APPLICATIONS OF THE LAPLACE-MELLIN INTEGRAL TRANSFORM TO DIFFERNTIAL EQUATIONS

Similar documents
TOPIC 5: INTEGRATION

Integration Continued. Integration by Parts Solving Definite Integrals: Area Under a Curve Improper Integrals

HIGHER ORDER DIFFERENTIAL EQUATIONS

(2) If we multiplied a row of B by λ, then the value is also multiplied by λ(here lambda could be 0). namely

INTEGRALS. Chapter 7. d dx. 7.1 Overview Let d dx F (x) = f (x). Then, we write f ( x)

CONTINUITY AND DIFFERENTIABILITY

The Z transform techniques

Research Scholar, Vinoba Bhave University, Hazaribag, Jharkhand

b. How many ternary words of length 23 with eight 0 s, nine 1 s and six 2 s?

Continuous Random Variables: Basics

Chapter 16. 1) is a particular point on the graph of the function. 1. y, where x y 1

CIVL 8/ D Boundary Value Problems - Rectangular Elements 1/7

1 Introduction to Modulo 7 Arithmetic

The Derivative of the Natural Logarithmic Function. Derivative of the Natural Exponential Function. Let u be a differentiable function of x.

Limits Indeterminate Forms and L Hospital s Rule

ECE COMBINATIONAL BUILDING BLOCKS - INVEST 13 DECODERS AND ENCODERS

Section 3: Antiderivatives of Formulas

Lecture 11 Waves in Periodic Potentials Today: Questions you should be able to address after today s lecture:

FSA. CmSc 365 Theory of Computation. Finite State Automata and Regular Expressions (Chapter 2, Section 2.3) ALPHABET operations: U, concatenation, *

Instructions for Section 1

Lecture contents. Bloch theorem k-vector Brillouin zone Almost free-electron model Bands Effective mass Holes. NNSE 508 EM Lecture #9

Ch 1.2: Solutions of Some Differential Equations

MASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS SEMESTER TWO 2014 WEEK 11 WRITTEN EXAMINATION 1 SOLUTIONS

Hardy Spaces, Hyperfunctions, Pseudo-Differential Operators and Wavelets

CONIC SECTIONS. MODULE-IV Co-ordinate Geometry OBJECTIVES. Conic Sections

( ) Differential Equations. Unit-7. Exact Differential Equations: M d x + N d y = 0. Verify the condition

DEVELOPING COMPUTER PROGRAM FOR COMPUTING EIGENPAIRS OF 2 2 MATRICES AND 3 3 UPPER TRIANGULAR MATRICES USING THE SIMPLE ALGORITHM

Constructive Geometric Constraint Solving

ME 522 PRINCIPLES OF ROBOTICS. FIRST MIDTERM EXAMINATION April 19, M. Kemal Özgören

VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS

Algorithmic and NP-Completeness Aspects of a Total Lict Domination Number of a Graph

Weighted graphs -- reminder. Data Structures LECTURE 15. Shortest paths algorithms. Example: weighted graph. Two basic properties of shortest paths

, between the vertical lines x a and x b. Given a demand curve, having price as a function of quantity, p f (x) at height k is the curve f ( x,

Paths. Connectivity. Euler and Hamilton Paths. Planar graphs.

CBSE 2015 FOREIGN EXAMINATION

Section 11.6: Directional Derivatives and the Gradient Vector

Garnir Polynomial and their Properties

MASSACHUSETTS INSTITUTE OF TECHNOLOGY HAYSTACK OBSERVATORY WESTFORD, MASSACHUSETTS

Hadamard-Type Inequalities for s-convex Functions

(1) Then we could wave our hands over this and it would become:

Introduction to the Fourier transform. Computer Vision & Digital Image Processing. The Fourier transform (continued) The Fourier transform (continued)

UNIT # 08 (PART - I)

On Gaussian Distribution

Properties of Hexagonal Tile local and XYZ-local Series

TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS

Practice Final Exam. 3.) What is the 61st term of the sequence 7, 11, 15, 19,...?

TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS

Double integrals on regions (Sect. 15.2) Review: Fubini s Theorem on rectangular domains

Engineering Differential Equations Practice Final Exam Solutions Fall 2011

Research Article On the Genus of the Zero-Divisor Graph of Z n

MERIDIAN SURFACE OF WEINGARTEN TYPE IN 4-DIMENSIONAL EUCLIDEAN SPACE E 4 arxiv: v1 [math.dg] 14 May 2013

Motivation (Cont d) Motivation (Cont d)

# 1 ' 10 ' 100. Decimal point = 4 hundred. = 6 tens (or sixty) = 5 ones (or five) = 2 tenths. = 7 hundredths.

1 Finite Automata and Regular Expressions

Fourier Transform Methods for Partial Differential Equations

MATHEMATICS Class 12 FOR JEE MAIN & ADVANCED. Indefinite Integration. Exhaustive Theory. Formula Sheet Problems

Problem solving by search

Laplace Transform. National Chiao Tung University Chun-Jen Tsai 10/19/2011

MAT 403 NOTES 4. f + f =

Binomials and Pascal s Triangle

Walk Like a Mathematician Learning Task:

The Similar Construction Method of Non-Homogeneous Boundary Value Problems for Second-Order Homogeneous Linear Differential Equations

The University of Sydney MATH2969/2069. Graph Theory Tutorial 5 (Week 12) Solutions 2008

COMPLEXITY OF COUNTING PLANAR TILINGS BY TWO BARS

A Study of the Solutions of the Lotka Volterra. Prey Predator System Using Perturbation. Technique

OpenMx Matrices and Operators

A Method for Interpolation Wavelet Construction Using Orthogonal Scaling Functions

Tutorial 2 Euler Lagrange ( ) ( ) In one sentence: d dx

DISCRETE TIME FOURIER TRANSFORM (DTFT)

Hermite-Hadamard Inequality for Geometrically Quasiconvex Functions on a Rectangular Box

The Equitable Dominating Graph

Register Allocation. How to assign variables to finitely many registers? What to do when it can t be done? How to do so efficiently?

Assignment 5-6 For Problems 1-4, find a general solution of each differential equation. *Solve for y in Problems 2 and 4.

cycle that does not cross any edges (including its own), then it has at least

Why the Junction Tree Algorithm? The Junction Tree Algorithm. Clique Potential Representation. Overview. Chris Williams 1.

Relation of Finite Mellin Integral Transform. with Laplace and Fourier Transforms

( x) On the Exponentiated Generalized Weibull Distribution: A Generalization of the Weibull Distribution. 1. Introduction.

PROOF OF FIRST STANDARD FORM OF NONELEMENTARY FUNCTIONS

INTRODUCTION TO AUTOMATIC CONTROLS INDEX LAPLACE TRANSFORMS

Mathematics. Mathematics 3. hsn.uk.net. Higher HSN23000

For more important questions visit :

Last time: introduced our first computational model the DFA.

SOME PARAMETERS ON EQUITABLE COLORING OF PRISM AND CIRCULANT GRAPH.

Q.28 Q.29 Q.30. Q.31 Evaluate: ( log x ) Q.32 Evaluate: ( ) Q.33. Q.34 Evaluate: Q.35 Q.36 Q.37 Q.38 Q.39 Q.40 Q.41 Q.42. Q.43 Evaluate : ( x 2) Q.

COHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim.

Designing A Concrete Arch Bridge

this is called an indeterninateformof-oior.fi?afleleitns derivatives can now differentiable and give 0 on on open interval containing I agree to.

Module 2 Motion Instructions

Analytic Solution of Nonlinear Schrödinger Equation by Means of A New Approach

CSE 373: More on graphs; DFS and BFS. Michael Lee Wednesday, Feb 14, 2018

Paths, cycles and flows in graphs

On Implicative and Strong Implicative Filters of Lattice Wajsberg Algebras

Basic Polyhedral theory

Minimum Spanning Trees

RELATIONS BETWEEN GABOR TRANSFORMS AND FRACTIONAL FOURIER TRANSFORMS AND THEIR APPLICATIONS FOR SIGNAL PROCESSING

Floating Point Number System -(1.3)

Floating Point Number System -(1.3)

QUESTIONS BEGIN HERE!

Vector Integration. Line integral: Let F ( x y,

Cycles and Simple Cycles. Paths and Simple Paths. Trees. Problem: There is No Completely Standard Terminology!

