IN THIS NOTE WE DISCUSS THE EXISTENCE AND STABILITY OF AN INVERSE PROBLEM ARISING FROM THE DETERMINATION OF THE REACTION COEFFICIENTS FOR AN SIS MODEL. THE STUDY IS MOTIVATED BY A REMARK REGARDING THE FINAL DISCUSSION OF THE RECENT PAPER BY XIANG AND LIU (2015). THE WEAK ASSUMPTION OF THE WORK OF H. XIANG AND B. LIU IS THAT THE PROOFS OF EXISTENCE AND STABILITY RESULTS ARE VALID ONLY FOR THE ONE-DIMENSIONAL CASE. HERE, WE INTRODUCE AN APPROPRIATE FRAMEWORK WHICH IS ALSO VALID IN THE MULTIDIMENSIONAL CASE AND THAT GENERALIZES THE PREVIOUS RESULTS.

IN THIS ARTICLE, WE PROPOSE A ONE-DIMENSIONAL HEAT CONDUCTION MODEL FOR A DOUBLE-PANE WINDOW WITH A TEMPERATURE-JUMP BOUNDARY CONDITION AND A THERMAL LAGGING INTERFACIAL EFFECT CONDITION BETWEEN LAYERS. WE CONSTRUCT A SECOND-ORDER ACCURATE FINITE DIFFERENCE SCHEME TO SOLVE THE HEAT CONDUCTION PROBLEM. THE DESIGNED SCHEME IS MAINLY BASED ON APPROXIMATIONS SATISFYING THE FACTS THAT ALL INNER GRID POINTS HAS SECOND-ORDER TEMPORAL AND SPATIAL TRUNCATION ERRORS, WHILE AT THE BOUNDARY POINTS AND AT INTER-FACIAL POINTS HAS SECOND-ORDER TEMPORAL TRUNCATION ERROR AND FIRST-ORDER SPATIAL TRUNCATION ERROR, RESPECTIVELY. WE PROVE THAT THE FINITE DIFFERENCE SCHEME INTRODUCED IS UNCONDITIONALLY STABLE, CONVERGENT, AND HAS A RATE OF CONVERGENCE TWO IN SPACE AND TIME FOR THE L?-NORM. MOREOVER, WE GIVE A NUMERICAL EXAMPLE TO CONFIRM OUR THEORETICAL RESULTS.

IN THIS PAPER WE STUDY THE INVERSE PROBLEM ARISING IN THE MODEL DESCRIBING THE TRANSPORT OF TWO-PHASE FLOW IN POROUS MEDIA. WE CONSIDER SOME PHYSICAL ASSUMPTIONS SO THAT THE MATHEMATICAL MODEL (DIRECT PROBLEM) IS AN INITIAL BOUNDARY VALUE PROBLEM FOR A PARABOLIC DEGENERATE EQUATION. IN THE INVERSE PROBLEM WE WANT TO DETERMINE THE COEFFICIENTS (FLUX AND DIFFUSION FUNCTIONS) OF THE EQUATION FROM A SET OF EXPERIMENTAL DATA FOR THE RECOVERY RESPONSE. WE FORMULATE THE INVERSE PROBLEM AS A MINIMIZATION OF A SUITABLE COST FUNCTION AND WE DERIVE ITS NUMERICAL GRADIENT BY MEANS OF THE SENSITIVITY EQUATION METHOD. WE START WITH THE DISCRETE FORMULATION AND, ASSUMING THAT THE DIRECT PROBLEM IS DISCRETIZED BY A FINITE VOLUME SCHEME, WE OBTAIN THE DISCRETE SENSITIVITY EQUATION. THEN, WITH THE NUMERICAL SOLUTIONS OF THE DIRECT PROBLEM AND THE DISCRETE SENSITIVITY EQUATION WE CALCULATE THE NUMERICAL GRADIENT. THE CONJUGATE GRADIENT METHOD ALLOWS US TO FIND NUMERICAL VALUES OF THE FLUX AND DIFFUSION PARAMETERS. ADDITIONALLY, IN ORDER TO DEMONSTRATE THE EFFECTIVENESS OF OUR METHOD, WE PRESENT A NUMERICAL EXAMPLE FOR THE PARAMETER IDENTIFICATION PROBLEM.

IN THIS SHORT NOTE THE AUTHORS GIVE ANSWERS TO THE THREE OPEN PROBLEMS FORMULATED BY WU AND SRIVASTAVA [{\IT APPL. MATH. LETT. 25 (2012), 1347--1353}]. WE DISPROVE THE PROBLEM 1, BY SHOWING THAT THERE EXISTS A TRIANGLE WHICH DOES NOT SATISFIES THE PROPOSED INEQUALITY. WE PROVE THE INEQUALITIES CONJECTURED IN PROBLEMS 2 AND 3. FURTHERMORE, WE INTRODUCE AN OPTIMAL REFINEMENT OF THE INEQUALITY CONJECTURED ON PROBLEM 3.

