PARTIAL DIFFERENTIAL EQUATIONS: THEORY, ANALYSIS AND APPLICATIONS -- PARTIAL DIFFERENTIAL EQUATIONS: THEORY, ANALYSIS AND APPLICATIONS -- CONTENTS -- PREFACE -- TIME-SPECTRAL SOLUTION OF INITIAL-VALUE PROBLEMS -- ABSTRACT -- 1. INTRODUCTION -- 2. THE METHOD -- 2.1. Multivariate Chebyshev Spectral Expansion -- 2.2. Weighted Residual Formulation -- 2.3. Integration and Differentiation -- 2.4. Minimax Considerations -- 2.5. Comparison with Least Square WRM Formulation -- 3. BOUNDARY AND INITIAL CONDITIONS -- 3.1. Implementation of Boundary and Initial Conditions Chebyshev Approximation -- 3.2. Example 1D Diffusion Equation -- 4. NONLINEARITY -- 5. ROOT SOLVER -- 6. SUBDOMAINS -- 7. APPLICATION - ACCURACY AND EFFICIENCY -- 7.1. Introductory Example -- the Match Equation -- 7.2. Accuracy -- Burger's Equation -- 7.2.1. Exact Solution -- 7.2.2. GWRM Solution -- 7.2.3. Finite Difference Solutions -- 7.2.4. Internal Gradients -- 7.3. Efficiency -- the Forced Wave Equation -- 7.3.1. GWRM Solution -- 7.3.2. Finite Difference Solutions -- 7.3.3. Conclusions on Forced Wave Equation -- 7.4. Large System of Initial-- Magnetohydrodynamic Stability -- 7.4.1. Benchmarking Using Simplified Model -- 7.4.2. GWRM Solution of MHD Stability Problem and Comparison with Eigenvalue Approach -- 7.5. Other Systems of Time- -- 7.6. Summary - Applications -- 8. CONCLUSION -- ACKNOWLEDGMENT -- REFERENCES -- A STOCHASTIC AGENT-BASED APPROACH TO THE FOKKER-PLANCK EQUATION IN HUMAN POPULATION DYNAMICS -- 1. INTRODUCTION -- 2. THE AGENT-BASED MODEL -- 3. THE MAIN RESULT -- 4. THE MASTER EQUATION -- 5. THE TRANSFORM OF THE MASTER EQUATION AND PROOF OF LEMMA 4.6 -- ACKNOWLEDGMENT -- REFERENCES -- TRAP-LIMITED DIFFUSION OF HYDROGEN IN AMORPHOUS SILICON THIN FILMS -- I. INTRODUCTION -- II. CONCEPTS ON DIFFUSION -- II.1. Diffusion through a Fixed Surface.

II.2. Some Special Cases -- II.3. Diffusion through a Moving Surface -- II.4. Trap-Limited Diffusion -- III. EXPERIMENTAL CONDITIONS FOR FILM DEPOSITION AND HYDROGEN PLASMA TREATMENT -- IV. TRAP-LIMITED HYDROGEN DIFFUSING IN AMOURPHOUS SILICON -- IV.1. Kinetics of Hydrogen Etching of Hydrogenated Amorphous Silicon -- IV.2. Hydrogen Evolution during Microcrystalline Silicon Deposition -- a) Before The Growth of the c-Si:H Layer ( T T1) -- b) During the Growth of the c-Si:H Layer ( T > T1) -- c) Hydrogen Out-Diffusion -- REFERENCES -- THE ROLE OF THE METHOD OF CHARACTERISTICS IN THE SOLUTION OF ESTIMATION AND CONTROL PROBLEMS FOR HYPERBOLIC PDE SYSTEMS -- ABSTRACT -- NOMENCLATURE -- ABBREVIATIONS -- 1. INTRODUCTION -- 2. METHOD OF CHARACTERISTICS: BASIC CONCEPTS -- 2.1. Dynamic Systems with One Characteristic Line -- 2.1.1. Study Case 1: Tubular Heat-Exchanger -- 2.1.2. Study Case 2: Non-Isothermal Plug Flow Reactor -- 2.2. Dynamic Systems with Multiple Characteristic Lines -- 2.2.1. Study Case 3: Solid-Waste Anaerobic Digestion -- 3. THE ROLE OF THE MC IN THE SOLUTION OF ESTIMATION PROBLEMS -- 3.1. General Dynamical Model -- 3.2. Design of a Robust State Observer -- 3.3. Robust State Observer Design for a Non-Isothermal Plug Flow Reactor -- State Observer Simulation -- 3.4. Robust State Observer Design for a Solid- Waste Anaerobic Digestion -- State Observer Simulation -- 4. THE ROLE OF THE MC IN THE SOLUTION OF CONTROL PROBLEMS -- 4.1. General Dynamical Model -- 4.2. Design of a Robust Control Law -- 4.2.1. Dynamical Representation for a Particular Axial Point -- 4.3. Robust Control Law Design for a Tubular Heat-Exchanger -- Robust Control Law Simulation -- 5. CONCLUSION -- ACKNOWLEDGMENT -- REFERENCES -- APPENDIX A -- APPENDIX B -- APPENDIX C -- SOLITON SOLUTIONS OF ONE KDV EQUATION -- ABSTRACT -- 1. INTRODUCTION -- 2. THE KDV EQUATION.