Transcription:

Intrntionl Journl o Sintii nd Rrh Publition Volum, Iu 5, M ISSN 5-353 APPLICATIONS OF THE LAPLACE-MELLIN INTEGRAL TRANSFORM TO DIFFERNTIAL EQUATIONS S.M.Khirnr, R.M.Pi*, J.N.Slun** Dprtmnt o Mthmti Mhrhtr Adm o Enginring, Alndi-45, Pun, Indi Mobil No.: 9853376,F No.: -785759, Emil-mhirnr@gmil.om Dprtmnt o Mthmti (A.S.& H.* R.G.I.T. Vrov, Andhri (W, Mumbi-53, Indi Mobil No: 9975954, F.No.-6776 Emil.pirm@rdimil.om nd Dprtmnt o Mthmti** North Mhrtr Univrit,Jlgon-Indi Mo.bil No. 9438998 Emil.drjnlun@gmil.om Abtrt- In thi wor, th Lpl.Mllin intgrl trnorm i uitd in th ininit rgion [,;, ]..Thi wor whih i put orwrd to undrtnd th proprti nd rltion thorm,drivtiv nd pplition or th Lpl - Mllin Intgrl Trnorm in [,] to [, ]. To illutrt th dvntg nd u o thi intgrl trnorm,lpl qution in rtin orm, th on dimnionl wv qution nd hr low qution nd Cuh linr dirntil qution r olv b uing thi intgrl trnorm olvd t th nd. Solution o th dirntil qution r grphill rprntd b uing Mtlb. Ind Trm- Lpl trnorm, Mllin trnorm, Doubl Lpl trnorm,intgrl trnorm AMS Mthmtil Cliition : 44A,33C99 (, 47D3( T I. INTRODUCTION h Lpl- Bi Ltrl Lpl trnorm i ud to ind th Lpl - Mllin intgrl trnorm in th rng [,] to, [ ], proprti li Linr proprt, Sling proprt, Powr proprt, thorm li invrion thorm, onvolution thorm, prvl thorm, hiting thorm.b uing thi intgrl trnorm w obtin th rult o drivtiv w. r. t., nd w ind th gnrlizd rult o th n drivtiv o th untion (, w. r. t... Uing th drivtiv w olv th Lpl qution in Crtin orm, on dimnionl wv nd ht low qution. It olution id rprntd b uing tool o Mtlb. II. PRELIMINARY RESULTS Rltion o th Mllin Intgrl Trnorm with Lpl Trnorm Th Lpl trnorm o th untion ( o i dnotd b L[(, nd dind ( d, L[(,] whnvr thi intgrl i it or r> i th prmtr Th invr o th Lpl trnorm i dnotd b L [ (, L ( [ (, ] nd dind ( i i L[ (, dr Th Mllin intgrl trnorm o th untion o ( o i dnotd b M[(] nd i dind M[( r ( d whnvr thi intgrl i it or > prmtr. Th invrion o th Mllin intgrl trnorm i dnotd b M M [ ( [ ( ( ( nd dind i r i i M[ ( ] d www.ijrp.org

Intrntionl Journl o Sintii nd Rrh Publition Volum, Iu 5, M ISSN 5-353 Th Lpl-Bi Ltrl Lpl trnorm L[ z, L[ z, nd dind ( rz z ddz i dnotd b whnvr thi doubl intgrl i it or r> nd > r prmtr. Subtitut z -log( thn z nd i z thn d, dz -,i z- thn (, p dd, thn (,+ g(, (,+ m[g(, ( B. Sling Proprt Sond vr importnt proprt i tht th Lpl Mllin Intgrl Trnorm i Sling Proprt:, tht i or th untion (, who Lpl -Mllin intgrl trnorm it, thn onidr th qution L[ z, ( rz z ddz (, p dd,thn ( r ( d log( d( rz z z ddz r z ddz q, d z d p (q,r,, (3 (,d C. Powr Proprt Third vr importnt proprt i tht th Lpl Mllin Intgrl Trnorm i Powr Proprt:, tht i or th untion (, who Lpl -Mllin intgrl trnorm it, thn onidr th qution (, p dd, thn r t L[ z, r dd (, r, r (,t,p/ (4 Thi i th rltion in btwn Mllin Intgrl Trnorm nd Lpl trnorm or (, with prmtr r> nd > in th rng [,;, ]..Thi i dnotd b L M[ r, ] r dd L M M[ whr nd nd >,p>. III. PRELIMINARY PROPERTIES A. Linr Proprt A vr importnt proprt i tht th Lpl Mllin Intgrl Trnorm i linr oprtion thorm:, tht i or th untion (, nd g(, who Lpl-Mllin Intgrl Trnorm it nd r ontnt thn onidr th qution ( IV. MAIN RESULTS A. Invrion Thorm Th Lpl-Mllin Intgrl Trnorm i (, (, (i i i i i p p p B. Convolution Thorm Th Lpl-Mllin Intgrl Trnorm i (, dd, thn ddp p dd, thn (5 www.ijrp.org