IN THIS NOTE, WE PROVE THE EXISTENCE AND UNIQUENESS OF WEAK SOLUTIONS FOR THE BOUNDARY VALUE PROBLEM MODELLING THE STATIONARY CASE OF THE BIOCONVECTIVE FLOW PROBLEM. THE BIOCONVECTIVE MODEL IS A BOUNDARY VALUE PROBLEM FOR A SYSTEM OF FOUR EQUATIONS: THE NONLINEAR STOKES EQUATION, THE INCOMPRESSIBILITY EQUATION, AND TWO TRANSPORT EQUATIONS. THE UNKNOWNS OF THE MODEL ARE THE VELOCITY OF THE FLUID, THE PRESSURE OF THE FLUID, THE LOCAL CONCENTRATION OF MICROORGANISMS, AND THE OXYGEN CONCENTRATION. WE DERIVE SOME APPROPRIATE A PRIORI ESTIMATES FOR THE WEAK SOLUTION, WHICH IMPLIES THE EXISTENCE, BY APPLICATION OF GOSSEZ THEOREM, AND THE UNIQUENESS BY STANDARD METHODOLOGY OF COMPARISON OF TWO ARBITRARY SOLUTIONS.

EXISTENCE OF ALMOST AUTOMORPHIC SOLUTIONS FOR ABSTRACT DELAYED DIFFERENTIAL EQUATIONS IS ESTABLISHED. USING ERGODICITY, EXPONENTIAL DICHOTOMY AND BI-ALMOST AUTOMORPHICITY ON THE HOMOGENEOUS PART, SUFFICIENT CONDITIONS FOR THE EXISTENCE AND UNIQUENESS OF ALMOST AUTOMORPHIC SOLUTIONS ARE GIVEN.

IN THIS WORK WE DEVELOP A STUDY OF POSITIVE PERIODIC SOLUTIONS FOR A MATHEMATICAL MODEL OF THE DYNAMICS OF COMPUTER VIRUS PROPAGATION. WE PROPOSE A GENERALIZED COMPARTMENT MODEL OF SEIR-KS TYPE, SINCE WE CONSIDER THAT THE POPULATION IS PARTITIONED IN FIVE CLASSES: SUSCEPTIBLE (S); EXPOSED (E); INFECTED (I); RECOVERED (R); AND KILL SIGNALS (K), AND ASSUME THAT THE RATES OF VIRUS PROPAGATION ARE TIME DEPENDENT FUNCTIONS. THEN, WE INTRODUCE A SUFFICIENT CONDITION FOR THE EXISTENCE OF POSITIVE PERIODIC SOLUTIONS OF THE GENERALIZED SEIR-KS MODEL. THE PROOF OF THE MAIN RESULTS ARE BASED ON A PRIORI ESTIMATES OF THE SEIR-KS SYSTEM SOLUTIONS AND THE APPLICATION OF COINCIDENCE DEGREE THEORY. MOREOVER, WE PRESENT AN EXAMPLE OF A GENERALIZED SYSTEM SATISFYING THE SUFFICIENT CONDITION.

IN THIS PAPER, WE STUDY THE INVERSE PROBLEM OF DETERMINING THE DENSITY FUNCTION MODELING THE VECTOR EXTERNAL SOURCE FOR THE LINEAR MOMENTUM OF PARTICLES, IN A MATHEMATICAL MODEL FOR THE BIOCONVECTIVE FLOW PROBLEM. THE MODEL CONSISTS OF THREE EQUATIONS: LINEAR MOMENTUM OF PARTICLES, A CONSERVATION LAW FOR THE MICROORGANISMS, AND THE INCOMPRESSIBILITY CONDITION. WE ANALYZE THE DIRECT PROBLEM OBTAINING RESULTS FOR THE WELL POSEDNESS. WE PROVE THE EXISTENCE OF WEAK SOLUTIONS UNDER GENERAL ASSUMPTIONS AND THE UNIQUENESS OF WEAK SOLUTIONS FOR A PARTICULAR CLASS OF DENSITY FUNCTIONS. TO SOLVE THE INVERSE PROBLEM, WE ASSUME THAT AN INTEGRAL OVERSPECIFICATION CONDITION IS GIVEN. THEN, WE PROVE THE LOCAL UNIQUENESS OF THE INVERSE PROBLEM. THE PROOF IS BASED ON THE CHARACTERIZATION OF THE INVERSE PROBLEM SOLUTIONS USING AN OPERATOR EQUATION OF SECOND KIND, THE INTRODUCTION OF SEVERAL A PRIORI ESTIMATES, AND THE APPLICATION OF THE TIKHONOV FIXED POINT THEOREM.

