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Partial Differential Equations (PDEs) are the mathematical language of change in space and time. They are used to describe a wide variety of real-world systems. Examples of their applications include describing how waves travel, how heat spreads, weather forecasting, how the fundamental laws of Nature work, the pricing of financial derivatives such as stock options, and many others.

The module begins with a review of ordinary differential equations (ODEs), discussing first- and second-order methods, boundary value problems, and eigenvalue problems. We then introduce Sturm-Liouville theory and Fourier Series, seeing that it is possible to express a general periodic function as a sum of sine and cosine functions.

Next, we introduce some of the basic concepts of PDEs. The three important classes of second order PDE appropriate for modelling different sorts of phenomena are introduced, and the appropriate boundary conditions for each of these are considered. The technique of separation of variables is used to reduce a PDE to a set of ODEs of the kind reviewed at the start of the module, and to derive the general solution using Fourier Series and Sturm-Liouville theory. Throughout the module there will be a strong emphasis on problem solving and examples.

The last part of the module is an introduction to integral transforms, comprising Laplace Transforms and Fourier Transforms. We show how Laplace transforms are a very powerful technique to solve ODEs and PDEs, and how Fourier Transforms are very useful to solve PDEs.

Partial Differential Equations (PDEs) are the mathematical language of change in space and time. They are used to describe a wide variety of real-world systems. Examples of their applications include describing how waves travel, how heat spreads, weather forecasting, how the fundamental laws of Nature work, the pricing of financial derivatives such as stock options, and many others.

The module begins with a review of ordinary differential equations (ODEs), discussing first- and second-order methods, boundary value problems, and eigenvalue problems. We then introduce Sturm-Liouville theory and Fourier Series, seeing that it is possible to express a general periodic function as a sum of sine and cosine functions.

Next, we introduce some of the basic concepts of PDEs. The three important classes of second order PDE appropriate for modelling different sorts of phenomena are introduced, and the appropriate boundary conditions for each of these are considered. The technique of separation of variables is used to reduce a PDE to a set of ODEs of the kind reviewed at the start of the module, and to derive the general solution using Fourier Series and Sturm-Liouville theory. Throughout the module there will be a strong emphasis on problem solving and examples.

The last part of the module is an introduction to integral transforms, comprising Laplace Transforms and Fourier Transforms. We show how Laplace transforms are a very powerful technique to solve ODEs and PDEs, and how Fourier Transforms are very useful to solve PDEs.

Partial Differential Equations (PDEs) are the mathematical language of change in space and time. They are used to describe a wide variety of real-world systems. Examples of their applications include describing how waves travel, how heat spreads, weather forecasting, how the fundamental laws of Nature work, the pricing of financial derivatives such as stock options, and many others.

The module begins with a review of ordinary differential equations (ODEs), discussing first- and second-order methods, boundary value problems, and eigenvalue problems. We then introduce Sturm-Liouville theory and Fourier Series, seeing that it is possible to express a general periodic function as a sum of sine and cosine functions.

Next, we introduce some of the basic concepts of PDEs. The three important classes of second order PDE appropriate for modelling different sorts of phenomena are introduced, and the appropriate boundary conditions for each of these are considered. The technique of separation of variables is used to reduce a PDE to a set of ODEs of the kind reviewed at the start of the module, and to derive the general solution using Fourier Series and Sturm-Liouville theory. Throughout the module there will be a strong emphasis on problem solving and examples.

The last part of the module is an introduction to integral transforms, comprising Laplace Transforms and Fourier Transforms. We show how Laplace transforms are a very powerful technique to solve ODEs and PDEs, and how Fourier Transforms are very useful to solve PDEs.

Partial Differential Equations (PDEs) are the mathematical language of change in space and time. They are used to describe a wide variety of real-world systems. Examples of their applications include describing how waves travel, how heat spreads, weather forecasting, how the fundamental laws of Nature work, the pricing of financial derivatives such as stock options, and many others.

The module begins with a review of ordinary differential equations (ODEs), discussing first- and second-order methods, boundary value problems, and eigenvalue problems. We then introduce Sturm-Liouville theory and Fourier Series, seeing that it is possible to express a general periodic function as a sum of sine and cosine functions.

Next, we introduce some of the basic concepts of PDEs. The three important classes of second order PDE appropriate for modelling different sorts of phenomena are introduced, and the appropriate boundary conditions for each of these are considered. The technique of separation of variables is used to reduce a PDE to a set of ODEs of the kind reviewed at the start of the module, and to derive the general solution using Fourier Series and Sturm-Liouville theory. Throughout the module there will be a strong emphasis on problem solving and examples.

The last part of the module is an introduction to integral transforms, comprising Laplace Transforms and Fourier Transforms. We show how Laplace transforms are a very powerful technique to solve ODEs and PDEs, and how Fourier Transforms are very useful to solve PDEs.

Relativistic wave equations with their predictions of anti-particles and fermion spin will be explored. The fundamental role of gauge symmetries in current theories of force will lead to the study of the standard model of particle physics, including the spontaneous electroweak symmetry breaking via the Higgs mechanism.
Finally theories of particle physics beyond the standard model will be briefly investigated concentrating on their motivations and testable consequences.

Relativistic wave equations with their predictions of anti-particles and fermion spin will be explored. The fundamental role of gauge symmetries in current theories of force will lead to the study of the standard model of particle physics, including the spontaneous electroweak symmetry breaking via the Higgs mechanism.
Finally theories of particle physics beyond the standard model will be briefly investigated concentrating on their motivations and testable consequences.

Relativistic wave equations with their predictions of anti-particles and fermion spin will be explored. The fundamental role of gauge symmetries in current theories of force will lead to the study of the standard model of particle physics, including the spontaneous electroweak symmetry breaking via the Higgs mechanism.
Finally theories of particle physics beyond the standard model will be briefly investigated concentrating on their motivations and testable consequences.

Relativistic wave equations with their predictions of anti-particles and fermion spin will be explored. The fundamental role of gauge symmetries in current theories of force will lead to the study of the standard model of particle physics, including the spontaneous electroweak symmetry breaking via the Higgs mechanism.
Finally theories of particle physics beyond the standard model will be briefly investigated concentrating on their motivations and testable consequences.

This unit entirely consists of a research-level project

Upon successful completion of the project the student will have completed the final year of their Physics with Particle Physics Year Abroad MPhys degree. The supervisor(s) at RAL and CERN will have directed completion of a significant research ideally of publishable quality.

This unit entirely consists of a research-level project

Upon successful completion of the project the student will have completed the final year of their Physics with Particle Physics Year Abroad MPhys degree. The supervisor(s) at RAL and CERN will have directed completion of a significant research ideally of publishable quality.