Intrntionl Journl o Sintii nd Rrh Publition Volum, Iu 5, M 3 ISSN 5-353 (,g(t-, (i i i i i p t z p g( t, ddp p d d (6 C. Prvl Thorm (Orthogonlit Th Lpl-Mllin Intgrl Trnorm i (, (,g(, (i (7 i i i i p t z p p D. Shiting Thorm Th Lpl-Mllin Intgrl Trnorm i (, dd, thn g(, ddp p dd, thm Lm], (8 V. LAPLACE MELLIN INTEGRAL TRANSFORM OF DERIVATIVES Thorm: Suppo tht (, i ontinuou or ll t nd z tiing or om t, vlu, nd m nd h drivtiv whih i piwi ontinuou on vr init intrvl in th rng o t. nd z Thn b uing th Lpl- Mllin intgrl trnorm, th drivtiv o (, it whn > nd p> nd t (, m z or ll nd or om ontnt A. Lpl-Mllin Intgrl Trnorm o irt ordr prtil drivtiv o (t w.r.t. Th Lpl- Mllin trnormtion i ( p d[[ d[ (, ] p p p t (, whr - ( dd dd - ( p (, d - d] d] p (, d - (9 B. Lpl-Mllin intgrl trnorm o n th ordr prtil drivtiv o (, w.r.t. Lpl-Mllin Intgrl Trnorm o irt ordr prtil drivtiv o (, w.r.t. Th Lpl- Mllin Intgrl Trnorm i (, (, p dd p dd, thn (, p dd, thn p d d (, p dd p d[[ ] ( d] www.ijrp.org

Intrntionl Journl o Sintii nd Rrh Publition Volum, Iu 5, M 4 ISSN 5-353 p d[ (, ( d] w. r. t. nd -ontnt. p d d,intgrt ( p dd - p (, d p d[[ ] ( d] p dd -(B uing DUIS p d[[ (, ( d] [ -] p (, d p d] (, (, (n (, - ( 3 - ( n - n ( Thi i th gnrlizd rult o h Lpl- Mllin Intgrl th Trnorm o n drivtiv o (,. VI. APPLICATIONS TO LIMIT A. On dimnionl wv qution i olvd b uing Lpl-Mllin Intgrl Trnorm. Th on dimnionl wv qution i whr ( t, z ( t, z Th Lpl- Mllin Intgrl Trnormtion i tht i (3 whr p (, d. + + p (, d p dd w. r. t. nd ontnt. p p d[[ d[[ (, p d ] ( ( d d] d], intgrt (, p dd, thn p (, d p d] p dd p (, d + www.ijrp.org

Intrntionl Journl o Sintii nd Rrh Publition Volum, Iu 5, M 5 ISSN 5-353 whr p (, d [ (DUIS + ppling Lpl- Mllin Intgrl Trnorm (,- D D D - ( D,w hv to Lm (4 [ D (5 Thi i th ordinr dirntil qution w. r. t. CF C C nd PI whr CS C C p (, d (6 To illutrt th u o th Lpl-Mllin intgrl trnorm in olving th rtin prtil dirntil qution.w propo to ind th olution (, o th qution,tiing th boundr ondition Th initil nd boundr ondition r ( I thn (,, ( I thn (,. Th olution o th on dimnionl wvl qution i CS C C I thn (, C + C - C + C - I thn (, W gt C C C C C C - ( ( ( whr C -. nd C ( ( (7 p (, d Grphil Rprnttion, thn (,thn. Empl www.ijrp.org

