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Abstract
THIS THESIS DEALS WITH THE NUMERICAL SOLUTION OF BURGER’S EQUATION.FINITE DIFFERENCE&FINITE ELEMENT METHODS ARE USED TO SOLVE THIS EQUATION NUMERICALLY.THE RESULTS OBTAINED IN EACH CASE COMARED WITH THE ANALYTICAL SOLUTION.THE THESIS CONSISTS OF THREE CHAPTERS,IN THE FIRST CHAPTER,WE GIVE SOME USEFUL INFORMATIONS ABOUT SHOCK WAVES &ITS RELATION TO BURGER’S EQUATION:A PHYSICAL &A MATHEMATHICAL PICTURE FOR SHOCK WAVES ARE GIVEN,AS IT APPEARS IN THE LITERATURE.THE SOLUTION IS MODIFIED TO INCLUDE THE CASE OF TWO INTERSECTING SHOCKS.IN THE 2ND CHAPTER,WE REVIEW SOME OF THE WELL KNOWN NUMERICAL METHODS THAT ARE USED FOR SOLVING HYPERBOLIC&PARAPOLIC EQUATIONS.THE CLASSICAL THOMMAN SCHEMES AS WELL AS THE GENERAL S^ SCHEMES ARE ALSO EXPLAINED.THE VON-NUMANN STABILITY ANALYSIS IS CARRIED OUT &THE RANGE OF STABILITY IS COMPUTED FOR EACH METHOD.BURGER’S EQUATION IS SOLVED NUMERICALLY FOR THE CASE OF TWO INTERSECTING SHOCK WAVES.THE NUMERICAL SOLUTIONS ARE COMPARED WITH THE ANALYTICAL ONE.AN ERROR ESTIMATE IS GIVEN FOR EACH NUMERICAL METHOD.IN THE 3RD CHAPTER,WE PRESENT A SURVEY OF THE METHODS OF WEIGHTED RESIGUALS(MWR).A DISCUSSION OF THE FINITE ELEMENT METHOD IS GIVEN &HOW TO INTERPOLATE THE APPROXIMATE SOLUTION WITHIN EACH ELEMENT.THE BUBNOV &PETROV-GALERKIN FINITE ELEMENT METHODS ARE APPLIED TO THE PROPAGATING TWO SHOCK PROBLEM GOVERNED BY BURGER’S EQUATION USING LINEAR &QUADRATIC FINITE ELEMENTS RESPECTIVELY.THE APPROXIMATE SOLUTIONS ARE COMARES WITH THE ANALYTICAL ONE.AN ERROR ESTIMATE IS GIVEN FOR EACH OF THESE TWO METHODS.A LISTING OF THE COMPUTER PROGRAMES,FOR THE FINITE DIFFERENCE&FINITE ELEMENT METHODS,ARE INCLUDED AT THE END OF THE THESIS.IT IS SHOWN THAT ,LERAT-PEYRET METHOD GIVES MINIMUM ERROR IN THE CASE OF FINITE DIFFERENCE WHILE THE CASE OF FINITE ELEMENT.GENERALLY IN THIS PROBLEM, TWO INTERSECTING SHOCKS, IT IS CONCLUDED THAT LERAT-PEYRET METHOD GIVES THE BEST RESULTS COMPARED TO THOSE OBTAINED EITHER FROM FINITE DIFFERENCE METHODS OR FINITE ELEMENT METHODS.