CRIME DETECTION MODELLING BY USING NUMERICAL METHODS

 CRIME DETECTION MODELLING BY USING NUMERICAL METHODS

1 CRIME:

 

In most countries the detection of crime is the responsibility of the police though special law enforcement agencies may be responsible for the discovery of the particular type of crime.

 

Example: Customs departments may be charged with computing smuggling and related offenses.

 

Crime detection falls in to three distinguishable phases the discovery that a crime has been committed the identification of a suspect and the collection of sufficient evidence to indict the suspect before a court.

 

Example: Electronic eavesdropping surveillance interception of communication and infiltration of gangs.

Fig: 1 FLOWCHAT OF CRIME DEFELATION

1.3.  STEPS OF CRIMINAL CASE:

 

The 5 basic steps of a criminal proceeding are

 

·         Arrest

·         Preliminary hearing

·         Grand jury investigation

·         Arraignment in criminal court

·         Trial by jury

 

1.4.  CAUSES OF CRIME:

 

·         Poverty this is perhaps one of the most concrete reasons why people

·         commit crime

·         Peer pressure new form of concern in the modern world

·         Drugs always been highly criticized by critics

·         Politics

·         Religion ,family conditions, the society ,unemployment.

 

1.5.  THE CRIME SCENE SKETCH:

 

·         Measure and outline area

·         North should be at the top of the paper

·         Determine the scale

·         Take the longest measurement at the scene and divide it by the longest measurement of the paper used for the Sketching.

Ø  1/2″ = 1 small rooms.

Ø  1/8″ = 1 very large rooms.

Ø  1/8″ = 10 large land area.

 

 

  2. CRIME PREVENTION AND TECHNIQUES

 

2.1.  INVOLVES:

 

Recognition.

                    ↓

 

Identification

 

↓

 

Individualization

 

                     ↓

 

Reconstruction

 

2.1.1  RECOGNITION:

 

•  Scene survey documentation collection

 

2.1.2  IDENTIFICATION:

 

•  Classification of evidence

 

2.1.3  INDIVIDUATION:

 

•  Comparison testing evaluation and interpretation

 

2.1.4  RECONSTRUCTION:

 

•  Sequencing events reporting and presenting

2.2.  THE CRIME SCENE SKETCH:

 

•         Measure and outline area

•         It doesn’t stretch

•         Use conventional units of measurements

•         Inches

•         Feet

•         Centimeters

•         Meters

 

2.3.  CRIME SCENE RECONSTRUCTION:

 

•         Data collection

•         Conjecture

•         Hypothesis formulation

•         Testing

•         Theory

 

3.FLOWCHART AND CRIME NOTES

 

The location of possible targets homes people as well as general environmental goes like levels of disorder housing density and surveillance researches also include knowledge offenders might have about target vulnerability for instance tricks learned from burglarizing nearby and what they found was two kinds of crime hotspots.

 

Supercritical which formed from a rapid chain reaction of law less ness and subcritical a large spike in a crime in an otherwise stable area in a supercritical situation small spikes in crime grow and spread when increased police force is applied new hotspots.

But off existing once they reform around small spikes in a crime to continue away from areas of police presence the crime is displaced police suppression is a cat and mouse game something very different happens when police are added to a sub critical hots

The surrounding are relatively stable so one suppression is applied the hotpots is reduced or eradicated completely and even when suppression efforts are removed the hotpots doesn’t really. Since strategy won’t work for both kinds of hotpots.

 

This key difference may help law enforcement tailor their crime strategies to different neighborliness ultimately thought hotpots maybe similar sizes and distribution

 

The research shows that looking at the dynamic of crime is important in gauging the best police response and that could be a fundamental step towards better prediction and crime prevention strategies and an even safer future for cities.

4.  NOTES OF CRIME:

 

Imagine someone is committing crime throughout local area on a regular basis may be they have committed 5 crimes in the last month at these locations how could we go about catching this person.

One thing we can do is try finding a patterns in crime location in order to make a prediction about the next one that can be tricky through since there likely a huge elements of randomness but decades ago Kim Ross mo. a PhD criminologist had another idea he tried to find a formula instead could find.

Where the criminal likely lives based on the past data he knows that criminals often don’t commit crimes right by their own home but also they don’t go too far away so from the data you can determine a quote hot zone fig★

Fig ★ HOT ZONE

Which is not too close or too far from the crime scene it has a high probability of the person living there this is his equation for determining those probabilities. I know it looks complex but its actually not as bad as you think like take this P_{i, j} parts.

To see the crime scene in map and put a gird over it any given squares will be in some row we will label i and column will label j, P_{i, j} is probability that the criminal lives in the square. How you calculate that value for any square is with this right side of the equation.

Formula for crime detection:

                      X function axis of the co-ordinates distance . Y= function axis of the co-ordinates distance . Where,

(| X_i-x_n|+|Y_j -y_n|)f → 1 𝑠𝑡 𝑡𝑒𝑟m

 

(2𝐵| X_i-x_n|+|Y_j -y_n|) g → 2 𝑛𝑑 𝑡𝑒𝑟𝑚

 

Now take this denominator equation, (|X_i-x_n |+|Y_j -y_n|) f this 1^stpart just means take the arbitrary grid you are analysing and one of the crime scene and subtract these x corps donets which gives you this distance.