THIS ARTICLE DEALS WITH THE ASYMPTOTIC BEHAVIOR OF FOURTH ORDER DIFFERENTIAL EQUATION WHERE THE COEFFICIENTS ARE PERTURBATIONS OF LINEAR CONSTANT COEFFICIENT EQUATION. WE INTRODUCE A CHANGE OF VARIABLE AND DEDUCE THAT THE NEW VARIABLE SATISFIES A THIRD ORDER DIFFERENTIAL EQUATION OF RICCATI TYPE. WE ASSUME THREE HYPOTHESIS. THE FIRST IS THE FOLLOWING: ALL ROOTS OF THE CHARACTERISTIC POLYNOMIAL ASSOCIATED TO THE FOURTH ORDER LINEAR EQUATION HAS DISTINCT REAL PART. THE OTHER TWO HYPOTHESIS ARE RELATED WITH THE BEHAVIOR OF THE PERTURBATION FUNCTIONS. UNDER THIS GENERAL HYPOTHESIS WE OBTAIN FOUR MAIN RESULTS. THE FIRST TWO RESULTS ARE RELATED WITH THE APPLICATION OF FIXED POINT THEOREM TO PROVE THAT THE RICCATI EQUATION HAS A UNIQUE SOLUTION. THE NEXT RESULT CONCERNS WITH THE ASYMPTOTIC BEHAVIOR OF THE SOLUTIONS OF THE RICCATI EQUATION. THE FOURTH MAIN THEOREM IS INTRODUCED TO ESTABLISH THE EXISTENCE OF A FUNDAMENTAL SYSTEM OF SOLUTIONS AND TO PRECISE FORMULAS FOR THE ASYMPTOTIC BEHAVIOR OF THE LINEAR FOURTH ORDER DIFFERENTIAL EQUATION.

IN THIS CONTRIBUTION WE CONSIDER THE PROBLEM OF FLUX IDENTIFICATION IN A SCALAR CONSERVATION LAW MODELLING THE PHENOMENON OF SEDIMENTATION. THE EXPERIMENTAL OBSERVATION DATA USED FOR THE CALIBRATION CONSIST OF A SOLID CONCENTRATION PROFILE AT A FIXED TIME. THE IDENTIFICATION PROBLEM IS FORMULATED AS AN OPTIMIZATION ONE, WHERE THE DISTANCE BETWEEN THE PROFILES OF THE MODEL SIMULATION AND OBSERVATION DATA IS MINIMIZED BY A LEAST SQUARES COST FUNCTION. THE DIRECT PROBLEM IS APPROXIMATED BY A MONOTONE FINITE VOLUME SCHEME. THE NUMERICAL SOLUTION OF THE CALIBRATION PROBLEM IS OBTAINED BY A CONTINUOUS GENETIC ALGORITHM. NUMERICAL RESULTS ARE PRESENTED IN ORDER TO VALIDATE THE EFFICIENCY OF THE PROPOSED ALGORITHM.

IN THIS PAPER, WE STUDY THE PROBLEM OF COEFFICIENTS IDENTIFICATION IN POPULATION GROWTH MODELS. WE CONSIDER THAT THE DYNAMICS OF THE POPULATION IS DESCRIBED BY A SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS OF SUSCEPTIBLE-INFECTIVE-RECOVERED (SIR) TYPE, AND WE ASSUME THAT WE HAVE A DISCRETE OBSERVATION OF INFECTIVE POPULATION. WE CONSTRUCT A CONTINUOUS OBSERVATION BY APPLYING TIME SERIES AND AN APPROPRIATE FITTING TO THE DISCRETE OBSERVATION DATA. THE IDENTIFICATION PROBLEM CONSISTS IN THE DETERMINATION OF DIFFERENT PARAMETERS IN THE GOVERNING EQUATIONS SUCH THAT THE INFECTIVE POPULATION OBTAINED AS SOLUTION OF THE SIR SYSTEM IS AS CLOSE AS TO THE OBSERVATION. WE INTRODUCE A REFORMULATION OF THE CALIBRATION PROBLEM AS AN OPTIMIZATION PROBLEM WHERE THE OBJECTIVE FUNCTION AND THE RESTRICTION ARE GIVEN BY THE COMPARISON IN THE L2-NORM OF THEORETICAL SOLUTION OF THE MATHEMATICAL MODEL AND THE OBSERVATION, AND THE SIR SYSTEM GOVERNING THE PHENOMENON, RESPECTIVELY. WE SOLVE NUMERICALLY THE OPTIMIZATION PROBLEM BY APPLYING THE GRADIENT METHOD WHERE THE GRADIENT OF THE COST FUNCTION IS OBTAINED BY INTRODUCING AN ADJOINT STATE. IN ADDITION, WE CONSIDER A NUMERICAL EXAMPLE TO ILLUSTRATE THE APPLICATION OF THE PROPOSED CALIBRATION METHOD.

