# PARTIAL DIFFERENTIAL EQUATIONS IN APPLIED SCIENCES

**Anno accademico 2020/2021**- 2° anno

**Docente:**

**Maria Alessandra Ragusa**

**Crediti:**6

**Organizzazione didattica:**150 ore d'impegno totale, 108 di studio individuale, 42 di lezione frontale

**Semestre:**1°

ENGLISH VERSION

## Obiettivi formativi

Knowing how to construct and understand mathematical models that describe qualitatively and quantitatively some phenomena related to the environment. Knowing how to use the main concepts of differential equation theory for application in the biological, geological and environmental fields. Knowing how to predict and justify the evolution of simple phenomena, described by ordinary differential equations, related to the biological, geological and environmental sciences.

## Modalità di svolgimento dell'insegnamento

Frontal class.

*Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programme planned and outlined in the syllabus.*

*Learning assessment may also be carried out on line, should the conditions require it.*

## Prerequisiti richiesti

Calculus, first course. Basic elements of geometry. Study do the graph of a varible

## Frequenza lezioni

Obligatory.

## Contenuti del corso

1. Ordinary differential equations and systems of ordinary differential equations. Differential equations and physical models: definition and terminology. Differential equations of the first order: equations with separable variables and linear equations of the first order. Method of variation of the arbitrary Lagrange constant. Cauchy's problem. Vector equations and systems of ordinary differential equations: definition and terminology. Existence and uniqueness of the solution of a system of ordinary differential equations. Systems of linear differential equations. Autonomous systems of linear differential equations. Homogeneous systems of linear differential equations, eigenvalues, eigenvectors and solutions. Systems of nonlinear ordinary differential equations: autonomous systems, stability of systems of linear ordinary differential equations, linearization and local stability, stability criteria for nonlinear first order differential equations and for autonomous plane systems.

2. Models in listening dynamics. Malthus model and its generalizations. Verhulst model and its generalizations. Study of the stability of the equilibrium solutions of the Malthus and Verhulst models.

3. Models for environmental systems. Model for the evaluation of the green quality. Model for the evolution of metastability in an environmental system. Model for the evaluation of the production and diffusion of biological energy in an environmental system.

4. Models for territorial sciences. Lotka-Volterra model. Duffing model. Study of the stability of the equilibrium solutions of the Lotka-Volterra and Duffing models. Lotka-Volterra type models for the study of interactions between social groups: cooperation models, competition models, prey-predator models. Dynamic model of mobility within a system of urban centers.

5. Physical models. Elementary dynamics of the Earth's Mantle. Climate and paleoclimate: types of models, global climate models, dynamics of glacial masses, climatic oscillators. Viscous gravitational currents: theory of lubrication and advancement of a lava front.

6. Dynamics of faults. Models for pollution. Models for atmospheric pollution: model of the transport of pollutants, model of transport and diffusion of pollutants, control of the propagation of pollutants in the atmosphere, hints on the model for the construction of plants and control of polluting emissions, interaction of pollutants. Models for water pollution: structure and classification of models, one-dimensional model and its analytical solutions, equation, hints on multi-dimensional models.

## Testi di riferimento

- S. Motta, M.A. Ragusa, A. Scapellato –
*Methods and mathematical models - ed.*CULC (20*20)* - N. Hritonenko, Y. Yatsenko -
*Mathematical Modeling in Economics, Ecology and the Environment. Second edition*– Springer (2013). - A. Fowler –
*Mathematical Geoscience*– Springer (2011*).*

## Programmazione del corso

Argomenti | Riferimenti testi | |
---|---|---|

1 | Knowing how to construct and understand mathematical models that describe qualitatively and quantitatively some phenomena related to the environment. Knowing how to use the main concepts of differential equation theory for application in the biological, geological and environmental fields. Knowing how to predict and justify the evolution of simple phenomena, described by ordinary differential equations, related to the biological, geological and environmental sciences. | S. Motta, M.A. Ragusa, A. Scapellato – Mathematical Models – ed. CULC (2020). |

2 | 1. Ordinary differential equations and systems of ordinary equations. Differential equations and physical models: definitions and terminology. First order differential equations: separable variable equations and first order linear equations. Method of variation of the arbitrary Lagrange constant. Cauchy's problem. Vector equations and systems of ordinary differential equations: definition and terminology. | |

3 | definition and terminology. Existence and uniqueness of the solution of a system of ordinary differential equations. Systems of linear differential equations. Autonomous systems of linear differential equations. Homogeneous systems of linear differential equations, eigenvalues, eigenvectors and solutions. Systems of nonlinear ordinary differential equations: autonomous systems, stability of systems of linear ordinary differential equations, linearization and local stability, stability criteria for nonlinear | |

4 | nonlinear differential equations of the first order and for more autonomous systems. | |

5 | 2. Models in listening dynamics. Malthus model and its generalizations. Verhulst model and its generalizations. Study of the stability of the equilibrium solutions of the Malthus and Verhulst models. | |

6 | 3. Models for environmental systems. Model for the evaluation of the green quality. Model for the evolution of metastability in an environmental system. Model for the evaluation of the production and diffusion of biological energy in an environmental system. | |

7 | 4. Models for territorial sciences. Lotka-Volterra model. Duffing model. Study of the stability of the equilibrium solutions of the Lotka-Volterra and Duffing models. Lotka-Volterra type models for the study of interactions between social groups: cooperation models, competition models, prey-predator models. Dynamic model of mobility within a system of urban centers. | |

8 | 5. Physical models. Elementary dynamics of the Earth's Mantle. Climate and paleoclimate: types of models, global climate models, dynamics of glacial masses, climatic oscillators. Viscous gravitational currents: theory of lubrication and advancement of a lava front. Dynamics of faults. | |

9 | 6. Models for pollution. Models for atmospheric pollution: model of the transport of pollutants, model of transport and diffusion of pollutants, control of the propagation of pollutants in the atmosphere, hints on the model for the construction of plants and control of polluting emissions, interaction of pollutants. Models for water pollution: structure and classification of models, one-dimensional model and its analytical solutions, equation, hints on multi-dimensional models. |

## Verifica dell'apprendimento

### Modalità di verifica dell'apprendimento

Frontal exam.Criteria for assigning the final grade: verification of the achievement of the educational objectives expressed through the European Descriptors of the qualification.

Verification of learning can also be carried out electronically, should the conditions require it

### Esempi di domande e/o esercizi frequenti

Ordinary differential equations and systems of ordinary differential equations; Models in population dynamics: study of models and stability analysis; Models for environmental systems: study of models and stability analysis; Models for territorial sciences: study of models and stability analysis; Physical models: study of models; Models for pollution: study of models.