 

The absolute value just ensure that its positive then |Y_j -y_n| parts just says do the same thing with the y axis which gives us this length add those up and we get the distance between the gird and the crime scene no it’s not the straight – line distance.

 

If you cannot move to agonal and as we saw this term is in the denominator which means as that distance goes up the entire fraction and thus the probability go down this is expected because like .I said these criminals usually don’t go super far to commit a crime so a larger distance between our square and the crime scene means a smaller Probability the criminal lies.

 

There at least for this laughter but remember criminals don’t commit these crime close to this homes either and that where this side of the equation comes.

 

 

You have that same distance down here subtracted from something known as a buffer zone which is just a 2B constant determined by what works best with known data or past crime so as

distance goes up the entire denominator actually decreasing making the whole fraction.

 

Probability go up so physically if you are too close to the crime scene probability is low that the criminal lives there but as you get further the probability increase that is of course until you pass a certain point which is where the left side of the formula comes in again after the distance and these 2 fractions change in opposite ways which is essentially.

 

The balancing act that creeps the hot zone of high probability that isn’t too close and isn’t too far from the crime scene for is sort of a constant that I am not going to go in to ∅ _{i j} and that g, f and B are constant. That just make certain parts of this equation matter more than other or they add more weights to certain parts then lastly this part.

 

We calculated these 2 fractions for every single crime committed and add the results do this for square on our gird and we create a heat map of probability. Actually is the equation output based on real crimes of a serial killer from the 70s named Richard chase you can see the crime locations in green and the formula predicts his residence to be somewhere in this dark region his actual residence is plotted here in purple exactly as expected so that least in the case Rasco’s formula works. 

 5.   CRIME SCENE CONTROL

The actions which the first arriving officer at the crime scene takes to make sure that the integrity of the scene maintained. Control also includes preventing people at the scene from becoming combatants and separating witnesses.

5.1  DRAWBACKS:

·         No bobby on the beat

·         Local police officer in the local community

·         Large amount of data but is there a significant impact on detecting and reducing crime overal

 

 

5.2  BENEFITS:

·         Administrative duties completed more quickly

·         Tools available to help track down criminals

·         Communication improved

·         Evidence collection tracking analysis and availability improved

·        

Amount of data available.

5.3  SCALING THE INVESTIGATION TO THE EVENT:

·         The crime scene must be secured preserved and recorded until evidence is collected

·         Existing contamination must be considered and recorded

·         Cross contamination must be prevented

·         Exhibits must be identified preserved collected and secured to pressure the chain of continuity

5.4  CRIME SCENE ANALYSIS PROCESS:

·         Crime analysts study crime reports arrests reports and police cells for service calls for service to identify emerging patterns series and trends as quickly as possible.

·         The analyse these phenomena for all relevant factors sometimes predict or forecast future occurrence and issue bulletins reports and alert to their agencies.

5.5  CRIME SCENE CONTROL:

 

·         The actions which the first arriving officer at the crime scene takes to make sure that the integrity of the scene is maintained.

·         Control also includes preventing people at the scene from becoming combatants and separating witnesses.

 

5.6.1  CRIME SCENE MAPPING:

 

·         Triangulation- uses to two points at the crime scene to map each piece of evidence

·         Coordinate or grid – divides the crime scene into squares for mapping

·         Suspended polar coordinate-for use in mapping evidence in a hole

·         Baseline- set a north or south line and measure each piece of evidence from this line.

 

5.6.2  MICROSCOPIC CRIME SCENE:

 

·         Based on the size

·         Trace evidence

·         Gunshot residue

·         Tire treads

·         Hair samples

·         Mites found in clothes

 

5.6.3  CRIME SCENE DOCUMENTATION:

 

·         Take notes at the crime scene

·         Videotape the crime scene

·         Photograph the crime scene

 

MOU ACTIVITY on RAPID MATHS Vs COMPETITIVE EXAMS Success At Your Door Steps

 

MARUDHAR KESARI JAIN COLLEGE FOR WOMEN, VANIYAMBADI

PG AND RESEARCH DEPARTMENT OF MATHEMATICS,

DEPARTMENT OF STATISTICS & LEE ACADEMY, TIRUPATTUR

Jointly Organizes

MOU ACTIVITY

on

RAPID MATHS Vs COMPETITIVE EXAMS

Success At Your Door Steps

Date: 23.04.2022

Elements of a mathematical model

 

Elements of a mathematical model

Mathematical models can take many forms, including differential or These and other types of models can overlap, with a given model involving a variety of abstract structures. In general, mathematical models may include In many cases, the quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments. Lack of agreement between theoretical mathematical models and experimental measurements often leads to important advances as better theories are developed.

