Physics and maths have undergone an intensely symbiotic period over the last two decades and are poised to even more exciting and productive relationship into the twentyfirst century. The mathematical physics is nothing but using the maths methods systematically for solving the physics problems and applying it for the physics concepts. There are various topics in mathematical physics like tensors, algebraic structures, matrices and dual spaces, eigenvalue problems, Riemann theorem, spherical harmonics, adjoint and unitary transformations etc. It is quite vast topic with various branches, topics etc. Lets see more about it.
Mathematical physics is an interdisciplinary branch of physics where the theoretical physics and mathematics concepts get related to solve out the concepts related to quantum mechanics, classical theory, relativistic theory etc and gives a new physical theory. There are several branches in mathematical physics namely
 Classical Mechanics
 Partial Differential Equations
 Quantum Theory
 Relativity and Quantum Relativistic Theories
 Statistical Mechanics
These branches were developed in the second half of eighteenth century almost in 1930's which rather contains different type of mathematical methods which proves some physics concepts.
A partial differential equation is the equation which consists of two or more independent variables with a unknown function and a quantity having a partial derivatives with respect to independent variables. The order of a partial differential equation has the highest order derivative. It formulates the solution giving out the understanding about velocity, heat, sound propagation, elasticity, flow of fluid, electrostatics, electrodynamics etc. There are first order, second order up to nth order derivatives in partial derivatives that gives the solution to any number of quantities.
It is the theoretical physics that explains the nature and behavior of matter and energy on the atomic and sub atomic particles. The interpretation of this theory came out in twentieth century by Bohr and Heisenberg in 1927. It explains in solids at most basic level why some solids are metals, some are insulators and others are semiconductors. It consists of various concepts like wave particle duality, wave quantization, Heisenberg uncertainty principle, Schrodinger's equation, expectation value and momentum operator, variational principle, energy in finite square well and infinite square well etc.
Mechanics is the study of how things move. The term classical mechanics is something really vague but gives the equivalent formulations of mechanics. Until the beginning of twentieth century, it seemed that classical mechanics to be the only kind of mechanics that described the possible motion. Then, in the twenty years from 1905 to 1925, it became very clear that classical mechanics did not correctly describe the motion of objects moving at speeds close to the speed of light, nor that of microscopic particles inside atoms and molecules. The result was development of two forms of mechanics namely the relativistic mechanics that described the high speed motions and quantum mechanics that described the motion of microscopic particles.
Matter consists of atoms which are distributed in the system. The statistical physics starts from the microscopic level which discusses not only thermalequilibrium states, but also non equilibrium states. It aims at understanding the properties of liquids, solids or gases on the basis of laws of motion of atoms and molecules.Statistical mechanics is a branch of physics that deals with the study of thermodynamic behavior of the systems containing large number of particles, free energy of the system etc. It consists of ergodic theory and many parts of probability theory.
The relativistic is the term often used when we are describing quantities appropriate to the particles moving at the speed close to speed of light. Eg: The particle moving at a velocity close to velocity of light is said to be moving at a relativistic velocity. The relativistic mechanics refers to special relativity and general relativity. It gives the mechanical description for a system having particles whose velocity is compared to the speed of light c.
Mathematical Physics Vs Theoretical Physics
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Lets see the difference between the mathematical physics Vs theoretical physics:

Theoretical physics 
Mathematical physics

1

Theoretical physics is the physics branch that deals with the mathematics to explain its natural phenomena 
Mathematical physics is the branch of applied mathematics that deals with the physics problems and solves it 
2 
It explains the experiment and even predicts the results what has not been physically tested

It explores the new phenomena, hypothesis and solutions by solving it and giving the result

3 
It is not in to the development of maths but applies maths for explaining physics theories 
It develops maths to apply it for physics problems 
4 
It motivates us to perform more experiments on the basis of physics theories

It gives the conclusion for physical theories. 