

)

,

n

$$y_{n} = f \star h(n),n \in \mathbb{Z}$$

.

P. Devaraj, S. Yugesh

?? 1 $$\mathcal{C}^{1}$$ -Rational Cubic Fractal Interpolation Surface Using Functional Values

Fractal interpolation is a modern technique for fitting of smooth/non-smooth data. In the present article, we develop the

??

1

$$ \mathcal{C}^{1} $$

-rational cubic fractal interpolation surface (FIS) as a fixed point of the Read-Bajraktarevi

ć

$$ \acute {c} $$

(RB) operator defined on a suitable function space. Our

??

1

$$ \mathcal{C}^{1} $$

-rational cubic FIS is effective tool to stich surface data arranged on a rectangular grid. Our construction needs only the functional values at the grids being interpolated, therefore implementation is an easy task. We first construct the

x

-direction rational cubic FIFs (

x

-direction fractal boundary curves) to approximate the data generating function along the grid lines parallel to

x

-axis. Then we form a rational cubic FIS as a blending of these fractal boundary curves. An upper bound of the uniform distance between the rational cubic FIS and an original function is estimated for the convergence results. A numerical illustration is provided to explain the visual quality of our rational cubic FIS. An extra feature of this fractal surface scheme is that it allows subsequent interactive alteration of the shape of the surface by changing the scaling factors and shape parameters.

A. K. B. Chand, N. Vijender

On Fractal Rational Functions

This article introduces fractal perturbation of classical rational functions via

α

-fractal operator and investigates some aspects of this new function class, namely, the class of fractal rational functions. Its specific aims are: (1) to define the fractal rational functions along the lines of the fractal polynomials (2) to extend the Weierstrass theorem of uniform approximation to fractal rational functions (3) to deduce a fractal version of the classical Müntz theorem on rational functions (4) to prove the existence of a Schauder basis for

??

(

I

)

$$\mathcal{C}(I)$$

consisting of fractal rational functions.

P. Viswanathan, A. K. B. Chand

Applications of Fractals and Wavelets

Frontmatter

Innovation on the Tortuous Path: Fractal Electronics

We describe the innovation environment that has limited the onset of innovations in technology, choosing as an example the use of fractals in electronics. There is a considerable time delay between the production of these innovations and their widespread implementation with end-users; this delay is typical in innovation; underlying and predictable pitfalls in the adoption process are inherent to the progress of innovation through a “tortuous path.” The paradox of rapid adoption in pure science and math versus delay in practical applications should be considered a normal aspect of innovation in fractal technology.

Nathan Cohen

Permutation Entropy Analysis of EEG of Mild Cognitive Impairment Patients During Memory Activation Task

Permutation Entropy (PE) statistic is a measure of self-similarity of the time series estimated from its ordinal patterns. This measure is used to detect the dynamical differences between patients with mild cognitive impairment (MCI) and normal controls. The comparison of PE values of Electroencephalograph (EEG) of the two groups in the resting eyes closed (EC) state and the short-term memory task (STM) state reveals altered efficiency of the different lobes of MCI brain in the compensational dynamical mechanism for task management. In resting EC state, PE values of MCI group is significantly (

p∠

0. 05) lower than that of controls in the frontal, temporal, and anterior parietal regions. In the STM task state, entropy levels of MCI group are significantly (

p∠

0. 05) lower than that of controls in the frontal region and the left parietal region. These findings suggest that nonlinear analysis of EEG using PE can provide important information about EEG characteristic of cognitively impaired condition that can lead to Alzheimer’s Disease(AD).

Leena T. Timothy, Bindu M. Krishna, Murali Krishna Menon, Usha Nair

A Multifractal-Based Image Analysis for Cervical Dysplasia Classification

This paper presents a study on microscopic images to classify cervical precancers by a multifractal analysis. Since internal structure of tissue is non-deterministic, multifractal spectrum is required to characterize such structure. The periodic structure of collagen present in the stromal region of cervical tissue gets disordered with progress in grade of dysplasia. This disorder is investigated through the multifractal study, enabling us to discriminate between normal and abnormal human cervical tissue sections. Holder exponent classifies normal from abnormal dysplasia by capturing local irregularities present in the image. While mean of Hausdorff–Besicovich dimension which describes global regularity are used to classify various grades of dysplasia. The box-counting method is used to estimate the fractal dimension. The results show, remarkably, the classification feature of multifractal analysis.