In the physical sciences, a traditional mathematical model contains most of the following elements:

1. Governing equations
2. Supplementary sub-models
   1. Defining equations
   2. Constitutive equations
3. Assumptions and constraints

Classifications

Mathematical models are usually composed of relationships and variables . Relationships can be described by operators , such as algebraic operators, functions, differential operators, etc. Variables are abstractions of system parameters  of interest, that can be quantified  Several classification criteria can be used for mathematical models according to their structure:

• Linear vs. nonlinear: If all the operators in a mathematical model exhibit , the resulting mathematical model is defined as linear. A model is considered to be nonlinear otherwise. The definition of linearity and nonlinearity is dependent on context, and linear models may have nonlinear expressions in them. For example,  it is assumed that a relationship is linear in the parameters, but it may be nonlinear in the predictor variables. Similarly, a differential equation is said to be linear if it can be written with linear  but it can still have nonlinear expressions in it. In a mathematical programming  model, if the objective functions and constraints are represented entirely by linear equations  then the model is regarded as a linear model. If one or more of the objective functions or constraints are represented with a nonlinear  equation, then the model is known as a nonlinear model.
  Linear structure implies that a problem can be decomposed into simpler parts that can be treated independently and/or analyzed at a different scale and the results obtained will remain valid for the initial problem when recomposed and rescaled.
  Nonlinearity, even in fairly simple systems, is often associated with phenomena such as chaos and irreversibility. Although there are exceptions, nonlinear systems and models tend to be more difficult to study than linear ones. A common approach to nonlinear problems is linearization , but this can be problematic if one is trying to study aspects such as irreversibility, which are strongly tied to nonlinearity.
• Static vs. dynamic: A dynamic model accounts for time-dependent changes in the state of the system, while a static (or steady-state) model calculates the system in equilibrium, and thus is time-invariant. Dynamic models typically are represented by differential equations or differnce equaions or difference equations.
• Explicit vs. implicit: If all of the input parameters of the overall model are known, and the output parameters can be calculated by a finite series of computations, the model is said to be explicit. But sometimes it is the output parameters which are known, and the corresponding inputs must be solved for by an iterative procedure, such as Newtons method  or broudens method . In such a case the model is said to be implicit. For example, a jet emgines  physical properties such as turbine and nozzle throat areas can be explicitly calculated given a design thermodynamic cycle (air and fuel flow rates, pressures, and temperatures) at a specific flight condition and power setting, but the engine's operating cycles at other flight conditions and power settings cannot be explicitly calculated from the constant physical properties.
• Discrete vs. continuous: A  discrete model  treats objects as discrete, such as the particles in a molecular model  or the states in a statistical model; while a continuous model  represents the objects in a continuous manner, such as the velocity field of fluid in pipe flows, temperatures and stresses in a solid, and electric field that applies continuously over the entire model due to a point charge.
• Deterministic vs. probabilistic (stochastic): A deterministic  model is one in which every set of variable states is uniquely determined by parameters in the model and by sets of previous states of these variables; therefore, a deterministic model always performs the same way for a given set of initial conditions. Conversely, in a stochastic model—usually called a "statistical model"—randomness is present, and variable states are not described by unique values, but rather by probality distributions.
• Deductive, inductive, or floating: A deductive model is a logical structure based on a theory. An inductive model arises from empirical findings and generalization from them. The floating model rests on neither theory nor observation, but is merely the invocation of expected structure. Application of mathematics in social sciences outside of economics has been criticized for unfounded models. Application of catastrophe theory  in science has been characterized as a floating model.
• Strategic vs non-strategic Models used in game theory  are different in a sense that they model agents with incompatible incentives, such as competing species or bidders in an auction. Strategic models assume that players are autonomous decision makers who rationally choose actions that maximize their objective function. A key challenge of using strategic models is defining and computing solution concepts  such as Nash equillibrium . An interesting property of strategic models is that they separate reasoning about rules of the game from reasoning about behavior of the players.

Construction

In  business  and  engineering , mathematical models may be used to maximize a certain output. The system under consideration will require certain inputs. The system relating inputs to outputs depends on other variables too: decision varibales, state variables, exogenous  variables, and random variables .

Decision variables are sometimes known as independent variables. Exogenous variables are sometimes known as parameters pr constants  The variables are not independent of each other as the state variables are dependent on the decision, input, random, and exogenous variables. Furthermore, the output variables are dependent on the state of the system (represented by the state variables).

Objectives and constraints  of the system and its users can be represented as function of the output variables or state variables. The objective functions  will depend on the perspective of the model's user. Depending on the context, an objective function is also known as an index of performance, as it is some measure of interest to the user. Although there is no limit to the number of objective functions and constraints a model can have, using or optimizing the model becomes more involved (computationally) as the number increases.

For example, economists  often apply linear algebra  when using input-output models. Complicated mathematical models that have many variables may be consolidated by use of  vector where one symbol represents several variables.

A priori information

Mathematical modeling problems are often classified into black box or white box models, according to how much a priori  information on the system is available. A black-box model is a system of which there is no a priori information available. A white-box model (also called glass box or clear box) is a system where all necessary information is available. Practically all systems are somewhere between the black-box and white-box models, so this concept is useful only as an intuitive guide for deciding which approach to take.

Usually it is preferable to use as much a priori information as possible to make the model more accurate. Therefore, the white-box models are usually considered easier, because if you have used the information correctly, then the model will behave correctly. Often the a priori information comes in forms of knowing the type of functions relating different variables. For example, if we make a model of how a medicine works in a human system, we know that usually the amount of medicine in the blood is an exponentially decaying innction. But we are still left with several unknown parameters; how rapidly does the medicine amount decay, and what is the initial amount of medicine in blood? This example is therefore not a completely white-box model. These parameters have to be estimated through some means before one can use the model.

