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 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.

IN THIS ARTICLE, WE PROPOSE A MATHEMATICAL MODEL FOR ONE-DIMENSIONAL HEAT CONDUCTION IN A THREE-LAYERED SOLID CONSIDERING THAT AN INTERFACIAL CONDITION IS PRESENT FOR THE TEMPERATURE AND HEAT FLUX CONDITIONS BETWEEN THE LAYERS. THE NUMERICAL APPROACH IS DEVELOPED BY CONSTRUCTING A FINITE DIFFERENCE SCHEME TO SOLVE THE INITIAL BOUNDARY?INTERFACE PROBLEM. THE NUMERICAL SCHEME IS DESIGNED BY CONSIDERING THE ACCURACY OF THE MODEL ON THE INNER PART OF EACH LAYER, THEN EXTENDING TO THE INTERFACES AND BOUNDARIES BY INCORPORATING THE CONTINUOUS INTERFACIAL CONDITIONS. THE FINITE DIFFERENCE SCHEME IS UNCONDITIONALLY STABLE, CONVERGENT, AND EASY TO IMPLEMENT SINCE IT CONSISTS OF THE SOLUTION OF TWO ALGEBRAIC SYSTEMS. WE PROVIDE THREE NUMERICAL EXAMPLES TO CONFIRM THAT OUR NUMERICAL APPROXIMATION IS CONSISTENT WITH THE ANALYTICAL SOLUTION AND THE PHYSICAL PHENOMENON.

WE RESEARCH A CONTROL PROBLEM FOR AN ECOLOGICAL MODEL GIVEN BY A REACTION?DIFFUSION SYSTEM. THE ECOLOGICAL MODEL IS GIVEN BY A NONLINEAR PARABOLIC PDE SYSTEM OF THREE EQUATIONS MODELLING THE INTERACTION OF THREE SPECIES BY CONSIDERING THE STANDARD LOTKA-VOLTERRA ASSUMPTIONS. THE OPTIMAL CONTROL PROBLEM CONSISTS OF THE DETERMINATION OF A COEFFICIENT SUCH THAT THE POPULATION DENSITY OF PREDATOR DECREASES. WE REFORMULATE THE CONTROL PROBLEM AS AN OPTIMAL CONTROL PROBLEM BY INTRODUCING AN APPROPRIATE COST FUNCTION. THEN, WE INTRODUCE AND PROVE THREE TYPES OF RESULTS. A FIRST CONTRIBUTION OF THE PAPER IS THE WELL-POSEDNESS FRAMEWORK OF THE MATHEMATICAL MODEL BY CONSIDERING THAT THE INTERACTION OF THE SPECIES IS GIVEN BY A GENERAL FUNCTIONAL RESPONSES. SECOND, WE STUDY THE DIFFERENTIABILITY PROPERTIES OF A COST FUNCTION. THE THIRD RESULT IS THE EXISTENCE OF OPTIMAL SOLUTIONS, THE EXISTENCE OF AN ADJOINT STATE, AND A CHARACTERIZATION OF THE CONTROL FUNCTION. THE FIRST RESULT IS PROVED BY THE APPLICATION OF SEMIGROUP THEORY AND THE SECOND AND THIRD RESULT ARE PROVED BY THE APPLICATION OF DUBOVITSKII AND MILYUTIN FORMALISM.