P. Singh, J. Jagtap, C. Pantola, A. Agarwal, A. Pradhan

Self-Similar Network Traffic Modelling Using Fractal Point Process-Markovian Approach

Several recent Internet traffic measurement studies reported that traffic in modern high-speed networks is a self-similar process. If the stochastic self-similar network traffic models do not accurately represent the real traffic, then the network performance may be over estimated or underestimated, and it causes degradation of Internet router performance. Therefore, it is decisive for an appropriate design of a router. In this paper, we investigate mean waiting time and tail probability of network router with pseudo self-similar traffic input. We use Fractal Point Process (FPP) as input process since it emulates self-similar traffic. However, this process is asymptotic in nature and has less effective in queueing based performance analysis. Therefore, for queueing analysis Markov modulated Poisson process (MMPP) is fitted for FPP. FPP involves another parameter Fractal Onset Time (FOT) besides Hurst parameter. Effect of FOT on tail probability and mean waiting time is examined.

Rajaiah Dasari, Ramesh Renikunta, Malla Reddy Perati

Validation of Variance Based Fitting for Self-similar Network Traffic

Most of the classical self-similar traffic models are asymptotic in nature. Hence, they are not suitable for queuing based performance evaluation. In this paper, we have validated further fitting method of CMMPP emulating self-similar traffic by means of IDC. We conclude from the numerical examples that self-similar traffic can be well represented by the proposed model.

Ramesh Renikunta, Rajaiah Dasari, Ranadheer Donthi, Malla Reddy Perati

Self-Similar Network Traffic Modeling Using Circulant Markov Modulated Poisson Process

Most of the classical self-similar traffic models are asymptotic in nature. Hence, they are not suitable for queuing based performance evaluation. In this paper, we propose a model for self-similar traffic using Circulant Markov modulated Poisson process (CMMPP). This model is to match the variance of self-similar traffic and that of CMMPP over a time-scale. The resultant CMMPP consists of several two-state CMMPPs. We conclude from the numerical examples that self-similar traffic can be well represented by the proposed model.

Ranadheer Donthi, Ramesh Renikunta, Rajaiah Dasari, Malla Reddy Perati

Investigation of Priority Based Optical Packet Switch Under Self-Similar Variable Length Input Traffic Using Matrix Queueing Theory

In this paper, queueing behavior of the optical packet switch (

OPS

) employing priority based partial buffer sharing (

PBS

) mechanism under asynchronous self-similar variable length packet input traffic is investigated. Markov modulated Poisson process (

MMPP

) emulating self-similar traffic is used as input process. In view of wavelength division multiplexing (

WDM

)

OPS

output port of switch is modeled as multi-server (

MMPP∕M∕c∕K

) queueing system. Service times (packet lengths) are assumed to be exponential distributed as traffic under consideration is unslotted asynchronous. Performance measures, namely, high priority packet loss probability and low priority packet loss probability against the system parameters and traffic parameters are computed by means of matrix-geometric solutions and approximate Markovian model. This kind of analysis is useful in dimensioning the switch employing

PBS

mechanism under self-similar variable length packet input traffic and to provide differentiated services (

DiffServ

) and quality of service (

QoS

) guarantee.

Ravi Kumar Gudimalla, Malla Reddy Perati

Computationally Efficient Wavelet Domain Solver for Florescence Diffuse Optical Tomography

Estrogen induced proliferation of mutant cells is a growth signal hallmark of breast cancer. Fluorescent molecule that can tag Estrogen Receptor (ER) can be effectively used for detecting cancerous tissue at an early stage. A novel target-specific NIRf dye conjugate aimed at measuring ER status was synthesized by ester formation between 17-

β

estradiol and a hydrophilic derivative of ICG, cyanine dye, bis-1,1-(4-sulfobutyl) indotricarbocyanine-5-carboxylic acid, sodium salt. In-vitro studies provided specific binding on ER+ve [MCF-7] cells clearly indicating nuclear localization of the dye for ER+ve as compared to plasma level staining for MDA-MB-231. Furthermore, cancer prone cells showed 4.5-fold increase in fluorescence signal intensity compared to control. A model of breast phantom was simulated to study the in-vivo efficiency of dye with the parameters of dye obtained from photo-physical and in-vitro studies. The excitation (754 nm) and emission (787 nm) equation are solved independently using parallel processing strategies. The results were obtained by carrying out wavelet transformation on forward and the inverse data sets. An improvisation of the Information content of system matrix was suggested in wavelet domain. The inverse problem was addressed using Levenberg–Marquardt (LM) procedure with the minimization of objective function using Tikhonov approach. The multi resolution property of wavelet transform was explored in reducing error and increasing computational efficiency. Our results were compared with the single resolution approach on various parameters like computational time, error function, and Normalized Root Mean Square (NRMS) error. A model with background absorption coefficient of 0.01 mm-1 with anomalies of 0.02 mm-1 with constant reduced scattering of 2.0 mm for different concentration of dye was compared in the result. The reconstructed optical properties were in concurrence with the tissue property at 787 nm. We intend our future plans on in-vivo study on developing a complete instrumentation for imaging a target specific lipophilic dye.