In black-box models one tries to estimate both the functional form of relations between variables and the numerical parameters in those functions. Using a priori information we could end up, for example, with a set of functions that probably could describe the system adequately. If there is no a priori information we would try to use functions as general as possible to cover all different models. An often used approach for black-box models are neural networks  which usually do not make assumptions about incoming data. Alternatively the NARMAX (Nonlinear AutoRegressive Moving Average model with eXogenous inputs) algorithms which were developed as part of nonlinear system identification can be used to select the model terms, determine the model structure, and estimate the unknown parameters in the presence of correlated and nonlinear noise. The advantage of NARMAX models compared to neural networks is that NARMAX produces models that can be written down and related to the underlying process, whereas neural networks produce an approximation that is opaque.

Subjective information

Sometimes it is useful to incorporate subjective information into a mathematical model. This can be done based on intution, experience, or expert  option,  or based on convenience of mathematical form.  bayesion statistics provides a theoretical framework for incorporating such subjectivity into a rigorous analysis: we specify a prior probability distribution (which can be subjective), and then update this distribution based on empirical data.

An example of when such approach would be necessary is a situation in which an experimenter bends a coin slightly and tosses it once, recording whether it comes up heads, and is then given the task of predicting the probability that the next flip comes up heads. After bending the coin, the true probability that the coin will come up heads is unknown; so the experimenter would need to make a decision (perhaps by looking at the shape of the coin) about what prior distribution to use. Incorporation of such subjective information might be important to get an accurate estimate of the probability.

 The concept of fuzzy set was developed in 1965. Since then, researchers have used this key set in many disciplines. As a brand-new conceptual system to support human-centric framework, fuzzy set has proved quite promising and effective in modeling human involvement in human-based intelligence to attain modernity in many departments like data analyzing, data mining, image coding and explaining, as well as in intelligence systems. Fuzzy set has also become an acknowledged research subject in both pure as well as in applied mathematics and statistics, showing how this theory is highly applicable and productive in many applications. Despite being a core subject for many years, fuzzy set still attracts researchers for putting forth solutions for prime issues with certain features questioned by these notions. Fuzzy set can effectively deal with a wide spectrum of problems of the physical world via cooperation, which may be beyond the capability of classical techniques. This means the fuzzy set could have the ability to handle a wide range of problems, for instance, decision making, intelligent data analysis, processing information, pattern recognition, and optimization.

The goal of this Special Issue is to dive deeper into the new trends of fuzzy set theory and the extension of fuzzy set theory with applications in group theory, ring theory, statistics, topological spaces, graph theory, decision making and other engineering applications.

Potential topics include but are not limited to the following:

• Fuzzy sets and their extensions
• Fuzzy subgroups and their extended forms
• Applications of fuzzy sets in ring theory
• Fuzzy sets and their extensions in graph theory
• Statistics on the application of fuzzy informatics
• Fuzzy sets and their extensions in topological spaces
• Decision making application with fuzzy logic

fuzzy logic, in mathematics, a form of logic based on the concept of a fuzzy set. Membership in fuzzy sets is expressed in degrees of truth—i.e., as a continuum of values ranging from 0 to 1. In a narrow sense, the term fuzzy logic refers to a system of approximate reasoning, but its widest meaning is usually identified with a mathematical theory of classes with unclear, or “fuzzy,” boundaries. Control systems based on fuzzy logic are used in many consumer electronic devices in order to make fine adjustments to changes . Fuzzy logic concepts and techniques have also been profitably used in linguistics, the behavioral sciences, the diagnosis of certain diseases, and even  analysis.

Fuzzy sets

Most concepts used in everyday language, such as “high temperature,” “round face,” or “aquatic animal,” are not clearly defined. In 1965 Lotfi Zadeh, an engineering professor at the University of California at Berkeley, proposed a mathematical definition of those classes that lack precisely defined criteria of membership. Zadeh called them fuzzy sets. Membership in a fuzzy set may be indicated by any number from 0 to 1, representing a range from “definitely not in the set” through “partially in the set” to “completely in the set.” For example, at age 45 a man is neither very young nor very old. This makes it difficult in traditional logic  to say whether or not he belongs to the set of “old persons.” Clearly he is “sort of” old, a qualitative assessment that can be quantified by assigning a value, or degree of membership, between 0 and 1—say 0.30—for his inclusion in a fuzzy set of old persons.

 

Catalan number

   In combinatorial mathematics, the Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the French-Belgian mathematician Eugène Charles Catalan (1814–1894).

The nth Catalan number can be expressed directly in terms of binomial coefficients The first Catalan numbers for n = 0, 1, 2, 3, ... are

1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, ...

Applications in combinatorics

There are many counting problems in combinatorics whose solution is given by the Catalan numbers. The book Enumerative Combinatorics: Volume 2 by combinatorialist Richard P. Stanley contains a set of exercises which describe 66 different interpretations of the Catalan numbers. Following are some examples, with illustrations of the cases C₃ = 5 and C₄ = 14

• C_n is the number of Dyck words^[2] of length 2n. A Dyck word is a string consisting of n X's and n Y's such that no initial segment of the string has more Y's than X's. For example, the following are the Dyck words of length 6:           XXXYYY     XYXXYY     XYXYXY     XXYYXY     XXYXYY.