2.1. Solution in Form of Exp-Function -- 2.2. 1-Soliton Solution by the Solitary Wave Ansatz -- 2.3. Solutions in Form of the Tanh Function -- 3. REMARKS -- 4. ACKNOWLEDGMENTS -- REFERENCES -- NUMERICAL SOLUTION OF FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS -- ABSTRACT -- 1. INTRODUCTION -- 2. THE UNIFIED DERIVATION METHOD OF OPERATIONAL MATRICES -- 3. THE OPERATIONAL MATRICES OF HAAR WAVELETS -- 4. NUMERICAL EVALUATION OF FRACTIONAL CALCULUS -- 5. NUMERICAL SOLUTION OF FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS -- 6. EXAMPLES AND RESULTS -- Example 1. Solve the Following Linear Homogeneous Partial Differential Equation -- Example 2. Solve the Following Nonhomogeneous Partial Differential Equation -- Example 3. Finally, We Use the Operational Method to Solve the Below Fractional Partial Differential Equation -- 7. CONCLUSIONS -- REFERENCES -- BOUNDARY CONTROL OF SYSTEMS DESCRIBED BY PARTIAL DIFFERENTIAL EQUATIONS BY INPUT-OUTPUT LINEARIZATION -- 1.Introduction -- 2.BoundaryControlProblemofPDESystem -- 3.ControlLawDesignandStrategies -- 3.1.CollocatedSensorandActuator -- 3.1.1.ControlofCounter-CurrentHeatExchanger -- 3.1.2.ControlLaw -- 3.1.3.SimulationResults -- 3.2.Anti-collocatedSensorandActuator -- 3.2.1.BoundaryOutputCase:Co-currentHeatExchanger -- ControlLaw -- SimulationResults -- 3.2.2.PunctualOutputCase:RodwithHeatExchange -- Controllaw -- SimulationResults -- 4.StabilityAnalysis -- 5.PracticalImplementation -- 6.Conclusion -- References -- ROBUST NO PARAMETRIC IDENTIFIER FOR A CLASS OF COMPLEX PARTIAL DIFFERENTIAL EQUATIONS -- Abstract -- 1.Introduction -- 2.ComplexValuedNeuralNetworks -- 3.ComplexPartialDifferentialEquations -- 3.1.SchrödingerEquation(HyperbolicType) -- 3.2.Ginzburg-LandauEquation(ParabolicType) -- 4.ApproximationofComplex-ValuedPartialDifferentialEquation -- 5.NeuralIdenti erforComplexPartialDifferentialEquations.

5.0.1.PracticalStability -- 6.NumericalResults -- 6.1.SchrödingerSimulation -- 6.2.LandauSimulation -- 7.Conclusions -- 8.Appendix -- References -- THE GENERALIZED WEIERSTRASS SYSTEM INDUCING SURFACES IN EUCLIDEAN THREE SPACE AND HIGHER DIMENSIONAL SPACES -- Abstract -- 1.Introdution -- 2.Two-DimensionalSigmaModelandCorrespondencewithGeneralizedWeierstrassSystem -- 2.1.Second-OrderSystemAssociatedwiththegeneralizedWeierstrassSystem -- 2.2.GroupInvariantSolutionsoftheSigmaModel -- 2.3.MultisolitonSolutions -- 3.ConditionalSymmetriesandaLinearSpectralProblem -- 4.EstimationoftheDegreeofIndeterminancyoftheGeneralAnalyticSolutiontotheWeierstrass-EnneperSystem -- 5.InducingSurfacesinHigherDimensionalSpaces -- 5.1.SurfacesinR4 -- 5.2.SurfacesinMulti-DimensionalSpaces -- 6.PhysicalApplicationstoStringTheoryInvolvingGWInducingRepresentations -- 7.Non-ConstantMeanCurvatureSurfaces -- References -- NATURAL CONVECTION AND ITS EFFECT ON DIFFUSION MEASURED WITH NUCLEAR MAGNETIC RESONANCE -- 1.Abstract -- 2.Introduction -- 3.NaturalConvection -- 4.NuclearMagneticResonanceandDiffusion -- References -- PARTIAL DIFFERENTIAL EQUATIONS AS A TOOL FOR EVALUATION OF THE CONTINUOUS WAVELET TRANSFORM -- Abstract -- 1.Introduction -- 1.1.BasicDefinitions -- 1.2.TheSimplestWaveletsBasedontheGaussian:TheDoGWaveletandtheMorletWavelet -- 1.3.TheGaussianandMorletFamiliesofWaveletswithHigherVanishingMoments -- 2.RealGaussianWaveletsandtheImageProcessing -- 3.ComplexContinuousWaveletTransformwiththeMorletWaveletasaCauchyProblemforPDE -- 3.1.HaaseEquationforCWTwiththeMorletWavelet -- 3.2.InitialValuefortheCauchyProblemEvaluatingCWTwiththeStan-dardMorletWavelet -- 3.3.AdvancesoftheNumericalEvaluationwiththeUsageofPDE-BasedAl-gorithm -- 3.3.1.ReviewoftheStandardMethodsofCWTCalculations -- 3.3.2.PDEUsage -- 3.4.CompositeNon-stationarySignal.

4.TheContinuousWaveletTransformwiththeWaveletsofGaussianandMorletFamiliesasaSuperpositionofPDESo-lutions -- 4.1.EvaluationofCWTwiththeDoGandtheExactMorletWavelets -- 4.2.EvaluationofCWTwiththeWaveletsofGaussianandMorletFamilies -- 5.TheCauchyProblemfortheCWTwiththeVariableTime-ScaleResolution -- 5.1.TheWaveletTransformwiththeFixedScaleasaCauchyProblem -- 5.2.ExampleofApplication:TheChaoticPhaseSynchronization -- 6.ConclusionandFuturePerspectives -- References -- THE BLOWUP MECHANISM IN NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS-SCALING AND VARIATION∗ -- Abstract -- 1.BlowupoftheSolution -- 2.VariationalStructure -- 3.ScalingInvariance -- 4.MeanFieldEquations -- 5.Smoluchowski-PoissonEquationinHigherDimension -- References -- INDEX.