K. J. Francis, I. Jose

Implementation of Wavelet Based and Discrete Cosine Based Algorithm on Panchromatic Image

In the past few years, there has been a tremendous increase in the need for the amount of information stored in the form of images especially from Remote Sensing Satellites. Recently there has been a exponentially conversion of conventional analog images to digital images. The volume of digitized image being very high will considerably slow down the transmission and storage of such images. Therefore there is strong need of compression of the images by extracting the visible elements which are encoded and transmitted. This paper compares different image compression techniques such as JPEG (Joint Picture Expert Group), JPEG2000 (Joint Picture Expert Group-2000), and SPIHT (Set Partitioning in Hierarchical Tree) using a set of objective picture quality measures like Peak Signal to Noise Ratio (PSNR) and Mean Square Error (MSE) have been used to measure the picture quality and comparison has been done based upon the results of these quality measures. Standard test images were assessed with different compression ratios. It is found that the JPEG2000 based compression has achieved better results as compared to SPIHT and JPEG for all compressions and images were produced showing better image quality.

Jyoti Sarup, Jyoti Bharti, Arpita Baronia

Trend, Time Series, and Wavelet Analysis of River Water Dynamics

Time series, trend, wavelet and statistical analysis of water quality parameters Chemical Oxygen Demand (COD), Biochemical Oxygen Demand (BOD), Dissolved Oxygen (DO) monitored for river Yamuna in India have been studied. It is observed that COD is highly correlated with BOD. For all auto regressive integrated moving average model (p,d,q) value of “d,” i.e. middle value is zero thus process is stationary. It is also observed that RMSE values are comparatively very low, thus dependent series is closed with the model predicted level. MAPE, MaxAPE, MAE, MaxAE, Normalized BIC are calculated and have low value for all parameters. Trend is calculated by using auto correlation function, partial auto correlation function, and lag. Thus the predictive model is useful at 95 % confidence limits. 1-D discrete and continuous Daubechies Wavelet analysis explains that the parameters COD, BOD, DO have the maximum value 120, 50, 8 and amplitude (a5) varies between 52 to 78, 10 to 30, 0.2 to 1.4, respectively. The scale values of Db5, i.e. d5, d4, d3, d2, and d1 range between − 20 and + 20 for all parameters. All parameters cross the prescribed limits of WHO/EPA, thus water is not fit for drinking, agriculture, and industrial use.

Kulwinder Singh Parmar, Rashmi Bhardwaj

An Efficient Wavelet Based Approximation Method to Film-Pore Diffusion Model Arising in Chemical Engineering

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

Pandy Pirabaharan, R. David Chandrakumar, G. Hariharan

A New Wavelet-Based Hybrid Method for Fisher Type Equation

In this paper, we have introduced a new wavelet-based hybrid method for solving the Fisher’s type equations. To the best of our knowledge, until now there is no rigorous wavelet solution has been addressed for the Fisher’s equations. With the help of wavelets operational matrices, the Fisher’s equations are converted into a system of algebraic equations. Some numerical examples are presented to demonstrate the validity and applicability of the method.

R. Rajaram, G. Hariharan
Weitere Informationen

Premium Partner

BranchenIndex Online

Die B2B-Firmensuche für Industrie und Wirtschaft: Kostenfrei in Firmenprofilen nach Lieferanten, Herstellern, Dienstleistern und Händlern recherchieren.

Whitepaper

- ANZEIGE -

Best Practices für die Mitarbeiter-Partizipation in der Produktentwicklung

Unternehmen haben das Innovationspotenzial der eigenen Mitarbeiter auch außerhalb der F&E-Abteilung erkannt. Viele Initiativen zur Partizipation scheitern in der Praxis jedoch häufig. Lesen Sie hier  - basierend auf einer qualitativ-explorativen Expertenstudie - mehr über die wesentlichen Problemfelder der mitarbeiterzentrierten Produktentwicklung und profitieren Sie von konkreten Handlungsempfehlungen aus der Praxis.
Jetzt gratis downloaden!

Bildnachweise