• Re-interpreting the symbol X as an open parenthesis and Y as a close parenthesis, C_n counts the number of expressions containing n pairs of parentheses which are correctly matched
•                     ((()))     ()(())     ()()()     (())()     (()())

• C_n is the number of different ways n + 1 factors can be completely parenthesized (or the number of ways of associating n applications of a binary operator, as in the matrix chain multiplication problem). For n = 3, for example, we have the following five different parenthesizations of four factors:       ((ab)c)d     (a(bc))d     (ab)(cd)     a((bc)d)     a(b(cd))

Also, the interior of the correctly matching closing Y for the first X of a Dyck word contains the description of the left subtree, with the exterior describing the right subtree.

• C_n is the number of non-isomorphic ordered (or plane) trees with n + 1 vertices.^[3] See encoding general trees as binary trees.
• C_n is the number of monotonic lattice paths along the edges of a grid with n × n square cells, which do not pass above the diagonal. A monotonic path is one which starts in the lower left corner, finishes in the upper right corner, and consists entirely of edges pointing rightwards or upwards. Counting such paths is equivalent to counting Dyck words: X stands for "move right" and Y stands for "move up".

The following diagrams show the case n = 4:

This can be represented by listing the Catalan elements by column height:

The dark triangle is the root node, the light triangles correspond to internal nodes of the binary trees, and the green bars are the leaves.

[0,0,0,0] [0,0,0,1] [0,0,0,2] [0,0,1,1]

[0,1,1,1] [0,0,1,2] [0,0,0,3] [0,1,1,2] [0,0,2,2] [0,0,1,3]

[0,0,2,3] [0,1,1,3] [0,1,2,2] [0,1,2,3]

• A convex polygon with n + 2 sides can be cut into triangles by connecting vertices with non-crossing line segments (a form of polygon triangulation). The number of triangles formed is n and the number of different ways that this can be achieved is C_n. The following hexagons illustrate the case n = 4:

• C_n is the number of stack-sortable permutations of {1, ..., n}. A permutation w is called stack-sortable if S(w) = (1, ..., n), where S(w) is defined recursively as follows: write w = unv where n is the largest element in w and u and v are shorter sequences, and set S(w) = S(u)S(v)n, with S being the identity for one-element sequences.
• C_n is the number of permutations of {1, ..., n} that avoid the permutation pattern 123 (or, alternatively, any of the other patterns of length 3); that is, the number of permutations with no three-term increasing subsequence. For n = 3, these permutations are 132, 213, 231, 312 and 321. For n = 4, they are 1432, 2143, 2413, 2431, 3142, 3214, 3241, 3412, 3421, 4132, 4213, 4231, 4312 and 4321.
• C_n is the number of noncrossing partitions of the set {1, ..., n}. A fortiori, C_n never exceeds the nth Bell number. C_n is also the number of noncrossing partitions of the set {1, ..., 2n} in which every block is of size 2. The conjunction of these two facts may be used in a proof by mathematical induction that all of the free cumulants of degree more than 2 of the Wigner semicircle law are zero. This law is important in free probability theory and the theory of random matrices.
• C_n is the number of ways to tile a stairstep shape of height n with n rectangles. Cutting across the anti-diagonal and looking at only the edges gives full binary trees. The following figure illustrates the case n = 4:

AN INTRODUCTION OF SUMUDU TRANSFORM

 AN INTRODUCTION OF SUMUDU

TRANSFORM

S. Jeevitha

PG and Research Department of Mathematics

Marudhar Kesari Jain College For Women, Vaniyambadi, Tamil Nadu, India.

S. Komala

PG and Research Department of Mathematics

Marudhar Kesari Jain College For Women, Vaniyambadi, Tamil Nadu, India.

S. Silambarasi

PG and Research Department of Mathematics

Marudhar Kesari Jain College For Women, Vaniyambadi, Tamil Nadu, India.

S. Susitha

PG and Research Department of Mathematics

Marudhar Kesari Jain College For Women, Vaniyambadi, Tamil Nadu, India.

R.Vanitha

PG and Research Department of Mathematics

Marudhar Kesari Jain College For Women, Vaniyambadi, Tamil Nadu, India.

ABSTRACT

In this paper, we see the definition, some basic properties and fundamental properties

of Sumudu transform, relationships between Laplace and Sumudu transforms and Existence

of Sumudu transform.

KEYWORDS

Sumudu Transform, Gamma Function, Laplace Transform.

INTRODUCTION

The Sumudu transform is introduced by Watugula. Sumudu transform may be used

to solve problems without resorting to a new frequency domain .Due to its simple

formulation and consequent special and useful properties, the Sumudu transform has

already shown much promise. It is revealed here in and elsewhere that it can help to solve

intricate problems in engineering mathematics and applied sciences. However, despite the

potential presented by this new operator, only few theoretical investigations have appeared

in the literature, over a fifteen-year period. Most of the available transform theory books, if

not all, do not refer to the Sumudu transform. Even in relatively recent well known

comprehensive handbooks, such as Debnath and Poularikas, no mention of the Sumudu

transform can be found.