IN THIS PAPER, WE DEVELOP A REVIEW OF THE RESEARCH FOCUSED ON THE TEACHING AND LEARNING OF ORDINARY DIFFERENTIAL EQUATIONS WITH THE FOLLOWING THREE PURPOSES: TO GET AN OVERVIEW OF THE EXISTING LITERATURE OF THE TOPIC, TO CONTRIBUTE TO THE INTEGRATION OF THE ACTUAL KNOWLEDGE, AND TO DEFINE SOME POSSIBLE CHALLENGES AND PERSPECTIVES FOR THE FURTHER RESEARCH IN THE TOPIC. THE METHODOLOGY WE FOLLOWED IS A COMBINATION OF A SYSTEMATIC LITERATURE REVIEW AND A BIBLIOMETRIC ANALYSIS. THE CONTRIBUTIONS OF THE PAPER ARE GIVEN BY THE FOLLOWING: SHED LIGHT ON THE LATEST RESEARCH IN THIS AREA, PRESENT A CHARACTERIZATION OF THE ACTUAL RESEARCH LINES REGARDING THE TEACHING AND LEARNING OF ORDINARY DIFFERENTIAL EQUATIONS, PRESENT SOME TOPICS TO BE ADDRESSED IN THE NEXT YEARS AND DEFINE A STARTING POINT FOR RESEARCHERS WHO ARE INTERESTED IN DEVELOPING RESEARCH IN THIS FIELD.

THIS ARTICLE IS CONCERNED WITH THE CONVERGENCE OF THE LEVEL-SET ALGORITHM INTRODUCED BY ASLAM (J COMPUT PHYS 167 (2001), 413?438) FOR TRACKING THE DISCONTINUITIES IN SCALAR CONSERVATION LAWS IN THE CASE OF LINEAR OR STRICTLY CONVEX FLUX FUNCTION. THE NUMERICAL METHOD IS DEDUCED BY THE LEVEL-SET REPRESENTATION OF THE ENTROPY SOLUTION: THE ZERO OF A LEVEL-SET FUNCTION IS USED AS AN INDICATOR OF THE DISCONTINUITY CURVES AND TWO AUXILIARY STATES, WHICH ARE ASSUMED CONTINUOUS THROUGH THE DISCONTINUITIES, ARE INTRODUCED. WE REWRITE THE NUMERICAL LEVEL-SET ALGORITHM AS A PROCEDURE CONSISTING OF THREE BIG STEPS: (A) INITIALIZATION, (B) EVOLUTION, AND (C) RECONSTRUCTION. IN (A), WE CHOOSE AN ENTROPY ADMISSIBLE LEVEL-SET REPRESENTATION OF THE INITIAL CONDITION. IN (B), FOR EACH ITERATION STEP, WE SOLVE AN UNCOUPLED SYSTEM OF THREE EQUATIONS AND SELECT THE ENTROPY ADMISSIBLE LEVEL-SET REPRESENTATION OF THE SOLUTION PROFILE AT THE END OF THE TIME ITERATION. IN (C), WE RECONSTRUCT THE ENTROPY SOLUTION USING THE LEVEL-SET REPRESENTATION. WE PROVE THE CONVERGENCE OF THE NUMERICAL SOLUTION TO THE ENTROPY SOLUTION IN FOR EVERY , USING -WEAK BOUNDED VARIATION (BV) ESTIMATES AND A CELL ENTROPY INEQUALITY. IN ADDITION, SOME NUMERICAL EXAMPLES FOCUSED ON THE ELEMENTARY WAVE INTERACTION ARE PRESENTED. © 2014 WILEY PERIODICALS, INC. NUMER METHODS PARTIAL DIFFERENTIAL EQ, 2014

IN THIS PAPER, ORDINARY AND EXPONENTIAL DICHOTOMIES ARE DEFINED IN DIFFERENTIAL EQUATIONS WITH EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT OF GENERAL TYPE. WE PROVE THE ASYMPTOTIC EQUIVALENCE BETWEEN THE BOUNDED SOLUTIONS OF A LINEAR SYSTEM AND A PERTURBED SYSTEM WITH INTEGRABLE AND BOUNDED PERTURBATIONS.