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 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.

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.

WE INTRODUCE NEW NECESSARY CONDITIONS FOR THE EXISTENCE AND UNIQUENESS OF STATIONARY WEAK SOLUTIONS AND THE EXISTENCE OF THE WEAK SOLUTIONS FOR THE EVOLUTION PROBLEM IN THE SYSTEM ARISING FROM THE MODELING OF THE BIOCONVECTIVE FLOW PROBLEM. OUR ANALYSIS IS BASED ON THE APPLICATION OF THE GALERKIN METHOD, AND THE SYSTEM CONSIDERED CONSISTS OF THREE EQUATIONS: THE NONLINEAR NAVIER?STOKES EQUATION, THE INCOMPRESSIBILITY EQUATION, AND A PARABOLIC CONSERVATION EQUATION, WHERE THE UNKNOWNS ARE THE FLUID VELOCITY, THE HYDROSTATIC PRESSURE, AND THE CONCENTRATION OF MICROORGANISMS. THE BOUNDARY CONDITIONS ARE HOMOGENEOUS AND OF ZERO-FLUX-TYPE, FOR THE CASES OF FLUID VELOCITY AND MICROORGANISM CONCENTRATION, RESPECTIVELY.