© 2021 JETIR July 2021, Volume 8, Issue 7 www.jetir.org (ISSN-2349-5162)

JETIR2107602 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org e772

SUMUDU TRANSFORM

Watugala introduced a new transform and named as Sumudu transform which is

defined by the following formula

𝐹(𝑢) = 𝒮[𝑓(𝑡); 𝑢] =

1

𝑢

∫ 𝑒−(

1

𝑢

) 𝑓(𝑡)𝑑𝑡, 𝑢 ∈ (−𝜏1, 𝜏2)

∞

0

BASIC SUMUDU TRANSFORM PROPERTIES

Sumudu transform for 𝑓 ∈ 𝐴:

𝐺(𝑢) = 𝒮[𝑓(𝑡)] = ∫ 𝑓(𝑢𝑡)𝑒−𝑡 𝑑𝑡, 𝑢 ∈ (−𝜏1, 𝜏2) ∞

0

Duality with Laplace transforms:

𝐺(𝑢) =

𝐹(1

⁄𝑢)

𝑢

, 𝐹(𝑠) =

𝐺(1

⁄𝑠)

𝑠

Linearity Property:

𝒮[𝑎𝑓(𝑡) + 𝑏𝑔(𝑡) = 𝑎𝒮[𝑓(𝑡)] + 𝑏𝒮[𝑔(𝑡)]]

Sumudu Transform of Function Derivatives:

𝐺1(𝑢) = 𝒮[𝑓,(𝑡)] =

𝐺(𝑢)−𝑓(0)

𝑢

=

𝐺(𝑢)

𝑢

−

𝑓(0)

𝑢

𝐺2(𝑢) = 𝒮[𝑓,,(𝑡)] =

𝐺(𝑢) − 𝑓(0)

𝑢2 =

𝐺(𝑢)

𝑢2 −

𝑓(0)

𝑢2 −

𝑓,(𝑜)

𝑢

𝐺𝑛(𝑢) = 𝒮[𝑓𝑛(𝑡)] =

𝐺(𝑢)

𝑢𝑛 −

𝑓(0)

𝑢𝑛 − ⋯ −

𝑓𝑛−1(𝑜)

𝑢

Sumudu transform of integral of a function:

𝒮 [∫ 𝑓(𝜏)𝑑𝜏

1

0 ] = 𝑢𝐺(𝑢)

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SUMUDU TRANSFORM FUNDAMENTAL PROPERTIES

THE DISCRETE SUMUDU TRANSFORM

Over the set of functions

𝐴 = {(𝑓(𝑡)|∃𝑀, 𝜏1,𝜏2 > 0, |𝑓(𝑡)| < 𝑀𝑒|𝑡|/𝜏𝑗 , 𝑖𝑓 𝑡 ∈ (−1)𝑗 × [0, ∞))}, (1)

the Sumudu transform is defined by

𝐺(𝑢) = 𝒮[𝑓(𝑡)] = ∫ 𝑓(𝑢𝑡)𝑒−𝑡 𝑑𝑡, 𝑢 ∈ (−𝜏1, 𝜏2) ∞

0

(2)

Among others, the Sumudu transform was shown to have units preserving properties and

hence may be used to solve problems without resorting to the frequency domain. As will

be seen below, this is one of many strength points for this new transform, especially with

respect to applications in problems with

physical dimensions. In fact, the Sumudu transform which is itself linear, preserves linear

functions, and hence in particular does not change units (see for instance Watugala or

Belgacem et al).Theoretically, this point may perhaps best be illustrated as an implication

of this more global result.

THEOREM:1

The Sumudu transform amplifies the coefficients of the power series function,

𝑓(𝑡) = Σ 𝑎𝑛𝑡𝑛 ∞

𝑛

=0 (1.1)

by sending it to the power series function,

𝐺(𝑢) = Σ 𝑛! 𝑎𝑛𝑢𝑛 ∞

𝑛

=0 (1.2)

PROOF :

Let f (t) be in A. If 𝑓(𝑡) = Σ 𝑎𝑛𝑡𝑛 ∞

𝑛

=0 in some interval I ⊂ R, then by Taylor’s

function expansion theorem,

𝑓(𝑡) = Σ 𝑓(𝑛)(𝑜)

𝑛!

∞𝑘

=0 𝑡𝑛

(1.3)

Therefore, by (2), and that of the gamma function Γ , we have

𝒮[𝑓(𝑡)] = ∫ Σ 𝑓(𝑛)(𝑜)

𝑛!

∞𝑘

=0 (𝑢𝑡)𝑛𝑒−𝑡 ∞

0 𝑑𝑡

= Σ 𝑓(𝑛)(𝑜)

𝑛!

∞𝑘

=0 𝑢𝑛 ∫ 𝑡𝑛 ∞

0 𝑒−𝑡 𝑑𝑡

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= Σ 𝑓(𝑛)(𝑜)

𝑛!

∞𝑘

=0 𝑢𝑛 Γ(n+1)

= Σ 𝑓(𝑛)(0)𝑢𝑛 ∞

𝑛

=0 (1.4)

Consequently, it is perhaps worth noting that since

𝒮[(1 + 𝑡)𝑚] = 𝒮 Σ 𝐶𝑛

𝑚𝑡𝑛 𝑚𝑛

=0

= 𝒮 Σ 𝑚!

𝑛!(𝑚−𝑛)!

𝑚𝑛

=0 𝑢𝑛

𝒮[(1 + 𝑡)𝑚] = Σ

𝑚!

(𝑚−𝑛)!