-i Intrntionl Journl o Sintii nd Rrh Publition Volum, Iu 5, M 6 ISSN 5-353 8 6 4 On Dimntionl.Wv Eqution lm[(t,z,..4.6.8 -i B. Lpl qution in Crtin orm i olvd b uing Lol-Mllin intgrl trnorm. Th on dimnionl wv qution i ( t, z ( t, z i Th Lpl- Mllin Intgrl Trnormtion i (, p dd +, thn Whr p (, d, thn (,- + ( D D + - (8 Thi i th ordinr dirntil qution o ond ordr in. It root r m i nd m i It Complmntr FuntionO.F. nd tht C o( C in( Prtiulr IntgrlP.I.-, th th Complt Solution i C o( C in( - (9. Empl whr p (, d To illutrt th u o th Lpl-Mllin intgrl trnorm in olving th rtin prtil dirntil qution.w propo to ind th olution (, o,tiing th boundr ondition An. ( I thn (,,( I thn (, Th olution o th prtil qution i givn b Lm [ C o( C in( ( I thn (,C-/ thn C / ( I thn (t,co(+cin(- / C o(-cot( o(+ ot( Th rquird olution i (, o(+[ o (+ ot(]in( - - ( whr p (, d Grphil Rprnttion www.ijrp.org

Intrntionl Journl o Sintii nd Rrh Publition Volum, Iu 5, M 7 ISSN 5-353 ( lm[(, whr C C - p (, d 3. Empl To illutrit th u o th Lpl-Mllin Intgrl Trnorm in olving th 6.3.6.3.On dimntionl Ht Slow Eqution i olvd b uing Lpl- Mllin Intgrl trnorm. Th on dimnionl qution i, whr Th Lpl- Mllin Intgrl Trnormtion i (, p dd, thn Lm [ whr, thn p (, d + D -, thn D D ( ( Thi i th ordinr dirntil qution in C.F. C P.I. - Th omplt olution i C + rtin prtil dirntil qution W propo to ind th olution tiing th boundr ondition It olution i (,p[ C C. - ( I thn (,, ( I thn '(,. Anwr. C ( I thn (,C+C- ( nd id thn (, thn ( ( C (( (, Whr Grphil Rprnttion + C C + (( p (, d (3., nd www.ijrp.org

Intrntionl Journl o Sintii nd Rrh Publition Volum, Iu 5, M 8 ISSN 5-353 VII. REMARKS B uing Lpl Mllin intgrl trnorm, olution o th Lpl qution in Crtin orm, on dimntionl wv qution nd Ht Flow qution r olvd nd it olution r rprntd grphill b uing tool o Mtl. VIII. CONCLUSION Rprnttion o olution o th dirntil qution b uing tool o Mtlb i th id i givn in thi ppr. REFERENCES [] Dr Nlor, On Mllin Tp Intgrl Trnorm Journl o Mthmti nd Mhni (963 vol., No. [] C.Fo, Applition o Mllin' Trnormtion to th intgrl qution,(933 [3] J.M.Mndz nd J.R.Ngrin, On th init Hnl-Shwrtz Trnormtion o Ditribution Gnit,(988, vol.39, No. [4] S. M. Khirnr, R. M. Pi, J. N. Slun,Bi Ltrl Lpl Mllin Intgrl Trnorm nd It Applition Imt. J. o Pur & Appl. Si. Th., (,, pp 4-6 [5] In N. Snddon,Th u o Intgrl TrnormTMH dition 974 [6] C. Fo Applition o Mllin Trnormtion to Intgrl Eqution, 3rd Mrh, 934,pp 495-5 [7] A.H.Zmnin, Gnrlizd Intgrl Trnormtion, Intrin Publition, Nw Yor,(968 [8] A. H.. ZmninTh Ditributionl lpl nd Mllin Trnormtion J. SIAM Vol 4. No.. Jn.98 Pritd in U.S.A. [9] S. M. Khirnr, R. M. Pi, J. N. Slun, Stud O Th Sumudu Mllin Intgrl Trnorm nd It Applition IJMSEA, Vol. 4 N. IV ( t. Pp 37-35 [] I.S. Rd, Th Mllin Tp Doubl Intgrl, Cmbridg, London. [] S. M. Khirnr, R. M. Pi, J. N. Slun, Rltion O Th Finit Mllin Intgrl Trnorm With Th Lpl nd Fourir Trnorm CES, Vol. 4,no. 6, 69-88 [] S. M. Khirnr, R. M. Pi, J. N. Slun, Applition O Th Mllin Tp Intgrl Trnorm In Th Rng [, ], IJMSA, Vol., No., Jn. Pp -7. www.ijrp.org

2024 © sciencedocbox.com Privacy Policy | Terms of Service | Feedback