𝑚 𝑛=

0

𝑢𝑛

=Σ 𝑃𝑛

𝑚𝑢𝑛 𝑚𝑛

=0 (1.5)

the Sumudu transform sends combinations, 𝐶𝑛

𝑚 into permutations, 𝑃𝑛

𝑚, and hence may

seem to incur more order into discrete systems.

Also, a requirement that 𝒮[𝑓(𝑡)] converges, in an interval containing u=0, is provided by

the following conditions when satisfied, namely, that

(𝑖)𝑓(𝑛)(0) → 0 𝑎𝑠 𝑛 → ∞,

(𝑖𝑖) lim

𝑛→∞

|

𝑓(𝑛+1)(0)

𝑓(𝑛)(0)

𝑢| < 1 (1.6)

This means that the convergence radius r of 𝒮[𝑓(𝑡)] depends on the sequence

𝑓(𝑛)(0), since

𝑟 = lim

𝑛→∞

|

𝑓(𝑛+1)(0)

𝑓(𝑛)(0)

| (1.7)

Clearly, the Sumudu transform may be used as a signal processing or a detection

tool,especially in situations where the original signal has a decreasing power tail .

However, care must be taken, especially if the power series is not highly decaying. This

next example may instructively illustrate the stated concern. For instance, consider the

function

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𝑓(𝑡) = {

𝐼𝑛 (𝑡 + 1) 𝑖𝑓 𝑡 ∈ (−1,1]

𝑜 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

(1.8)

Since 𝑓(𝑡) = Σ (−1)𝑛−1 ∞

𝑛

=1

𝑡𝑛

⁄𝑛 the expect for u=0

𝒮[𝑓(𝑡)] = Σ (−1)𝑛−1(𝑛 − 1)! 𝑢𝑛 ∞

𝑛

=1 (1.9)

Diverges throught , because its convergens radius

𝑟 = lim

𝑛→∞

|

(−1)𝑛−1(𝑛−1)!

(−1)𝑛𝑛!

|

lim

𝑛→∞

1

𝑛

= 0 (1.10)

RELATION BETWEEN SUMUDU AND LAPLACE TRANSFORM

In our study, we use the following convolution notation: double convolution

between two continuous functions F (x, y) and G(x, y) given by

𝐹1(𝑥, 𝑦) ∗∗ 𝐹2(𝑥, 𝑦) = ∫ ∫ 𝐹1(𝑥 − 𝜃1, 𝑦 − 𝜃2)𝐹2(𝜃1, 𝜃2)𝑑𝜃1𝑑𝜃2

𝑥

0

𝑦

0

The single Sumudu transform is defined over the set of the functions

𝐴 = {(𝑓(𝑡)|∃𝑀, 𝜏1,𝜏2 > 0, |𝑓(𝑡)| < 𝑀𝑒|𝑡|/𝜏𝑗 , 𝑖𝑓 𝑡 ∈ (−1)𝑗 × [0, ∞))}

by 𝐺(𝑢) = 𝒮[𝑓(𝑡)] = ∫ 𝑓(𝑢𝑡)𝑒−𝑡 𝑑𝑡, 𝑢 ∈ (−𝜏1, 𝜏2) ∞

0

A sufficient condition for the existence of the Sumudu transform of a

function f is of exponential order, that is, there exist real constants

M > 0, 𝐾1, and 𝐾2 , such that |𝑓(𝑡, 𝑥)| ≤ 𝑀𝑒

𝑡

𝐾1

+

𝑥

𝐾2

EXISTENCE OF THE SUMUDU TRANSFORM

THEOREM:2

If f is of exponential order, then its Sumudu transform 𝒮[𝑓(𝑡, 𝑥)] = 𝐹(𝑣, 𝑢)exists and is

given by

𝐹(𝑣, 𝑢) = ∫ ∫ 𝑒−

𝑡

𝑣

−

𝑥

𝑢𝑓(𝑡, 𝑥)

∞

0

∞

0

𝑑𝑡𝑑𝑥

© 2021 JETIR July 2021, Volume 8, Issue 7 www.jetir.org (ISSN-2349-5162)

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where

1

𝑢

=

1

ɳ

+

𝑖

𝜏

,

1

𝑣

=

1

𝜇

+

𝑖

𝜉

The defining integral for F exists at points

1

𝑢

+

1

𝑣

=

1

ɳ

+

1

𝜇

+

𝑖

𝜏

+

𝑖

𝜉

in the right half plane

1

ɳ

+

1

𝜇

>

1

𝐾1

+

1

𝐾2

.

PROOF:

Using

1

𝑢

=

1

ɳ

+

𝑖

𝜏

𝑎𝑛𝑑

1

𝑣

=

1

𝜇

+

𝑖

𝜉

We can express 𝐹(𝑣, 𝑢) as

𝐹(𝑣, 𝑢) = ∫ ∫ 𝑓(𝑡, 𝑥)𝑐𝑜𝑠(

𝑡

𝜏

+

𝑥

𝜉

)𝑒

−

𝑡ɳ

−

𝑥𝜇

∞

0

∞

0

𝑑𝑡𝑑𝑥

−𝑖 ∫ ∫ 𝑓(𝑡, 𝑥)𝑠𝑖𝑛(

𝑡

𝜏

+

𝑥

𝜉

)𝑒

−

𝑡

ɳ

−

𝑥

𝜇

∞

0

∞

0 𝑑𝑡𝑑𝑥

Then for values of 1

ɳ

+

1

𝜇

>

1

𝐾1

+

1

𝐾2

we have

∫ ∫ |𝑓(𝑡, 𝑥)| |𝑐𝑜𝑠(

𝑡

𝜏

+

𝑥

𝜉

)| 𝑒

−

𝑡ɳ

−

𝑥𝜇

∞

0

∞

0

𝑑𝑡𝑑𝑥 ≤ 𝑀 ∫ ∫ 𝑒

(

1

𝐾1

−

1

ɳ

)𝑡+(

1

𝐾2

−

1

𝜇

)𝑥

∞

0

∞

0

𝑑𝑡𝑑𝑥

≤ 𝑀 (

ɳ𝐾1

ɳ−𝐾1

) (

ɳ 𝐾2

𝜇− 𝐾2

)

and

∫ ∫ |𝑓(𝑡, 𝑥)| |𝑠𝑖𝑛(

𝑡

𝜏

+

𝑥

𝜉

)| 𝑒

−

𝑡ɳ

−

𝑥𝜇

∞

0

∞

0

𝑑𝑡𝑑𝑥 ≤ 𝑀 ∫ ∫ 𝑒

(

1

𝐾1

−

1

ɳ

)𝑡+(

1

𝐾2

−

1

𝜇

)𝑥

∞

0

∞

0

𝑑𝑡𝑑𝑥

≤ 𝑀 (

ɳ𝐾1

ɳ−𝐾1

) (

ɳ 𝐾2

𝜇− 𝐾2

)

which imply that the integrals defining the real and imaginary parts of F exist for value of

𝑅𝑒 (

1

𝑢

+ 1

𝜇

) > 1

𝐾1

+

1

𝐾2

, and this completes the proof.

Thus, we note that for a function f, the sufficient conditions for the existence of the

Sumudu transform are to be piecewise continuous and of exponential order.

We also note that the double Sumudu transform of function f(t, x) is defind by

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𝐹(𝑣, 𝑢) = 𝒮2[𝑓(𝑡, 𝑥); (𝑣, 𝑢)] =

1

𝑢𝑣

∫ ∫ 𝑒−(

𝑡

𝑣

+

𝑥

𝑢

)𝑓(𝑡, 𝑥)

∞

0

∞

0

𝑑𝑡𝑑𝑥

where, 𝑠2 indicates double Sumudu transform and 𝑓(𝑡, 𝑥 ) is a function which can be

expressed as a convergent infinite series. Now, it is well known that the derivative of

convolution for two functions 𝑓 and 𝑔 is given by

𝑑

𝑑𝑥

(𝑓 ∗ 𝑔)(𝑥) =

𝑑

𝑑𝑥

𝑓(𝑥) ∗ 𝑔(𝑥)𝑜𝑟 𝑓(𝑥) ∗

𝑑

𝑑𝑥

𝑔(𝑥)

and it can be easily proved that Sumudu transform is

𝒮 [

𝑑

𝑑𝑥

(𝑓 ∗ 𝑔)(𝑥); 𝑣] = 𝑢𝒮 [

𝑑

𝑑𝑥

𝑓(𝑥); 𝑢] 𝒮[𝑔(𝑥); 𝑢] or

= 𝑢𝒮[𝑓(𝑥); 𝑢]𝒮 [

𝑑

𝑑𝑥

𝑔(𝑥); 𝑢].

The double Sumudu and double Laplace transforms have strong relationships that may be

expressed either as

(𝐼) 𝑢𝑣𝐹(𝑢, 𝑣) = £2 (𝑓(𝑥, 𝑦); (

1

𝑢

,

1

𝑣

))

Or (𝐼𝐼) 𝑝𝑠𝐹(𝑝, 𝑠) = £2 (𝑓(𝑥, 𝑦); (

1

𝑝

,

1

𝑠

))

where £2 represents the operation of double Laplace transform. In particular, the double

Sumudu and double Laplace transforms interchange the image of sin(x + t) and cos(x + t).

It turns out that

𝑠2[sin(𝑥 + 𝑡)] = £2[cos(𝑥 + 𝑡) =

𝑢 + 𝑣

(1 + 𝑢)2(1 + 𝑣)2

And

𝑠2[cos(𝑥 + 𝑡)] = £2[sin(𝑥 + 𝑡) =

1

(1 + 𝑢)2(1 + 𝑣)2

REFERENCE:

1. G. K. Watugala, Sumudu Transform: a new integral transform to solve differential

equations and control engineering problems, Internat. J. Math. Ed. Sci.Tech. 24 (1993) 35-

43.

2.G. K. Watugala, The Sumudu transform for functions of two variables, Math.

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JETIR2107602 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org e778

Engrg. Indust. 8 (2002) 293-302.

3. M.A.Asiru,Sumudu transform and the solution of integral equations of onvolution type,

International Journal of Mathematical Education in Science and Technology 32 (2001),

no. 6, 906–910.

4. Further properties of the Sumudu transform and its applications, International Journal of

Mathematical Education in Science and Technology 33 (2002), no.3, 441–449.

5. Classroom note: application of the Sumudu transform to discrete dynamic systems,

International Journal of Mathematical Education in Science and Technology 34 (2003),

no. 6,944–949.